Hecke Operators and Hall Algebras for Global Function Fields

Instituto Nacional de Matemática Pura e Aplicada

Hecke operators and Hall algebras for global function ﬁelds

Roberto Alvarenga

Advisor: Oliver Lorscheid

Thesis presented to the Post-graduate Program in Mathematics at Instituto Nacional de Matemática Pura e Aplicada as partial fulﬁllment of the requirements for the degree of Doctor in Mathematics.

Rio de Janeiro April 11, 2018 Abstract

The subject of this thesis is arithmetic geometry. In particular, we aim to describe explicitly some graphs, which we call graphs of Hecke operators, for function fields of curves of small genus. The first two chapters are dedicated to the introduction the graphs of Hecke operators and its first properties. The Chapter 1 is dedicated to introduce the prerequisites from number theory and algebraic geometry necessaries to define the graphs of Hecke operators. In the Chapter 2 we give the number theoretic interpretation of graphs of Hecke operators, which were originally introduced by Lorscheid in his Phd thesis [30]. These graphs encode the complete data of the action of Hecke operators on automorphic forms over a global function field. After giving the number theoretic definition of graphs of Hecke operators we translate that definition in terms of algebra- geometric tools. We dedicate a section of Chapter 2 to show a necessary condition for two vertices of these graphs to be connected by an edge and end the chapter with the solution of how to obtain these graphs in the case of the function field of the projective line. Most of the content of this first part appears in the following pre-print:

• [1] R. Alvarenga. On graphs of Hecke operators. http://front.math.ucdavis. edu/1709.09243, 2017.

In the chapters 3 and 4 we aim to solve our problem of giving a description for the graphs of Hecke operators in the case of an elliptic function ﬁeld. The theory of Hall algebras plays an important role in the elliptic case, for that we dedicate Chapter 3 to give a friendly introduction to that theory. We develop in Chapter 4 an algorithm to describe the graphs of Hecke operators using the elliptic Hall algebra. Namely, the edges in the graphs of Hecke operators can be recovered from certain products in the Hall algebra, from that our algorithm uses such information to calculate these products, returning the edges of the graphs. Most of the content of this two last chapters appears in the following pre-print:

• [2] R. Alvarenga. Hall algebra and graphs of Hecke operators for elliptic curves. In preparation, 2018. Keywords: Hecke operators, automorphic forms, elliptic curves, arithmetic geometry, vector bundles, symmetric functions, Hall algebras.

3 Resumo

O tópico dessa tese é geometria aritmética. Em particular, nosso objetivo é descrever explicitamente certos grafos, denominados grafos de operadores de Hecke, para curvas de gênero pequeno. Os primeiros dois capítulos são dedicados a introduzir os grafos de operadores de Hecke e suas primeiras propriedades. No Capítulo 1 introduzimos os pré-equisitos necessários da teoria dos números e da geometria algébrica. No Capítulo 2 começamos com a interpretação dos grafos de operadores de Hecke na linguagem da teoria dos números, que é originalmente introduzida por Lorscheid em sua tese de doutorado [30]. Esse grafos codificam todas as informações da ação dos operadores de Hecke nas formas automórficas definidas sobre um corpo de funções global. Depois da abordagem teórico numérica dos grafos de operadores de Hecke, traduzimos aquela definição em termos de objetos da geometria algébrica. Dedicamos uma seção do Capítulo 2 para provar uma condição necessária para dois vertices desses grafos serem conectados por uma aresta e terminamos o capítulo com a solução de como obter tais grafos no caso em que o corpo de funções é o corpo de funções da reta projetiva. Quase todo o conteúdo dessa primeira parte aparece no seguinte preprint: • [1] R. Alvarenga. On graphs of Hecke operators. http://front.math.ucdavis. edu/1709.09243, 2017. Nos capítulos 3 e 4 objetivamos a solução do nosso problema de descrever os grafos no caso do corpo de funções elíptico. A teoria de álgebras de Hall desempenha um papel fundamental no caso elíptico, por isso dedicamos o Capítulo 3 a uma amigavel introdução a esta teoria. Desenvolvemos no Capítulo 4 um algoritmo para descrever os grafos de operadores de Hecke usando a álgebra de Hall elíptica. A saber, as arestas nos grafos de operatodes de Hecke podem ser determinadas de certos produtos na álgebra de Hall, sabendo disto, nosso algoritmo calcula esses produtos que por sua vez nos retornam as arestas dos grafos. Boa parte destes dois últimos capítulos pode ser encontrada no seguinte preprint • [2] R. Alvarenga. Hall algebra and graphs of Hecke operators for elliptic curves. In preparation, 2018.

Palavras-chave: Operadores de Hecke, formas automórﬁcas, curvas elípticas, geometria aritmética, ﬁbrados vetoriais, funções simétricas, álgebras de Hall. Dedicated to my family. Acknowledgment

First of all, I would like to express my highest gratitude to my advisor Oliver Lorscheid, without him this thesis would not be possible. His advising, about life or math, was ever very helpful for me. Moreover, I would to tank Oliver for suggesting the main problem of this thesis and for many stimulating conversations. I also wish to express my thank to my parents Elza and Roberto, which ever given me all the necessary support and their love. Even if I continuous to say ”thank you” for ever, it will not be enough. I thank my brother Weiner and my sister Isabella for all, hoping that this thesis serves of inspiration. I would like to thank my beloved (future) wife Natália, for her comprehension, support and love. I thank all my family. I thank all my IMPA’s friends, it was a pleasure to share these years with you. I also would like to thank my friend Victor Hugo. For many reasons I thank Parham Salehyan and Alex Massarenti. I would like to express my thank to professor Olivier Schiffmann for hosting me for a term in Paris and Dragos Fratila for fruitful discussions. I wish express my gratitude to all IMPA’s staff, in particular Antônio Carlos, Kênia, Isabel, Andréia and Josenildo. Also my gratitude to the IMPA’s researches, in particular Carolina Araujo, Eduardo Esteves, Karl-Otto Störn, Reimundo Heluani, Misha Belolipetsky and Henrique Bursztyn. To the thesis defence committee Eduardo Esteves, Reimundo Heluani, Karl-Otto Störn, Cecília Salgado and Olivier Schiffmann, thank you. For the financial support I would like to thank Faperj and Capes. ”I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.” Isaac Newton Contents

Introduction 9

1 Background 16 1.1 Notation ...... 16 1.2 Hecke operators ...... 19 1.3 Automorphic forms ...... 20 1.4 Coherent sheaves ...... 22

2 Graphs of Hecke operators for GLn 26 2.1 Graphs of Hecke operators ...... 26 2.2 Graphs of unramiﬁed Hecke operators ...... 29 2.3 Geometry of graphs of unramiﬁed Hecke operators ...... 34 2.4 The δ−invariant ...... 37 2.5 Graphs for the projective line ...... 44

3 Hall algebras 58 3.1 The classical Hall algebras ...... 59 3.2 The Hall algebras of a ﬁnitary category ...... 64 3.3 The Hall algebra of a curve ...... 69

4 Graphs of Hecke operators for elliptic curves 72 4.1 Introduction ...... 72 4.2 Hall algebras and graphs of Hecke operators ...... 77 4.3 Graphs of Hecke operators for elliptic curves ...... 79 4.4 The algorithm ...... 87 4.5 Calculating structure constants ...... 93 4.6 The case of rank 2 ...... 96

8 Introduction

Hecke operators have been proved to be a powerful tool in number theory. Before Hecke had published his theory in 1937 about these operators in [20] and [21], they were used by Mordel in 1917 to show that the Ramanujan tau function is multiplicative, cf. [34]. Hecke operators play an important role in the theory of modular forms (in the classical context) and more generally in the study of automorphic forms and automorphic representations. Autormorphic forms in their turn, are a central ingredient of the Langlands program and play an important role in modern number theory. Zagier develops in [52] the theory of toroidal automorphic forms over Q and ties it up with the Riemann hypothesis. In the end of this article, he asks questions about an analogous theory for global function ﬁelds and the potential connections to the Hasse-Weil theorem. Lorscheid introduces in [30] graphs of Hecke operators as a computational device to determine the spaces of toroidal automorphic forms for function ﬁelds. In this thesis, we continue the study of graphs of Hecke operators and generalize some results of Lorscheid in [32] from PGL2 to GLn. In particular, we describe how to determine these graphs for curves of small genus, namely: for the projective line and elliptic curves. We summarize our main results in the following.

General theory The first two chapters are dedicated to generalizing the original definition of those graphs, given by Lorscheid in [32], from PGL2 to GLn and give its first properties. Moreover, we prove some structure results about those graphs and end Chapter 2 with a solution of how to obtain the graphs of Hecke operators for the rational function field. Let us define the central object of our investigation, the graph of Hecke operator. Let X be a smooth projective and geometrically irreducible curve whose function field is F . We say that two exact sequences of coherent sheaves

0 0 0 −→ F1 −→ F −→ F2 −→ 0 and 0 −→ F1 −→ F −→ F2 −→ 0

9 0 0 are isomorphic with ﬁxed F if there are isomorphism F1 → F1 and F2 → F2 such that 0 / F1 / F / F2 / 0 =∼ =∼ 0 0 0 / F1 / F / F2 / 0 commutes. Let BunnX be the set of rank n vector bundles on X. For x a closed point on X, we denote by κ(x) the residue ﬁeld at x. Consider the exact sequences of the form

0 ⊕r 0 −→ E −→ E −→ Kx −→ 0

0 ⊕r where E , E are rank n-vector bundles on X, x is a closed point of X, and Kx is ⊕r 0 the skyscraper sheaf on x whose stalk is κ(x) . Let mx,r(E, E ) be the number of isomorphism classes of exact sequences

00 ⊕r 0 −→ E −→ E −→ Kx −→ 0 (0.0.1)

00 ∼ 0 0 0 with ﬁxed E such that E = E . We denote by Vx,r(E) the set of E, E , mx,r(E, E ) 0 such that exists an exact sequence of the type as above, i.e. mx,r(E, E ) 6= 0. We deﬁne the graphs of Hecke operators Gx,r as follows.

Deﬁnition. Let x be a closed point in X. The graph Gx,r is deﬁned as a Vert Gx,r = BunnX and Edge Gx,r = Vx,r(E).

E∈BunnX

The graphs of Hecke operators have this name because they might be seen as graphs which encode the action of Hecke operators on automorphic forms over the global function field F . This is what we call the number theoretical definition of graphs of Hecke operators. See Section 2.1 for this approach and Theorem 2.3.4 for the connection between the two definitions. Our first main theorem is the following. Theorem (Theorem 2.2.7). There exits a correspondence between the set of isomorphism classes of short exact sequences

00 ⊕r 0 −→ E −→ E −→ Kx −→ 0 with ﬁxed E and the Grassmaniann Gr(n − r, n)(κ(x)). Moreover, the multiplicity of an edge of E to E 0 equals the number of sequences of the above type with E 00 ∼= E 0. The multiplicities of the edges originating in E sum up to #Gr(n − r, n)(κ(x)).

For a subbundle E 0 of a bundle E, we deﬁne

δ(E 0, E) := rk(E) deg(E 0) − rk(E 0) deg(E)

10 and 0 δk(E) := sup δ(E , E) E0,→E k−subbundle for k = 1, . . . , n − 1, where n = rk(E).

0 Theorem (Theorem 2.4.15). Let E be a rank n−bundle on X. If mx,r(E, E ) 6= 0, then

0 δk(E ) ∈ δk(E) − k|x|(n − r), δk(E) − k|x|(n − r) + n, . . . , δk(E) + r|x|k for every k = 1, . . . , n − 1.

Rational function fields In section 2.5, we explain an algorithmic way to calculate the graphs of unramified Hecke operators for a rational function field. This case is particularly simple because we may represent a vector bundle on P1 as the matrix

 d1  πx  ..  g =  .  , dn πx

(recall Theorem 2.3.1) where d1, . . . , dn are positive integers in increasing order and x is a closed point of degree one. Thus we may proceed with matrix calculations, following the correspondence of Theorem 2.2.7, to obtain the isomorphism classes corresponding to the product of g by an element of Gr(n − r, n)(κ(x)). All the results in the first four sections of Chapter 2 are available for any smooth projective and geometrically irreducible curve over a finite field. Only in the last section 2.5, we specialize to the projective line.

Hall algebras With the description of the graphs of Hecke operators for the projective line, we have left to describe those graphs for an elliptic curve. This is the subject of chapters 3 and 4. In order to describe the graphs of Hecke operators for elliptic curves, we use the theory of Hall algebras. Hall algebras were first considered by Steinitz in 1901 in his forgotten paper [49], before Hall studied them in the 1950’s in [17]. Ringel formalized in the early 1990s the notion of the Hall algebra of a finitary category, and Kapranov studied in the late 1900s in his paper [26] the Hall algebra of a smooth projective curve over a finite field. The Hall algebra HX of a smooth projective curve X over a

11 ﬁnite ﬁeld Fq encodes the extensions of coherent sheaves on X. Fix a square root v of q−1. The Hall algebra of X is the vector space M HX := C[F] [F]∈Coh(X) equipped with the product

−hF,Gi X H [F][G] = v hF,G[H] H

0 1 where hF,Gi := dimFq Ext (F, G) − dimFq Ext (F, G) and #0 −→ G −→ H −→ F −→ 0 hH := F,G #Aut(F) #Aut(G)

The main observation, which links the theory of Hall algebras with the graphs of 0 Hecke operators, is that we can recover the multiplicities mx,r(E, E ) from the product ⊕r 0 Kx E in the Hall algebra of X (Lemma 4.2.1). Thus, for a fixed n, the graphs of ⊕r 0 Hecke operators can be described by calculating explicitly the products Kx E where E 0 runs through the set o rank n vector bundles on X. By what we have explained in the previous paragraph, our problem is reduced to ⊕r calculating the products Kx E for all E ∈ BunnX. To do so, we use some structure results from Burban and Schiffmann [10] and Dragos [12] about the elliptic Hall ⊕r algebra. Our strategy is to make a ”base change” and write the product Kx E in terms of elements in some subalgebras of the whole Hall algebra, called twisted spherical Hall algebras (see Definition 4.3.10). Since the twisted spherical Hall algebras are well understood and have a characterization in terms of the path algebra (see Theorem 4.3.15), the base change allows us to explicitly calculate these products. However, these calculation depends closely on our initial data, i.e. the degree of x, the choice of r and the vector bundle E. Hence, what we do in the Chapter 4, Section 4.4, is to develop an algorithm which calculate these products step-by-step. This algorithm is, without doubt, the main contribution of this thesis for the development of the theory of graphs of Hecke operators. We outline the algorithm in the following.

The algorithm for elliptic function ﬁelds

⊕r In order to calculate the products Kx E, we will use the twisted spherical Hall algebras, which are subalgebras of the whole Hall algebra. Let Z := {(r, d) ∈ Z2 | r > 0 or r = 0 and d > 0} and for (r, d) ∈ Z we deﬁne the slope of (r, d) by d/r if r 6= 0 and by ∞ if r = 0, and denote that by µ(r, d). The twisted spherical Hall algebra d ρe Eσ,σ has an explicit description in terms of generators Tv (cf. Deﬁnition 4.3.10) and 0 relations (see Theorem 4.3.15), where v ∈ Z, ρe is a character on Pic Xn modulo the

12 n Frobenius action, and Xn = X ×Spec Fq Spec Fq . These algebras were introduced by Burban and Schiﬀmann in [10] and generalized by Fratila in [12].

Input. Let x ∈ X be a closed point, E ∈ BunnX and r an integer such that 1 ≤ r ≤ n.

Base change.

(i) Let Cµ be the category of semistable coherent sheaves of slope µ. Write

E = E1 ⊕ · · · ⊕ Es

where Ei ∈ Cµi and µ1 < ··· < µs (Harder-Narasimhan decomposition).

(ii) Let Tor(X) be the category of torsion sheaves. For each Ei ∈ Cµi , use the

equivalence Cµi ≡ Tor(X) (cf. Theorem 4.1.1) to associate Ei with Ti ∈ TorX 0 0 and write Ti = Txi1 ⊕ · · · ⊕ Txim with Txij ∈ Torxij and xij 6= xij for j 6= j

(iii) Each Txij corresponds to a Hall-Littlehood symmetric function Pλij in the Mac-

donald ring of symmetric functions. Write Pλij as sum of products of power- sums functions.

(iv) Taking the inverse image of the power-sums yields an expression of Txij in terms of sum of products of

[m]|xi| X (λ) T(0,m),x := nu (l(λ) − 1)K i m xi xi |λ|=m/|xi|

(λ) where Kxi is the unique torsion sheaf with support at xi associated to the partition λ. See Section 4.3 for the deﬁnitions of the constants in the above

deﬁnition. By Proposition 4.3.6, we can write Txij as a sum of products of elements X ρe Tv := ρe(x)Tv,x. x∈X 0 where ρe is a character in Pic Xd modulo the Frobenius action, for some extension of base ﬁeld Xd of X and µ(v) = ∞.

(v) Since E = a E1 ···Es for some a ∈ C and Ti = Txi1 ···Txim we may write

ρ ρ e1jk esjk X ρe1j1 j ρesj1 j E = aijk Tv1j ··· Tv1j ··· Tvsj ··· Tvsj , 1 kj 1 kj ρ eijk a ∈ ρ for some ijk C, eijk orbits of characters in some Picard groups.

13 (vi) Therefore, m X ρi ρi ρi K⊕r E = a T e 0 T e 1 ··· T e ` x i vi0 vi1 vi` i=1 a ∈ ρ for some i C where eij orbits of characters in some Picard groups and where vij ∈ Z are such that µ(vi0 ) = ∞ and µ(vi1 ) ≤ · · · ≤ µ(vi` ). Order by slopes d (i) Aiming to use the structure of Eσ,σ (cf. Deﬁnition 4.3.11) to write the above ⊕r product in increasing order of slopes, we consider the commutator Kx , E vec ⊕r vec ⊕r vec since π (Kx E) = π Kx , E . Where π means that we are considering the vector bundles which appear in the product. ρ ρ T ρei ,T ej = 0 T ρei ,T ej (ii) Observe that vi vj unless vi vj belong to same twisted spherical Hall algebra (cf. Deﬁnition 4.3.10). By Theorem 4.3.15 the problem of ordering the d slopes is reduced to Eσ,σ. d (iii) Using the structure of Eσ,σ, we can calculate [tv, tw] explicitly. If µ(v) > µ(w) we replace tvtw by tv, tw + twtv. (iv) Combining the previous steps we may write m X ρi ρi K⊕r E = a T e 1 ··· T e ` x i vi1 vi` i=1

where µ(vi1 ) ≤ · · · ≤ µ(vi` ). Base change back and calculations ρ eij P 0 Tv 0 ρi (x )Tv ,x0 . Tv ,x0 (i) We replace ij by its deﬁnition x e j ij The elements ij are given

explicitly by sums of semistable sheaves of slope µ(vij ) via Atiyah’s classiﬁca- tion (cf. Theorem 4.1.1). Therefore we are left with calculating the product

Tv1,x1 Tv2,x2 .

(ii) If µ(v1) < µ(v2) or x1 6= x2, then

Tv1,x1 Tv2,x2 = Tv1,x1 ⊕ Tv2,x2 vec ⊕r and we are able to calculate explicitly the product π (Kx E).

(iii) If µ(v1) = µ(v2) and x1 = x2, the product Tv1,x1 Tv2,x2 can be calculated as the product of power-sums in the Macdonald ring of symmetric functions

(cf. Proposition 4.3.1). In this case, we write the product Tv1,x1 Tv2,x2 in the Macdonald ring of symmetric functions as linear combination of Hall-Littlehood P functions i Pλi and come back to the Hall algebra via Proposition 4.3.1, which

associates Pλi with the unique semistable sheaf in Cµ(v1) that corresponds to (λi) Kx1 via Cµ(v1) ≡ TorX.

14 Results for elliptic function ﬁelds We conclude chapter 4, and thus this thesis, with applying our algorithm to obtain structure results for graphs of Hecke operators. Via Atyiah’s classiﬁcation of vector bundles on an elliptic curve (cf. Theorem 4.1.1), an irreducible vector bundle E is determined by its rank n, degree d, a closed (r,d) point x and weight `. We denote such E by E(x,`) .

Theorem (Theorem 4.5.1). Fix n ≥ 1 and integer. Let x be a closed point in X with (n,d) 0 |x| = 1. Let E = E(y,1) with n|d be a stable vector bundle on X. Then mx,1 E , E 6= 0 0 ∼ (n,d+1) 0 0 if and only if E = E(y0,1) where y = x+y. In this case mx,1 E , E = 1. Graphically,

(n,d+1) (n,d) E(y0,1) E(y,1)

(n,d) are all incoming arrows at E(y,1) in Gx,1.

Theorem (Theorem 4.5.2). Let x be a closed point in X and L1,..., Ln line bundles 0 on X. If E = L1 ⊕ · · · ⊕ Ln with deg(Li) > deg(Li−1) + |x| for i = 1, . . . , n, then

n 0 [ 0 k−1 Vx,1(E ) = E , L1 ⊕ · · · ⊕ Lk(−x) ⊕ · · · ⊕ Ln, qx . k=1

In section 4.6, we calculate the complete graph Gx,1 for GL2 and provide an explicitly example for the elliptic curves with only one rational point.

15 Chapter 1

Background

This ﬁrst chapter states our basic notation and recalls relevant notions and facts from the theory of automorphic forms, Hecke operators and coherent sheaves which will be used throughout this thesis. One of the main objects of this work are Hecke operators. These operators are the elements of the Hecke algebras, which are convolution algebras of functions with compact support on the adelic points of the group under consideration. A representation of the group in a complex vector space corresponds to a representation of the Hecke algebra by assigning an integral operation to each of its elements. Here we concentrate in the case in which the group under consideration is the general linear group GLn. The concept of automorphic forms used nowadays can be applied to a large class of algebraic groups. Here, however, we will restrict to GLn, i.e. we consider automorphic forms on GLn with trivial central character. These are nothing else but automorphic forms on PGLn the projective general linear group, but for technical reasons, it is more convenient to work with GLn. We refer to [8, Chapter 12], [35] and [42] as main references for this part, but we can ﬁnd this subject in classical references [9], [24] and [50] as well.

1.1 Notation

In this first section, we set up the notation and basic facts of number theory that is used throughout the thesis. Let N be the natural numbers, Z the integers, Q the rationals, R the reals and C the complex numbers together with the usual absolute value | |C and usual topology. Fix Fq a finite field of q elements, where q is a prime power. Let F be a global function field over Fq, i.e. the function field of a geometrically irreducible smooth projective curve X over Fq. Let g be the genus of X. Let |X| the set of closed points of X or, equivalently, the set of places in F . For x ∈ |X|, we denote

16 Fx the completion of F at x, by Ox its integers, by πx ∈ Ox an uniformizer (where we ∼ suppose πx ∈ F ) and by qx the cardinality of the residue ﬁeld κ(x) := Ox/(πx) = Fqx .

With the choice of πx, we can identify Fx with the field of Laurent series Fqx ((πx)) in πx and Ox with the ring of formal power series Fqx [[πx]]. Let |x| be the degree of |x| x, which is defined as the extension field degree [κ(x): Fq]; in other words qx = q . The field Fx comes with a valuation vx that satisfies vx(πx) = 1 and an absolute value −vx −1 | · |x := qx , which satisfies |πx|x = qx . ∗ Q Let A be the adele ring of F and A the idele group. Put OA = Ox where the product is taken over all places x of F . The idele norm is the quasi-character ∗ ∗ ∗ Q | · | : A → C that sends an idele (ax) ∈ A to the product |ax|x over all local norms. By the product formula, this defines a quasi-character on the idele class group ∗ ∗ A /F . We think of Fx being embedded into the adele ring A by sending an element a ∈ Fx to the adele (ay)y∈|X| with ax = a and ay = 0 for y 6= x. Not quite compatible ∗ with this embedding, we think of the unit group Fx as a subgroup of the idele group ∗ ∗ A by sending an element b of Fx to the idele (by) with bx = b and by = 1 for y 6= x. We will explain in case of ambiguity, which of these embeddings we use. An idele a = (ax) is characterised by the vanishing of vx(ax) for all but finitely P Q many places x. The degree deg a = x∈|X| |x|·vx(ax) and the norm |a| = x∈|X| |ax|x ∗ of an idele are thus well-defined functions. Denote by A0 the ideles of degree 0, or equivalently, of norm 1. A divisor of F is an element

M ∼ ∗ ∗ D = (Dx) ∈ Z · x = A /OA x∈|X| with Dx ∈ Z for all x ∈ |X|. The latter isomorphism is obtained by sending the ∗ ∗ ∗ divisor x to πx, where we interpret πx as an idele via the inclusion F ⊂ Fx ⊂ A . idele class group F ∗ \ ∗ divisor class group Cl F F ∗ \ ∗/O∗ . Define the as A and the as A A D ∈ Cl F D ∈ ∗/O∗ If we write , then we always mean the class of the divisor A A in F ∗ \ ∗/O∗ . A A By embedding an element a ∈ F diagonally into A along the canonical inclusions Q F,→ Fx, we may regard F as a subring of A. The product formula x∈|X| |a|x = 1 F ∗ ⊂ ∗. O∗ a = (a ) can be reformulated as A0 Since A consists of the ideles x with v (a ) = 0 x O∗ ⊂ ∗. x x for all places , this yields A A0 Therefore we can define the degree of a divisor (and of a divisor class) as the degree of a representative in A∗. The class group Cl0 F = F ∗ \ ∗/O∗ h A0 A is a finite group (cf. [29, Thm. 7.13]), whose order F is called the class number. These groups fit into an exact sequence

0 deg 0 −→ Cl F −→ Cl F −→ Z −→ 0, which splits non-canonically (cf. [30, 2.1.2]). For the surjectivity of the degree map

17 cf [29, Prop. 6.2]. In particular, there are always ideles of degree 1, even when F has no place of degree 1. Local fields and adeles come with a natural topology, which tuns them into locally compact rings. Hence all algebraic groups over these rings turn into locally compact groups, which carry a Haar measure. The topology of Fx is given by the neighbour- i hood basis {πxOx}i∈N of 0, which turns Fx into a locally compact field, since Ox is a compact neighbourhood of 0. Note that Fx is totally disconnected and Hausdorff. By its definition as [ Y Y A = Fx × Ox, S⊂|X| x∈S x6∈S finite we can endow A with a canonical topology, in which the subsets above are embedded as open subspaces carrying the product topology. By Tychonoff’s theorem OA is compact as product of compact spaces, and thus A is locally compact, totally disconnected and Hausdorff. If V is an algebraic group over F , then the group law turns V (A) into a locally compact group. A locally compact group has a left and a right Haar measure, i.e. a non-trivial measure that is invariant by left or right translations, respectively. Every time that an algebraic group appears we assume the adelic points to carry a Haar measure. A Haar measure is unique up to a constant multiple. Rather than fixing the constant, we point out that constructions are independent of the choice of constant. The Haar measure defines a Lebesgue integral for measurable functions with compact support. The Haar measure of the product H1×H2 of two locally compact groups equals, up to a multiple, the product of the Haar measures of the factors H1 and H2. ∼ Thus we can apply Fubini’s theorem if we have an isomorphism H = H1 ×H2 of topological groups, quietly assuming that the Haar measures are suitably normalized. Let G(A) := GLn(A),Z(A) be the center of G(A), G(F ) := GLn(F ) and K := GLn(OA) the standard maximal compact open subgroup of G(A). From the previous discussion, note that G(A) comes together with an adelic topology that turns G(A) into a locally compact group. We fix the Haar measure on G(A) for which vol(K) = 1. The topology of G(A) has a neighbourhood basis V of the identity matrix that is given by all subgroups 0 Y 0 Y K = Kx < Kx = K x∈|X| x∈|X| 0 where Kx := GLn(Ox), such that for all x ∈ |X| the subgroup Kx of Kx is open and 0 consequently of finite index and such that Kx differs from Kx only for a finite number of places.

18 1.2 Hecke operators

This section is dedicated to introduce, and describe first properties of a fundamental object of this work, namely the Hecke operators. Definition 1.2.1. The complex vector space H of all smooth (i.e. locally constant) compactly supported functions Φ: G(A) → C together with the convolution product Z −1 Φ1 ∗ Φ2 : g 7−→ Φ1(gh )Φ2(h)dh G(A) for Φ1, Φ2 ∈ H is called the Hecke algebra for G(A). Its elements are called Hecke operators. The zero element of H is the zero function, but there is no multiplicative unit. For 0 0 K ∈ V, we define HK0 to be the subalgebra of all bi-K -invariant elements. These subalgebras have multiplicative units. Namely, the normalized characteristic function 0 −1 0 K0 := (volK ) charK0 acts as the identity on HK0 by convolution. When K = K we call HK the unramified part of H (or spherical Hecke algebra)and its elements are called unramified Hecke operators. There exists a friendly description of the unramified part of H, cf. Theorem 2.2.1. In this thesis we will be interested in the unramified Hecke operators. Lemma 1.2.2. Every Φ ∈ H is bi-K0-invariant for some K0 ∈ V. Proof. Since Φ is locally constant and V is a system of neighbourhoods of the identity, we can cover the support of Φ with sets of the form giKi with gi ∈ G(A) and Ki ∈ V, where i varies in some index set, such that Φ is constant in each giKi. But the support 00 T of Φ is compact, so we may assume that a finite index set. Then K = i Ki ∈ V, and Φ is right K00-invariant. Mutatis mutandis, we find a K000 ∈ V such that Φ is left K000-invariant. Then K0 = K00 ∩ K000 satisfies the assertion of the lemma. The above lemma implies directly the following proposition. S Proposition 1.2.3. H = K0∈V HK0 .

Proposition 1.2.4. If Φ1 ∈ H is left K1-invariant and Φ2 is right K2-invariant for K1,K2 ∈ V, then Φ1 ∗ Φ2 is left K1-invariant and right K2-invariant.

Proof. For g ∈ G(A), k1 ∈ K1 and k2 ∈ K2, we have that Z −1 Φ1 ∗ Φ2(k1gk2) = Φ1(k1gk2h )Φ2(h)dh G(A) Z 0−1 0 0 = Φ1(gh )Φ2(h k2)dh G(A) = Φ1 ∗ Φ2(g) 0 −1 by the change of variables h = hk2 .

19 Let C0(G(A)) be the space of continuous functions f : G(A) → C. The Hecke algebra H acts on C0(G(A)) by Z Φ(f): g 7−→ Φ(h)f(gh)dh. G(A)

A function f ∈ C0(G(A)) is H-ﬁnite if the space H.f is ﬁnite dimensional.

0 Proposition 1.2.5. For every f ∈ C (G(A)) and every Φ ∈ HK0 , Φ(f) is right K0-invariant.

Proof. Let g ∈ G(A) and k0 ∈ K0, then Z Z Φ(f)(gk0) = Φ(h)f(gk0h)dh = Φ(k0−1h0)f(gh0) = Φ(f)(g), G(A) G(A) where the second equality is given by the change of variables h0 = k0h.

1.3 Automorphic forms

Consider the space C0(G(A)) of continuous functions. Such a function is called smooth if it is locally constant. The group G(A) acts on C0(G(A)) through the right regular representation ρ : G(A) → Aut(C0(G(A))), that is defined by right translation of the argument: (g.f)(h) := (ρ(g)f)(h) := f(hg) for g, h ∈ G(A) and f ∈ C0(G(A)). A function f ∈ C0(G(A)) is called K−finite if the complex vector space that is generated by {k.f}k∈K is finite dimensional. Let H be a subgroup of G(A). We say that f ∈ C0(G(A)) is left or right H- invariant if for all h ∈ H and g ∈ G(A), f(hg) = f(g) or f(gh) = f(g), respectively. If f is right and left H-invariant, it is called bi-H-invariant. n2+1 −1 We embed G(A) ,→ A via g 7→ (g, det(g) ). We define a local height ||gx||x on G(Fx) := GLn(Fx) by restricting the height function

(v1, . . . , vn2+1) 7→ max{|v1|x,..., |vn2+1|x}

n2+1 on Fx . We note that ||gx||x ≥ 1 and that ||gx||x = 1 if gx ∈ Kx. We deﬁne the global height ||g|| to be the product of the local heights. We say that f ∈ C0(G(A)) is of moderate growth if there exist constants C and N such that

N |f(g)|C ≤ C||g|| for all g ∈ G(A).

20 Definition 1.3.1. The space of automorphic forms A (with trivial central character) is the complex vector space of all functions f ∈ C0(G(A)) which are smooth, K−finite, of moderate growth, left G(F )Z(A)-invariant and H-finite. Its elements are called automorphic forms.

Remark 1.3.2. Note that for g ∈ G(A) and f smooth, K-ﬁnite, of moderate growth, or left G(F )Z(A)-invariant, g · f is also smooth, K-ﬁnite, of moderate growth, or left G(F )Z(A)-invariant, respectively. Thus the right regular representation restricts to A.

Lemma 1.3.3 ([30] Lemma 1.3.2). A function f ∈ C0(G(A)) is smooth and K−ﬁnite if and only if there is a K0 ∈ V such that f is right K0−invariant.

Proof. If f is smooth, then we find for every g ∈ G(A) a Kg ∈ V such that for all k ∈ Kg, f(gk) = f(g). If f is K-finite, span{k·f}k∈K admits a finite basis {f1, . . . , fr} and K acts on this finite-dimensional space. Let

r F = (f1, . . . , fr): G(A) −→ C .

By linear independence of basis elements, we ﬁnd g1, . . . , gr ∈ G(A) such that the r elements in {F (g1),...,F (gr)} ⊂ C are linearly independent. is linearly indepen- 0 0 0 dent. Put K = Kg1 ∩ · · · ∩ Kgr , then for all k ∈ K and i = 1, . . . , r, this yields 0 0 0 k F (gi) = F (gik ) = F (gi). Thus K acts trivially on span{k · f}k∈K , and in particular, f is right K0-invariant. The reverse implication follows from the deﬁnition of the neighbourhood basis V.

0 For every subspace V ⊆ A, let V K be the subspace of all f ∈ V that are right 0 K0-invariant. By the previous lemma, functions in AK can be identiﬁed with the functions on G(F )Z(A) \ G(A)/K0 that are of moderate growth. In particular, AK is called the space of unramiﬁed automorphic forms. A consequence of the last lemma is:

S K0 Proposition 1.3.4. V = K0∈V V for every H-invariant subspace V ⊆ A. The fundamental fact for deﬁning the main object of this thesis is the following proposition.

0 Proposition 1.3.5. Fix Φ ∈ HK0 . For all [g] ∈ G(F )\G(A)/K , there is a unique set 0 of pairwise distinct classes [g1],..., [gr] ∈ G(F )\G(A)/K and numbers m1, . . . , mr ∈ ∗ such that C r X Φ(f)(g) = mif(gi). i=1 for all f ∈ AK0 .

21 Proof. The existence is as follows. Since Φ is K0−bi-invariant and compactly supported, it is a ﬁnite linear combination of characteristic functions on double cosets 0 0 of the form K hK with h ∈ G(A). So we may reduce the proof to Φ = charK0hK0 . Again, since K0hK0 is compact, it equals the union of a ﬁnite number of pairwise 0 0 distinct cosets h1K , . . . , hrK , and thus R 0 0 0 Φ(f)(g) = charK0hK0 (h )f(gh )dh G(A) Pr R 0 0 0 = charh K0 (h )f(gh )dh i=1 G(A) i Pr 0 = i=1 vol(K )f(ghi) Pr = i=1 mif(gi) for any g ∈ G(A), where for the last equality we have taken care of putting together values of f in the same classes of ∈ G(F ) \ G(A)/K0 and throwing out zero terms. 0 Uniqueness follows from the construction and the fact that f ∈ AK .

1.4 Coherent sheaves

Let us just recall a few basic facts on coherent sheaves on an algebraic curve, the reference here is the classical [19]. Let fix X be a smooth projective curve defined over a field k which will be either C or a finite field Fq. A torsion sheaf on X is a coherent sheaf whose support is a finite set of points. There exists a one-to-one correspondence between the set of isomorphism classes of locally free sheaves of rank n on X and the set of isomorphism classes of vector bundles of rank n on X (cf. [19, II Ex. 5.18]). Because of that, we sometimes interchange the words ”locally free sheaf” and ”vector bundles” if there is no misunderstanding. The category Coh(X) of coherent sheaves on X, contrary to the category of BunX of vector bundles on X, is abelian. The category BunX is an exact subcategory of Coh(X). Any coherent sheaf F on X has a canonical maximal torsion subsheaf T ⊆ F and a canonical quotient vector bundle E := F T such that 0 −→ T −→ F −→ E −→ 0 is exact and splits. Hence, any coherent sheaf can be decomposed as a direct sum F = E ⊕ T . Let Tor(X) stand for the abelian full subcategory of Coh(X) consisting of torsion sheaves. It decomposes as a direct product of blocks Y Tor(X) = Torx x∈|X| where Torx is the category of torsion sheaves supported at x. This last category is equivalent to the category of finite length modules over the local ring OX,x at x

22 which is a discrete valuation ring since X is smooth. If M is a OX,x-module of finite r length, then (πx) M = 0 for all sufficiently large r and thus we may consider M r as an OX,x (πx) -module. Hence M may be considered as Ox-module (note that Ox = κ(x)[[t]] is the (πx)-adic completion of OX,x), and therefore, in the end, the category Torx is equivalent to the category of modules of finite length over the discrete valuation ring κ(x)[[t]]. In particular, there is a unique simple sheaf Kx supported at x, namely the skyscraper in x with stalk κ(x). The rank (rk) of a coherent sheaf F is the rank of its canonical quotient vector bundle E. The degree (deg) of a coherent sheaf is the only invariant satisfying deg(OX ) = 0, deg(Kx) = |x| = [κ(x): k] and which is additive on short exact sequence. Let ωX be the line bundle of differential forms. For F, G ∈ Coh(X), Serre duality yields a canonical isomorphism 1 ∨ ∼ Ext (F, G) = Hom(G, F ⊗ ωX ). (1.4.1)

As a consequence, we have for any T ∈ Tor(X) and E ∈ BunX that

Hom(T , E) = Ext1(E, T ) = {0}. (1.4.2)

1 ∼ |x| In particular, Ext (Kx, OX ) = k for any closed point x of X, and up to isomorphism there is a unique non-trivial line bundle extension of Kx by OX , which is denoted by L(x). Next we recall the notion of semistability, an important notion for studying the categories of coherent sheaves on smooth projective curves. In particular, this will play an important role in Chapter 4. For F ∈ Coh(X), observe that if rk(F) = 0, then deg(F) > 0. We deﬁne the slope of F to be the quotient deg(F) µ(F) := ∈ ∪ {∞}, rk(F) Q where µ(F) = ∞ if rk(F) = 0 and we convention to be zero the slope of the zero sheaf. A sheaf F is called semistable of slope µ if µ(F) = µ and if µ(G) ≤ µ for all nonzero subsheaves G ⊂ F. It is called stable if in addition µ(G) < µ for all proper nonzero subsheaves G ⊂ F. Example 1.4.1. A line bundle L is stable of slope µ(L) = deg(L). Any torsion sheaf is semistable of slope ∞, and is stable if it is simple. The following lemma states a very useful property of slopes in exact sequences. Lemma 1.4.2 ([18] Lemma 1.3.12). Let F, G, H be coherent sheaves on X. If 0 −→ F −→ G −→ H −→ 0

23 is an exact short sequence, then

min µ(F), µ(H) ≤ µ(G) ≤ max µ(F), µ(H).

In particular, if G is semistable µ(H) ≥ µ(G).

Lemma 1.4.3 ([18] Lemma 1.3.11). For any coherent sheaf F there exists a unique ﬁltration 0 ⊂ F1 ⊂ · · · ⊂ Fs = F such that Fi Fi−1 is semistable and µ(F1) > µ(F2 F1) > ··· > µ(F Fs−1).

The above filtration is called the Harder-Narasimhan filtration of the sheaf F. It is a refinement of the canonical filtration 0 ⊂ T ⊂ F. Fix µ ∈ Q ∪ {∞} and let Cµ be the full subcategory of Coh(X) whose objects are the semistable sheaves of slope µ. By Example 1.4.1, C∞ = Tor(X).

0 Lemma 1.4.4. If µ > µ , then Hom(Cµ, Cµ0 ) = {0}.

Proof. Let F ∈ Ob(Cµ) and G ∈ Ob(Cµ0 ). If f : F → G is a non-zero morphism, by Lemma 1.4.2 µ(Im(f)) ≥ µ(F) = µ > µ0 = µ(G), a contradiction with the fact that G is semistable.

The above lemma joint with Serre duality gives.

0 1 Lemma 1.4.5. If µ > µ + deg(ωX ), then Ext (Cµ0 , Cµ) = {0}. The previous lemma implies the following important result. Proposition 1.4.6. Assume that X is a smooth projective curve of genus at most one. Then

1 0 (i) Ext (Cµ, Cµ) = {0} if µ < µ. (ii) The Harder-Narasimhan ﬁltration splits.

(iii) Any indecomposable sheaf in semistable.

Proof. Clearly, (i) implies (ii) which implies (iii). To prove (i) simply observe that under the hypothesis, deg(ωX ) ≤ 0.

We end the section summarizing the properties of the category Cµ.

Proposition 1.4.7. For any µ ∈ Q∪{∞}, the category Cµ is abelian and close under extension. Moreover, the simple objects of Cµ are the stable sheaves.

24 Proof. Let us ﬁrst show that Cµ is abelian. Let F, G be two objects of Cµ and let f : F → G be a morphism. As F and G are semistable, we have µ(Im(f)) ≥ µ (Lemma 1.4.2) and µ(Im(f)) ≤ µ. Thus µ(Im(f)) = µ and it follows, by semitability of G again, that Im(f) is also semistable. Similarly, ker(f) is semistable of slope µ. For showing that Cµ is closed under extension, let

0 −→ F −→ H −→ G −→ 0 be an exact sequence where F, G ∈ Ob(Cµ). By Lemma 1.4.2, it is clear that µ(H) = µ. If J is a subsheaf of H, then J is an extension of

0 −→F∩J −→J −→J 0 −→ 0

0 where J is the image of J under the projection H → H F ∼= G. By the semistability of F and G we have µ(J ∩ F) ≤ µ and µ(J 0) ≤ µ. Therefore, µ(J ) ≤ µ and H is semistable of slope µ. Since the slope is ﬁxed, the last assertion follows from the fact that the degree of any sheaf in Cµ is determined by its rank.

25 Chapter 2

Graphs of Hecke operators for GLn

The graph of a Hecke operator encodes all information about the action of this operator on automorphic forms. Let X be a smooth and projective curve over Fq, F its function ﬁeld and A the adele ring of F . In this chapter we will exhibit the ﬁrst properties for the graph of Hecke operators for GLn(A), for every n ≥ 1. This includes a description of the graph in terms of coherent sheaves on X. We provide a numerical condition for two vertices to be connected by an edge. Moreover, we describe how to 1 calculate these graphs in the case of the projective line X = P (Fq).

2.1 Graphs of Hecke operators

In this section we deﬁne the main object of this thesis, the graphs of Hecke operators. To each Hecke operator we associate a certain graph which encodes the action of this Hecke operator on automorphic forms. For that we start recalling the Proposition 1.3.5. 0 Proposition (Proposition 1.3.5). Fix Φ ∈ HK0 . For all [g] ∈ G(F ) \ G(A)/K , 0 there is a unique set of pairwise distinct classes [g1],..., [gr] ∈ G(F ) \ G(A)/K and ∗ numbers m1, . . . , mr ∈ C such that r X Φ(f)(g) = mif(gi). i=1 for all f ∈ AK0 . 0 For [g], [g1],..., [gr] ∈ G(F ) \ G(A)/K , as in the above proposition, we denote VΦ,K0 ([g]) := {([g], [gi], mi)}i=1,...,r.

Deﬁnition 2.1.1. Using the notation of Proposition 1.3.5, we deﬁne the graph GΦ,K0 of Φ relative to K0 whose vertices are 0 VertGΦ,K0 = G(F ) \ G(A)/K

26 and the oriented weighted edges [ EdgeGΦ,K0 = VΦ,K0 ([g]).

[g]∈VertGΦ,K0

0 The classes [gi] are called the Φ−neighbours of [g] (relative to K ).

Remark 2.1.2. We adopt the following terminology convention. For the triples in the above deﬁnition of edges of the graph GΦ,K0 , we call the ﬁrst entry by origin, the second entry by terminus and third by the weight of the edge.

We make the following drawing conventions to illustrate the graph of a Hecke operator: vertices are represented by labelled dots, and an edge ([g], [g0], m) together with its origin [g] and its terminus [g0] is drawn as

m [g] [g0] [g]

where the second ﬁgure is the case [g] = [g0]. 0 By Proposition 1.3.5 and the deﬁnition of the graph of Φ, we have for f ∈ AK and [g] ∈ G(F ) \ G(A)/K0 that X Φ(f)(g) = mif(gi).

([g],[gi],mi) ∈EdgeGΦ,K0

Hence one can read oﬀ the action of a Hecke operator on the value of an automorphic form from the illustration of the graph:

m1 [g1]

[g]

[gr]

27 S 0 Observe that, since H = HK0 , with K running over all compact opens in G(A), the notion of the graph of a Hecke operator applies to any Φ ∈ H. 0 The set of vertices of the graph of a Hecke operator Φ ∈ HK0 only depends on K , while the edges depend on the particular choice of Φ. There exists at most one edge for each two vertices, and the weight of an edge is always non-zero. Each vertex is connected with only a ﬁnitely many other vertices. The next proposition is an easy consequence of the algebra structure of HK0 .

Proposition 2.1.3. For the zero element 0 ∈ HK0 , the multiplicative unit 1 ∈ HK0 ∗ and arbitrary Φ1, Φ2 ∈ HK0 , r ∈ C , we obtain that

(i) Edge G0,K0 = ∅;

(ii) Edge G1,K0 = ([g], [g], 1) ; [g]∈VertG1,K0

n 0 X 0 X 00 o (iii) Edge G 0 = ([g], [g ], m) m = m + m 6= 0 ; Φ1+Φ2,K ([g],[g0],m0) ([g],[g0],m00) ∈EdgeG 0 ∈EdgeG 0 Φ1,K Φ2,K

0 0 0 0 (iv) Edge GrΦ1,K = ([g], [g ], rm) ([g], [g ], m) ∈ EdgeGΦ1,K , and

n 0 X 0 00 o (v) Edge G 0 = ([g], [g ], m) m = m · m 6= 0, . Φ1∗Φ2,K 00 0 ([g],[g ],m )∈EdgeG 0 Φ1,K 00 0 00 ([g ],[g ],m )∈EdgeG 0 Φ2,K

00 0 We end this section with the following observation. If K < K and Φ ∈ HK0 , then also Φ ∈ HK00 . This implies that we have a canonical map π : GΦ,K00 → GΦ,K0 , which is given by

00 π 0 Vert GΦ,K00 = G(F ) \ G(A)/K −→ G(F ) \ G(A)/K = Vert GΦ,K0 and

π Edge GΦ,K00 −→ Edge GΦ,K0 ([g], [g0], m0) 7−→ (π([g]), π([g0]), m0).

28 2.2 Graphs of unramiﬁed Hecke operators

In this section, we investigate the structure around a vertex in the graph of unramified Hecke operators. The main result of this section is that the neighbours of a vertex, counted with multiplicities, correspond to the points of an appropriate Grassmannian. Notation: We adopt the convention that the empty entry in a matrix means a zero entry. In the following given an explicit description of the unramified Hecke algebra. Fix n ≥ 1 an integer. For x a place of F, let Φx,r be the characteristic function of π I K x r K In−r where Ik is the k ×k identity matrix. Observe that Φx,n is invertible and its inverse is −1 given by the characteristic function of K(πxIn) K. The well-known theorem about the polynomial structure of unramified Hecke algebra due by Tamagawa and Satake is:

Theorem 2.2.1 ([8] Chapter 12, 1.6). Identifying K with 1 ∈ C yields ∼ −1 HK = C[Φx,1,..., Φx,n, Φx,n]x∈|X|.

In particular, HK is commutative. Remark 2.2.2. By Proposition 2.1.3, it is enough to determine the graphs for the −1 algebra generators Φx,1,..., Φx,n, Φx,n with x ∈ |X| (Theorem 2.2.1) in order to understand the graph of any Hecke operator. We use the shorthand notation Gx,r for the graph GΦx,r,K and Vx,r([g]) for Φx,r−neighbourhood VΦx,r,K ([g]) of [g], for x ∈ |X| and r = 1, . . . , n. The following considerations will be used in the proof of Theorem 2.2.7. For the standard Borel subgroup B < G of upper triangular matrices, we have the local and global form of Iwasawa decomposition, respectively:

G(Fx) = B(Fx)Kx and G(A) = B(A)K with Kx = G(Ox), see [9, Proposition 4.5.2]. Schubert Cell Decomposition. Let Gr(k, n) denote the Grassmannian that parametrizes the k−dimensional linear subspaces of a ﬁxed n−dimensional vector space. As a set, the Grassmannian has a decomposition as the disjoint union a Gr(k, n) = Cλ λ∈J(k,n) where J(k, n) = {(j1, . . . , jk)|1 ≤ j1 < ··· < jk ≤ n}, and Cλ is the set of n × n−matrices (aij)n×n of the following form:

29 • ajj = 1 if j ∈ λ.

• aij = 0 if j 6∈ λ, or j < i, or i ∈ λ and j 6∈ λ.

Where we denote j ∈ λ = (j1, . . . , jn) if j ∈ {j1, . . . , jn}. The sets Cλ are called Schubert cells and the disjoint union above is the Schubert cell decomposition of Gr(k, n). See [22, Theorem 11 ] and [14] for more details. We associate with each w ∈ Gr(k, n)(Fq) a matrix ξw ∈ G(A). By the Schubert cell decomposition, each w ∈ Gr(k, n)(Fq) lies in precisely one Schubert cell Cλ. Thus we can identify w with a matrix (aij)n×n in Cλ. We deﬁne ξw = (bij) ∈ G(A) by just replace 0’s on the diagonal of (aij)n×n by πx. We identify πx with the idele a where ay = 1 if y 6= x and ax = πx in y = x, and we can consider α ∈ κ(x) as the adele whose component at x is α and whose others components are 0. With these identiﬁcations, we can consider ξw as an adelic matrix in G(A). The Schubert cell decomposition can be reformulated as follows. Lemma 2.2.3. There is a bijection of Gr(k, n)(κ(x)) with the set    1 b12 ··· b1n i ∈ {1, πx},  . .   .. .  #{i|i = 1} = k and bij ∈ κ(x)   .   b  with bij = 0 if either  n−1 n−1n   n j = πx or i = 1  Remark 2.2.4. For n = 2 and k = 1, we have 1 0 1 ξw1 = where w1 = [1 : 0] ∈ P (κ(x)) 0 πx and π ∗ ξ = x w = [∗ : 1] ∈ 1(κ(x)), w2 0 1 where 2 P which are the same matrices as considered by Lorscheid in [30]. As the next step, we describe a method to distinguish the double cosets Kx \ G(Fx)/Kx, which will be useful in the proof of the next lemma. Let Vk : GLn(Fx) −→ GL n (Fx) (k) g 7−→ ∧kg be the k−th exterior power representation, i.e. the entries of ∧kg are the k×k−minors k k of g. Let I(∧ g) be the fractional ideal of Ox generated by the entries in ∧ g, i.e. k the fractional ideal generated by these minors. It is clear that ∧ Kx is a subset of k GL n (Ox), thus I(∧ g) is invariant under left and right multiplication by Kx. Also (k) note that I(∧1g) = I(g) is the fractional ideal generated by the entries of g and that I(∧ng) is generated by determinant of g. Thus the following lemma.

30 Lemma 2.2.5. Let g1, g2 ∈ GLn(Fx). Two double cosets Kxg1Kx and Kxg2Kx are k k equal, and thus I(∧ g1) = I(∧ g2) for k = 1, . . . , n. The next lemma is the key observation for the proof of the main theorem of this section. Lemma 2.2.6. There is a decomposition of sets πxIr a K K = ξwK. In−r w∈Gr(n−r,n)(κ(x))

Proof. Consider the map

K × K −→ G(A) πxIr (k1, k2) 7−→ k1 k2 In−r which is continuous because it is induced by the group multiplication. By Tychonoff’s theorem, K × K is compact. As the continuous image of the compact set K × K, the double coset π I K x r K In−r is compact. Since K is open, the quotient π I . K x r K K In−r is finite. Therefore, π I G K x r K = ξK In−r ξ∈I where I is a finite set of coset representatives. Thus we have to show that I = ξw| w ∈ Gr(n − k, n)(κ(x)) as in the Lemma 2.2.3. By the Iwasawa decomposition, we can take ξ ∈ I to be upper triangular. The question can be solved component-wise at each place y ∈ |X|. If y 6= x, then KyInKy = Ky. Thus we can use I w = n−r n×(n−r) and obtain In−r ξw = with (ξw)y = In, πxIr as desired.

31 If y = x, observe that πxIr Kx Kx ⊆ Mat(Ox). In−r

Thus ξx is an upper triangular matrix with entries in Ox. Let π I g = x r . In−r

k k Lemma 2.2.5 states that I(∧ g) = I(∧ ξx) for every k = 1, . . . , n. By a simple 2 n−r−1 n−r calculation, we have I(g) = Ox,I(∧ g) = Ox,...,I(∧ g) = Ox,I(∧ g) = n−r+1 n r Ox,I(∧ g) = (πx),...,I(∧ g) = (πx). Let

 1 bij  . ξx =  ..  n

li ∗ where i = uiπx , li ≥ 0 and ui ∈ Ox. By right multiplication with  −1  u1 .  ..  ∈ K, −1 un

li we can assume that i = πx , li ≥ 0, thus

 l1  πx bij . ξx =  ..  with bij ∈ Ox. ln πx n n r As I(∧ ξx) = I(∧ g) = (πx), we have l1 + ··· + ln = r. 0 Suppose that there is some lj > 1. As li ≥ 0, there are at least n − r + 1 lis equal n−r+1 n−r+1 to zero. Thus I(∧ ξx) = Ox, but I(∧ g) = (π ), a contradiction. Therefore, x all li ∈ {0, 1} and #{i ∈ {1, . . . , n} li = 1} = r. This show that the diagonal of ξx is of the predicted shape. We are left with showing that the bij are as claimed. First we can assume that bij = 0 if i = 1. This follows since we can change the representative of ξK by means of the product

 1 b12 ··· b1n   1  . . .  .. .   ..       1 bii+1 ··· bin   1 −bii+1 · · · −bin       .. .  ·  ..  .  . .   .   . .   .   .. .   ..  n 1

32 Doing this step for every i ∈ J, we obtain a matrix ξx which is zero in entries on 0 right and left of i = 1. Observe that these multiplications change the original bijs, for convenience we shall continuous using the same notation for those elements. n−r+1 If i = j = πx, then bij ∈ I(∧ ξx). Indeed, let Bij the (n − r + 1) × (n − r + 1) submatrix of ξx by removing the r − 1 columns 6= j with πx in the diagonal and removing the r − 1 rows 6= i with πx in the diagonal. Observe that bij is an entry of Bij and the column of Bij which contains bij has all entries, unless bij, equal to zero. Moreover, when we remove the row and the column of Bij which contains bij, we get the identity matrix. Thus, from Laplace’s formula, bij = ± det(Bij) and hence n−r+1 bij ∈ I(∧ ξx). n−r+1 n−r+1 Since (πx) = I(∧ g) = I(∧ ξx), we have πx | bij if i = j = πx. Therefore we can eliminate these entries by the following multiplication

 1   1  . .  ..   ..       π π b˜   1 −b˜   x x ij   ij   ..   ..   .  ·  .       πx   1   .   .   ..   ..  n 1 ˜ where πxbij = bij. P ` Finally, for bij = `≥0 bij(`)πx where j = 1 and i = πx, we multiply ξx with the matrix λ = (crs) fro the right, where crr = 1, and for r 6= s, crs = 0 unless P `−1 r = i, s = j, in which case cij = − `≥1 bij(`)πx . This allow to consider bij equal to bij(0) ∈ κ(x), therefore ξx is as claimed in the lemma 2.2.3.

Theorem 2.2.7. The Φx,r−neighbours of [g] are the classes [gξw] with ξw as in the previous lemma and the multiplicity of an edge of [g] to [g0] equals the number of 0 w ∈ Gr(n−r, n)(κ(x)) such that [gξw] = [g ]. The multiplicities of the edges originating in [g] sum up to #Gr(n − r, n)(κ(x)).

Proof. Let Φ := char(KhK), for h ∈ G(A). Since KhK is compact and K is open, KhK is equals to the union of a ﬁnite number of pairwise distinct cosets h1K, . . . , hrK. Hence, by Proposition 1.3.5 and deﬁnition of GΦ,K the neighbours of [g] are [ghi], i = 1, . . . , r. Therefore the theorem follows from preceding lemma.

Corollary 2.2.8. The edges set of the graph of Φx,n is EdgeGx,n = ([g], [g(πxIn)], 1) . [g]∈G(F )\G(A)/K

Proof. Observe that K(πxIn)K = (πxIn)K.

33 In the calculation of the graph of Φx,r, the following well-known formula is very useful. (qn − 1)(qn−1 − 1)(qn−2 − 1) ··· (qn−k+1 − 1) Lemma 2.2.9. #Gr(k, n)( ) = . Fq (qk − 1)(qk−1 − 1)(qk−2 − 1) ··· (q − 1)

2.3 Geometry of graphs of unramiﬁed Hecke operators

Recall that X is the geometrically irreducible smooth projective curve over Fq whose function ﬁeld is F. A well-known theorem by Weil states that G(F ) \ G(A)/K stays in bijection to the set BunnX of isomorphism classes of rank n-vector bundles on X. This allows us to give a interpretation of Gx,r in geometric terms. We begin with a review of Weil’s theorem. The bijection

F ∗ \ ∗/O∗ = ClF ←→1:1 PicX = Bun X A A 1 [a] 7−→ La where La = LD if D is the divisor determined by a, generalises to all vector bundles as follows, cf. [13, Lemma 3.1] and 2.1 in [8]. A rank n bundle E can be described by choosing bases

∼ n n ∼ n n Eη = OX,η = F and Ex = OX,x = (Ox ∩ F ) for all stalks, where η is the generic point of X, and the inclusion maps Ex ,→ Eη for all x ∈ |X|. After tensoring with Fx, for each x ∈ |X| we obtain

n ∼ n ∼ ∼ ∼ n ∼ n Fx = Ox ⊗Ox Fx = Ex ⊗OX,x Ox ⊗Ox Fx = Ex ⊗OX,x F ⊗F Fx = F ⊗F Fx = Fx which yields an element g of G(A). A change of bases for Eη and Ex corresponds to multiplying g by an element of G(F ) from the left and by an element of K from the right, respectively. Since the inclusion F ⊂ Fx is dense for every place x, and G(OA) is open in G(A), every class in G(F ) \ G(A)/G(OA) is represented by a g = (gx) ∈ G(A) such that gx ∈ G(F ) for all places x. This means that the above construction can be reversed. Weil’s theorem asserts the following.

Theorem 2.3.1 ([13] Lemma 3.1). For every n ≥ 1, the above construction yields a bijection 1:1 G(F ) \ G(A)/K ←→ BunnX. g 7−→ Eg

34 The last theorem identiﬁes the set of vertices of Gx,r with geometric objects BunnX. The next task is to describe the edges of Gx,r in geometric terms. We say that two exact sequences of sheaves

0 0 0 −→ F1 −→ F −→ F2 −→ 0 and 0 −→ F1 −→ F −→ F2 −→ 0

0 0 are isomorphic with fixed F if there are isomorphism F1 → F1 and F2 → F2 such that 0 / F1 / F / F2 / 0 =∼ =∼ 0 0 0 / F1 / F / F2 / 0 commutes. Let Kx be the torsion sheaf that is supported at x and has stalk κ(x) at x, i.e. the skyscraper torsion sheaf at x. Fix E ∈ BunnX. For r ∈ {1, . . . , n}, 0 0 and E ∈ BunnX we define mx,r(E, E ) as the number of isomorphism classes of exact sequences 00 ⊕r 0 −→ E −→ E −→ Kx −→ 0 with fixed E and with E 00 ∼= E 0.

Deﬁnition 2.3.2. Let x ∈ |X|. For a vector bundle E ∈ BunnX we deﬁne

0 0 Vx,r(E) := {(E, E , m)|m = mx,r(E, E ) 6= 0},

0 0 0 and we call E a Φx,r-neighbour of E if mx,r(E, E ) 6= 0, and mx,r(E, E ) its multiplicity.

We will show that this concept of neighbours is the same as the one deﬁned for classes in G(F ) \ G(A)/K in Deﬁnition 2.1.1. According to Theorem 2.2.7,the Φx,r−neighbours of a class [g] ∈ G(F ) \ G(A)/K are of the form [gξw] for a w ∈ Gr(n − r, n)(κ(x)).

Lemma 2.3.3. For every x ∈ |X|, the map

Vx,r([g]) −→ Vx,r([Eg]) 0 ([g], [g ], m) 7−→ (Eg, Eg0 , m) is a well-deﬁned bijection. ∼ ⊕n Proof. Fix a base (Eg)y → OX,y, for each y ∈ |X|. Note that by the deﬁnition of ξw, ⊕n multiplying an element of OX,y with the component (ξw)y from the right yields an ⊕n element of OX,y. Thus we obtain an exact sequence of Fq−modules

Y ⊕n ξw Y ⊕n ⊕r 0 −→ OX,y −→ OX,y −→ κ(x) −→ 0 y∈|X| y∈|X|

35 where r is the number of πx in the diagonal of (ξw)x. The correspondence in Theorem 2.3.1 implies the following exact sequence of sheaves

⊕r 0 −→ Egξw −→ Eg −→ Kx −→ 0. This maps w ∈ Gr(n − r, n)(κ(x)) to the isomorphism class of ⊕r 0 → Egξw → Eg → Kx → 0 with ﬁxed Eg. On the other hand, an isomorphism class of exact sequences 0 ⊕r 0 −→ E −→ E −→ Kx −→ 0 (2.3.1) of sheaves yields an isomorphism class of exact sequence of OX,x-modules

0 ⊕r 0 −→ Ex −→ Ex −→ κ(x) −→ 0 by taking stalks. After tensoring by OX,x mX,x, where mX,x is the maximal ideal of OX,x, we obtain an exact sequence of κ(x)−vector spaces

0 OX,x OX,x ⊕r Ex ⊗ −→ Ex ⊗ −→ κ(x) −→ 0. mX,x mX,x

Since 0 0 OX,x ∼ Ex OX,x ∼ Ex Ex ⊗ = 0 and Ex ⊗ = mX,x mX,xEx mX,x mX,xEx (see [6] page 31) we have 0 Ex Ex ⊕r 0 −→ −→ κ(x) −→ 0. mX,xEx mX,xEx

0 0 Let wx be the image of Ex mX,xEx in the above sequence. Then

Ex ⊕r 0 −→ wx −→ −→ κ(x) −→ 0 mX,xEx and dim(wx) = n − r. Therefore we have associated with the exact sequence (2.3.1) an element wx in Gr(n − r, n)(κ(x)). Theorem 2.3.1 and Lemma 2.3.3 imply:

Theorem 2.3.4. Let x ∈ |X|. The graph Gx,r of Φx,r is described in geometric terms as a VertGx,r = BunnX and EdgeGx,r = Vx,r(E).

E∈BunnX

36 Proposition 2.3.5. Let L1,..., Ln be invertible sheaves on X. Then for every S ⊆ {1, . . . , n}, with #S = r

0 0 m = mx,r(E, E ) 6= 0, i.e. (E, E , m) ∈ Vx,r(E)

0 L L where E = L1 ⊕ · · · ⊕ Ln and E = j∈S Lj(−x) ⊕ j6∈S Lj. Proof. Consider the exact sequence

0 −→ L(−x) −→ OX −→ Kx −→ 0.

Tensoring the above sequence with the locally free sheaf Lj, does neither aﬀect the sheaf Kx nor the exactness. Thus we get

0 −→ Lj(−x) −→ Lj −→ Kx −→ 0.

Combining these exact sequences for j ∈ S with

0 −→ Li −→ Li −→ 0 −→ 0 for i 6∈ S shows that

M M ⊕r 0 −→ Lj(−x) ⊕ Lj −→ L1 ⊕ · · · ⊕ Ln −→ Kx −→ 0 j∈S j6∈S is an exact sequence.

2.4 The δ−invariant

Let E be a locally free sheaf and E 0 a subsheaf. Note that the quotient E/E 0 is not necessarily locally free. We will call E 0 a subbundle if the quotient E/E 0 is still a vector bundle i.e. a locally free sheaf. 0 0 0 0 0 Let E := ker E → E/E → (E/E ) (E/E )tors , where (E/E )tors denotes the 0 0 0 0 torsion subsheaf of E/E , i.e. (E/E ) (E/E )tors is torsion free. By deﬁnition E is a 0 0 0 0 ∼ 0 subbundle of E, and E ⊂ E . It is easy to see that E /E = (E/E )tors. Hence

rk(E 0) = rk(E 0) and deg(E 0) ≥ deg(E 0).

Therefore we can extend a subsheaf E 0 of E to a subbundle E 0 of E. In what follows, k-(sub)bundle means a (sub)bundle of rank k. For a subbundle E 0 of a n-bundle E, we deﬁne

δ(E 0, E) := rk(E) deg(E 0) − rk(E 0) deg(E)

37 and for k = 1, . . . , n − 1,

0 δk(E) := sup δ(E , E). E0,→E k−subbundle

Moreover, we deﬁne δ(E) := max{δ1(E), . . . , δn−1(E)}.

0 0 Deﬁnition 2.4.1. We say that E ,→ E is a maximal k−subbundle if δ(E , E) = δk(E) and that E 0 is a maximal subbundle if δ(E 0, E) = δ(E).

Remark 2.4.2. A vector bundle E is (semi)stable if and only if

δ(E) < 0 (δ(E) ≤ 0).

Lemma 2.4.3. Let E be a locally free sheaf of rank n on X. Suppose that H0(X, E) is non-trivial. Then there is an exact sequence

0 −→ L −→ E −→ E 00 −→ 0 with E 00 locally free of rank r − 1, L locally free of rank 1 and deg(L) ≥ 0.

0 Proof. Let s ∈ H (X, E) be non-zero. The global section s deﬁnes a map OX → E by OX (U) −→ E(U), 1 7−→ s|U This map is injective and yields the short exact sequence

0 −→ OX −→ E −→ E/sOX −→ 0

00 Since we do not know if F = E/sOX is locally free, take E = F/Ftors, which is a locally free sheaf of rank r − 1 where Ftors is the subsheaf of torsion elements. Consider the surjective map ψ : E −→ E 00 and deﬁne L = ker(ψ). By construction, we have a short exact sequence

0 −→ L −→ E −→ E 00 −→ 0 with L locally free of rank 1 and E 00 locally free of rank r − 1. Concerning the degree of L, note that

00 deg(L) = deg(E) − deg(E ) = deg(E) − deg(F) + deg(Ftors) = deg(Ftors) ≥ 0 where the last inequality holds for every torsion sheaf.

38 Proposition 2.4.4. For every rank n bundle E,

−ng ≤ δ(E) < ∞

Proof. Recall that for coherence sheaves F, G ∈ Coh(X),

deg(F ⊗ G) = rk(F) deg(G) + deg(F)rk(G) and rk(F ⊗ G) = rk(F)rk(G).

Let L be an invertible sheaf and E 0 ⊆ E a subbundle. Then δ(L ⊗ E 0, L ⊗ E) = n deg(E 0) + rk(E 0) deg(L) − rk(E 0) deg(E) + deg(L)n = n deg(E 0) − rk(E 0) deg(E) = δ(E 0, E).

Thus δ(−, −) is invariant under tensoring with line bundles. As deg(L⊗E) = deg(E)+ n deg(L), the multiplication by a line bundle changes the degree of E by a multiple of n. Thus we can assume that n(g − 1) < deg(E) ≤ ng.

By the Riemann-Roch theorem, dim H0(X, E) = deg(E) + n(1 − g) + dim H1(X, E) ≥ deg(E) + n(1 − g) > 0.

By Lemma 2.4.3, there is a line subbundle L of E with deg(L) ≥ 0. Thus δ(L, E) = rk(E) deg(L) − deg(E) ≥ − deg(E) ≥ −ng, which implies δ(E) ≥ −ng. This establishes the ﬁrst inequality. Let E 0 ⊆ E be a subbundle. Again by Riemann-Roch, deg(E 0) ≤ rk(E 0)(g − 1) + dim H0(X, E 0) ≤ rk(E)(g − 1) + dim H0(X, E).

This proves the second inequality. We derive some immediate consequences from the proof of the above proposition.

Corollary 2.4.5. The same is true in the last proposition for δ1(E), i.e.,

−ng ≤ δ1(E) < ∞.

Corollary 2.4.6. For every k = 1, . . . , n − 1, δk(E) < ∞. 0 Corollary 2.4.7. If g = 0, 1, then δk(E) is at most n dim H (X, E), for every k = 1, . . . , n − 1. In particular, δ(E) ≤ n dim H0(X, E).

39 Proof. Let E 0 be a subbundle of E. As in the proof of last proposition deg(E 0) ≤ dim H0(X, E), and we can assume deg(E) ≥ 0. Thus δ(E 0, E) = n deg(E 0) − deg(E)rk(E 0) ≤ n deg(E 0) ≤ n dim H0(X, E).

Lemma 2.4.8. Let E be a locally free and (semi)stable sheaf over X such that deg(E) ≥ rk(E)(2g − 2), (deg(E) > rk(E)(2g − 2)), then H1(X, E) = 0. Proof. We will prove by contradiction that H1(X, E) = 0. Suppose that H1(X, E) 6= 0. Using Serre duality, 1 ∼ 0 ∨ ∼ 0 ∨ ∼ 0 H (X, E) = H (X, E ⊗ ωX ) = Ext (OX , E ⊗ ωX ) = Ext (E, ωX ) = Hom(E, ωX ).

Hence there is a non-zero morphism ϕ : E → ωX . Let K be the subbundle of E, which extends ker(ϕ). Since for line bundles L, L0 we have Hom(L, L0) = 0 if deg(L) > 0 deg(L ), it is clear that deg(im(ϕ)) ≤ deg(ωX ). Thus

deg(K) ≥ deg(ker(ϕ)) ≥ deg(E) − deg(ωX ) = deg(E) − (2g − 2). By (semi)stability of E, deg(E) − (2g − 2) deg(K) deg(E) ≤ = µ(K) ≤ µ(E) = rk(E) − 1 rk(E) − 1 rk(E) where we have a strict inequality if E is stable. Therefore, deg(E) < rk(E)(2g − 2) in the stable case and deg(E) ≤ rk(E)(2g − 2) in the semistable case. A contradiction with the degree of E. We conclude that H1(X, E) = 0. Proposition 2.4.9. Let E 0 be a subbundle of the bundle E on X, such that (E/E 0)∨⊗E 0 is (semi)stable. If δ(E 0, E) ≥ rk(E 0)(rk(E) − rk(E 0))(2g − 2) (with strict inequality in the stable case), then E ∼= E 0 ⊕ E/E 0. Proof. We have an exact sequence 0 −→ E 0 −→ E −→ E/E 0 −→ 0. Since, Ext1(E/E 0, E 0) ∼= H1(X, (E/E 0)∨ ⊗ E 0) and deg((E/E 0)∨ ⊗ E 0) = rk(E)(deg(E 0) − deg(E)) + (rk(E) − rk(E 0))(deg(E)) = rk(E) deg(E 0) − rk(E 0) deg(E) = δ(E 0, E) ≥ rk(E 0)(rk(E) − rk(E 0))(2g − 2) (with strict inequality in the stable case), we have H1(X, (E/E 0)∨ ⊗ E 0) = 0 by the previous lemma. Thus Ext1(E/E 0, E 0) = 0 and E ∼= E 0 ⊕ E/E 0.

40 This implies immediately the following. Corollary 2.4.10. Let E be a 2-bundle and L ,→ E a line subbundle with δ(L, E) > ∼ 2g − 2, then E = L ⊕ E/L.

Lemma 2.4.11. Let L, L1,..., Ln be invertible sheaves. If

ι : L −→ L1 ⊕ · · · ⊕ Ln is a non-zero morphism, then deg(L) ≤ deg(Lj) for some j ∈ {1, . . . , n}.

Proof. Write ι = ι1 ⊕ · · · ⊕ ιn, as ι is non-zero, there is some j ∈ {1, . . . , n} such that ιj : L → Lj is non-zero and thus injective. Therefore deg(L) ≤ deg(Lj).

0 0 Proposition 2.4.12. Let L1,..., Ln, L1,..., Lm be invertible sheaves and

0 0 ι : L1 ⊕ · · · ⊕ Ln → L1 ⊕ · · · ⊕ Lm be an injective morphism. Then there is an injection {1, . . . , n} → {1. . . . , m}, i 7→ ji, such that deg(L ) ≤ deg(L0 ). i ji

0 0 Proof. Let di = deg(Li) and dj = deg(Lj) for i = 1, . . . , n and j = 1, . . . , m. Suppose 0 0 0 that d1 ≤ d2 ≤ · · · ≤ dn and d1 ≤ d2 ≤ · · · ≤ dm. We proceed by induction on n. The case n = 1 follows from the previous lemma. 0 0 If dj < d1 for all j ∈ {1, . . . , m}, then Hom(Li, Lj) = 0 for all j = 1, . . . , m, and ι 0 0 is not injective. Let j0 = min{j|dj > d1}, hence Hom(L1 ⊕ · · · ⊕ Ln, Lj0 ) 6= 0 and m − j0 ≥ n. Consider the commutative diagram with exact rows and columns

0 0 0

∗ 0 f1 f2 0 / ι Lj0 / L1 ⊕ · · · ⊕ Ln / Q / 0 (A)

ι0 ι ϕ m 0 g1 0 0 g2 M 0 0 / Lj0 / L1 ⊕ · · · ⊕ Lm / Lj / 0 (B) j=1 j6=j0

∗ 0 where Q := (L1 ⊕ · · · ⊕ Ln)/ι Lj0 . Next we prove the existence and injectivity of ϕ. First observe that, since the diagram commutes, g2 ◦ ι ◦ f1 = 0. The universal property of cokernels implies the existence of ϕ. It is enough to prove the injectivity on stalks ([19] ex II.1.2). Let x ∈ X and suppose ϕx(q) = 0 for some q ∈ Qx. Since (f2)x is surjective, q = (f2)x(a) for some a ∈ (L1 ⊕ · · · ⊕ Ln)x, and we conclude successively

41 • (g2)x((ι)x(a)) = ϕx(q) = 0;

0 • (ι)x(a) ∈ ker((g2)x) = im((g1)x) = (Lj0 )x;

0 • a ∈ (L1 ⊕ · · · ⊕ Ln)x and (ι)x(a) ∈ (Lj0 )x;

∗ 0 • a ∈ (ι Lj0 )x;

• q = (g2)x(a) = 0.

This shows that ϕ is injective. The snake lemma implies that the following diagram is commutative with exact rows and columns

0 0 0

∗ 0 f1 f2 0 / ι Lj0 / L1 ⊕ · · · ⊕ Ln / Q / 0 (A)

ι0 ι ϕ m 0 g1 0 0 g2 M 0 0 / Lj0 / L1 ⊕ · · · ⊕ Lm / Lj / 0 (B) j=1 j6=j0 0 / coker(ι0) / coker(ι) / coker(ϕ) / 0

0 0 0

0 0 0 0 Since (B) splits, there is π1 : L1 ⊕ · · · ⊕ Lm → Lj0 such that π1 ◦ g1 = idL . The j0 0 0 above diagram implies im(π1 ◦ι) ⊂ im(ι ). Indeed, (on stalks) if y ∈ im(π1 ◦ι)\im(ι ), then y goes to non-zero element in coker(ι). Hence g1(y) 6∈ im(ι) which implies that y 6∈ im(π1 ◦ ι) since π1 ◦ g1 = idL0 . j0 Deﬁne ∗ 0 π2 : L1 ⊕ · · · ⊕ Ln −→ ι Lj0 0 −1 0 by π2 = (ι ) ◦π1◦ι. Since ι is injective, π2 is well deﬁned. Moreover, π2◦f1 = idι∗L0 , j0 thus (A) also is a split sequence and

∗ 0 L1 ⊕ · · · ⊕ Ln = ι Lj0 ⊕ Q

∗ 0 0 . Therefore ι Lj0 = Li for some i ∈ {1, . . . , n}. By Lemma 2.4.11, di ≤ dj0 . Thus the claim follows from the induction hypothesis.

42 Corollary 2.4.13. Let E = L1 ⊕ · · · ⊕ Ln be an n-bundle where L1,..., Ln are line bundles and deg(L1) ≤ · · · ≤ deg(Ln). Then k−1 r X X δk(E) ≥ n deg(Lr−i) − k deg(Li) i=0 i=1 for every k = 1, . . . , n − 1. In particular, δk(E) ≥ 0 for k = 1, . . . , n − 1. Corollary 2.4.14. If r = 2 and k = 1 in the last corollary, then equality holds, i.e.

δ1(E) = deg(L2) − deg(L1). We use the δ−invariants to investigate vertices in the graph of an unramiﬁed Hecke operator which are connected by edge, i.e. to investigate sequences of the form 0 ⊕r 0 −→ E −→ E −→ Kx −→ 0 for n−bundles E and E 0, r = 1, . . . , n − 1, and x ∈ X a closed point. If E 00 ,→ E is a subbundle, then we say that it lifts to E 0 if there exists a morphism E 00 → E 0, such that the diagram E 00

| E 0 / E commutes. In this case, E 00 → E 0 is indeed a subbundle, since otherwise it would extend nontrivially to a subbundle E 00 → E 0 ,→ E and would contradict the hypothesis that E 00 is a subbundle of E. By exactness of the above sequence, a subbundle E 00 → E 0 00 ⊕r lifts to E , if and only if the image of E in Kx is 0. 0 Theorem 2.4.15. Let E be an n−bundle. If E is a neighbour of E in Gx,r, then 0 δk(E ) ∈ δk(E) − k|x|(n − r), δk(E) − k|x|(n − r) + n, . . . , δk(E) + k|x|r for k = 1, . . . , n − 1. Proof. Let d = deg(E) and d0 = deg(E 0). By the exactness of the sequence 0 ⊕r 0 −→ E −→ E −→ Kx −→ 0, we have d0 = d − r|x|. Let F ,→ E and F 0 ,→ E 0 be maximal k-subbundles. Note that every subbundle of E 0 is a locally free subsheaf of E and thus extends to a subbundle of E. Let F 0 be the subbundle of E that extends F 0. We know that deg(F 0) ≥ deg(F 0). Thus 0 0 0 δk(E ) = n deg(F ) − kd ≤ n deg(F 0) − k(d − r|x|) 0 = δk(F , E) + k|x|r

≤ δk(E) + k|x|r.

43 If F ,→ E lifts to E 0, then F ,→ E 0 is a maximal k-subbundle. Thus we have 0 equalities in the above estimation, i.e. δk(E ) = δk(E) + k|x|r. Let L(−x) be the ideal sheaf of {x}. For every k-subbundle F ,→ E, we may think of L(−x) ⊗ F as a subsheaf of F. If F ,→ E does not lift to E 0, then L(−x) ⊗ F ⊆ F ,→ E lifts to a subsheaf of E 0 since L(−x) = ker(OX → Kx). Hence

L(−x) ⊗ F ⊆ F

E 0 / E

Therefore,

0 0 0 δk(E ) = δk(F , E ) 0 ≥ δk(L(−x) ⊗ F, E ) = n deg(L(−x) ⊗ F) − kd0 ≥ n(deg(F) − k|x|) − k(d − r|x|)

= δk(F, E) − nk|x| + k|x|r

= δk(E) − k|x|(n − r).

Furthermore,

0 0 δk(E ) ≡ −kd ≡ (d − r|x|)k ≡ δk(E) + k|x|r ≡ δk(E) − (n − r)k|x| (mod n).

This concludes the proof of the theorem. Corollary 2.4.16. Let E be an n−bundle and F ,→ E a maximal k-subbundle. If E 0 0 is a neighbour of E in Gx,r such that F ,→ E lifts to E , then

0 δk(E ) = δk(E) + k|x|r.

Proof. This follows from last theorem’s proof.

2.5 Graphs for the projective line

With the theory from the previous sections, we are able to calculate all graphs for unramified Hecke operators over a rational function field. For the rest of the section, we fix F = Fq(T ), i.e. X is the projective line over Fq. We intend to determine the 1 graphs of Φx,r for every n ∈ Z>0, r = 1, . . . , n and x ∈ P a closed point.

44 First of all, we know by a theorem due to Grothendieck1 (see [15, Theorem 11.51]) that every rank n-vector bundle over P1 is isomorphic to

OP1 (d1) ⊕ · · · ⊕ OP1 (dn) for some d1 ≥ · · · ≥ dn. By Proposition 2.1.3 and Grothendieck’s theorem, the graphs for the zero element are given by a subset of the standard n−lattice Zn consisting of vertices without edges n and the graphs for identity 1 in HK is a subset of the standard n−lattice Z consisting of vertices with loops of multiplicity one. We draw these two cases below for n = 1, 2. In certain cases, for a better reading we will denote by (d1, . . . , dn) the rank n-vector bundle OP1 (d1) ⊕ · · · ⊕ OP1 (dn).

1 OP1 (−2) OP1 (−1) OP OP1 (1) OP1 (2)

The graph of zero element in HK for n = 1.

(1, 1) (2, 1)

(0, 0) (1, 0) (2, 0)

(−1, −1) (0, −1) (1, −1) (2, −1)

The graph of zero element in HK for n = 2.

1 1 1 1 1

1 OP1 (−2) OP1 (−1) OP OP1 (1) OP1 (2)

The graph of the identity in HK for n = 1.

1actually it was already known by Dedekind and Weber, see [11].

45 1 1

(1, 1) (2, 1) 1 1 1

(0, 0) (1, 0) (2, 0) 1 1 1 1

(−1, −1) (0, −1) (1, −1) (2, −1)

The graph of the identity in HK for n = 2.

We continue with the calculation of the graph of Φx,1, for every closed point x in P1 of degree |x| and n = 1. By the correspondence of Theorem 2.3.4, a neighbourhood of OP1 (d) is given in terms of exact sequences 0 0 −→ E −→ OP1 (d) −→ Kx −→ 0. By Grothendieck’s classiﬁcation of vector bundles on P1, E 0 is completely determined 0 0 by its degree. By additivity of the degree, deg E = d − |x|, thus E = OP1 (d − |x|) and

mx,1(OP1 (d), OP1 (d − |x|)) = 1 (Corollary 2.2.8). The graph is illustrated as in the following ﬁgure.

1 1

OP1 (d) OP1 (d − |x|) OP1 (d − 2|x|) 1 1

OP1 (d − 1) OP1 (d − 1 − |x|) OP1 (d − 1 − 2|x|)

1 1

OP1 (d − |x| + 1) OP1 (d − 2|x| + 1) OP1 (d − 3|x| + 1)

The graph of Φx,1 in HK for n = 1.

46 Our next aim is to explain an algorithm to calculate the graphs for Φx,r and n > 1. First we treat the case |x| = 1 and afterwards we treat arbitrary degrees. The closed points y ∈ P1 are in one-yo-one correspondence with irreducible and monic polynomials πy in Fq[T ] and the inﬁnite valuation. Moreover |y| = deg πy and πy is the uniformizer of y. For z = f(T )/g(T ) ∈ F \{0} with (f, g) = 1,

[F : Fq(z)] = max{deg(f), deg(g)}.

If x is a degree one place, then [F : Fq(πx)] = deg(πx) = 1. Therefore we can suppose 1 + T = πx. Hence we can cover P by two open aﬃne subschemes U0 := SpecR and − U1 := SpecR , where

+ − −1 ± −1 R := Fq[πx],R := Fq[πx ] and R := Fq[πx, πx ].

We have bijections

+ ± − ∼ 1 ∼ G(R )\G(R )/G(R ) = BunnP = G(F )\G(A)/K

gx −→ (gx, id) where (gx, id) means the class of an adelic matrix which equals gx at place the x and the identity matrix at all places y 6= x (see [15] Chapter 11 for more details).

n n Lemma 2.5.1 ([15] Lemma 11.50). Let (Z )+ be the set of d = (di)i ∈ Z with n d ± d1 ≥ d2 ≥ · · · ≥ dn. For each d ∈ (Z )+, let πx ∈ G(R ) be the diagonal matrix with d1 dn entries πx , . . . , πx . Then the following map is bijective

n − ± + (Z )+ −→ G(R )\G(R )/G(R ). − d + d 7−→ G(R )πx G(R )

In the case of a degree one place x, we can determine the graph of Φx,r, for n > 1 − d 0 + and r ∈ {1, . . . , n} by ﬁnding the representative G(R )πx G(R ) for the double d cosets of πx ξw, cf. Lemma 2.2.3 and Theorem 2.2.7. We shall use the symbol "∼" when two matrices represent the same class in G(R−)\G(R±)/G(R+).

Example 2.5.2. In the following, for n = 2 i.e. rank two bundles, we describe the graph of Φx,1 where x a degree one place. By Theorem 2.2.7 and Lemma 2.5.1, we d n have to ﬁnd the unique diagonal matrix πx with d ∈ (Z )+ that represents gξw in G(R−)\G(R±)/G(R+) where

d1 πx [g] = d2 πx

47 with d1 ≥ d2 which corresponds to the vector bundle OP1 (d1) ⊕ OP1 (d2), and where π b ξ = x w = [1 : b] ∈ Gr(1, 2)(κ(x)) = 1( ) w 1 with P Fq or 1 1 ξw = with w = [0 : 1] ∈ Gr(1, 2)(κ(x)) = P (Fq). πx

For d1 > d2, w = [1 : b] and b 6= 0, we have

d1 d1+1 d1 πx πx b πx bπx gξw = d2 = d2 πx 1 πx

d1+1 d1 d2 1 πx bπx 1 πx ∼ d2 = d1 d1+1 1 πx 1 bπx πx

−1 d2−d1 d2 −1 d2+1 1 −b πx πx −b πx ∼ d1 d1+1 = d1 d1+1 1 bπx πx bπx πx

−1 d2+1 d1 d1+1 1 −b πx bπx πx ∼ d1 d1+1 = −1 d2+1 1 bπx πx −b πx

d1 d1+1 −1 d1 bπx πx 1 −b πx bπx ∼ −1 d2+1 = −1 d2+1 −b πx 1 −b πx

d1 −1 d1 bπx b πx ∼ −1 d2+1 = d2+1 =: g1. −b πx −b πx

Note that the number of classes [gξw] with w = [1 : b] and b 6= 0 is q − 1. For b = 0, we have d1+1 πx gξw = d2 =: g2. πx

Thus [g2] is a neighbour of [g] of at least multiplicity 1. If d1 > d2 and w = [0 : 1], then

d1 d1 πx 1 πx gξw = d2 = d2+1 = g1 πx πx πx with multiplicity 1. We conclude that for d1 > d2, the neighbourhood of [g] is

q [g1]

[g] 1

[g2]

48 Observe that the sum of multiplicities in the vertex [g] is q +1 = #Gr(1, 2)(κ(x)), in accordance with Theorem 2.2.7. If d1 = d2, then [g1] = [g2], and we have

q + 1

[g] [g1]

Example 2.5.3. Next we illustrate the graph of Φx,2, for rank three bundles and x a degree one place. Let  d1  πx d2 g :=  πx  , d3 πx which corresponds to OP1 (d1) ⊕ OP1 (d2) ⊕ OP1 (d3), with d1 ≥ d2 ≥ d3. 2 For w = [1 : a : b] ∈ Gr(1, 3)(κ(x)) = P (Fq), a 6= 0 we have

 d1     d1+1 d1  πx πx a πx aπx d2 d2+1 d2 gξw =  πx   πx b  =  πx bπx  d3 d3 πx 1 πx

   d1+1 d2   d3  1 πx aπx πx d2+1 d2 d2+1 d2 ∼  1   πx bπx  =  πx bπx  d3 d1+1 d1 1 πx πx aπx

 d3    πx −a d2+1 d2 ∼  πx bπx   1  d1+1 d1 πx aπx πx 1

 d3+1 d3   −1 d3−d1   d3+1 d3  πx πx 1 −a πx πx πx d2+1 d2+1 d2 d2+1 d2+1 d2 =  bπx πx bπx  ∼  1   bπx πx bπx  d1 d1 aπx 1 aπx

 d3+1     d3+1  πx 1 πx d2+1 d2+1 d2 −1 d2−d1 d2+1 d2+1 d2 =  bπx πx bπx  ∼  1 −a bπx   bπx πx bπx  d1 d1 aπx 1 aπx

 d3+1   d3+1    πx πx 1 d2+1 d2+1 d2+1 d2+1 =  bπx πx  ∼  bπx πx   −b 1  d1 d1 −1 aπx aπx a

 d3+1   d1  πx πx d2+1 d2+1 =  πx  ∼  πx  =: g1 d1 d3+1 aπx πx

49 with multiplicity q(q − 1). For a = 0 and b 6= 0

 d1+1   d1+1  πx aπx d2+1 d2 d2 d2+1 gξw =  πx bπx  ∼  bπx πx  d3 d3 πx πx

 d1+1     d1+1  aπx 1 πx aπx d2 d2+1 d2 ∼  bπx πx   −b  =  bπx  d3 d3 d3+1 πx 1 πx πx

   d1+1   d1+1  1 aπx πx d2 d2 ∼  1   bπx  =  bπx  −1 d3−d2 d3 d3+1 d3+1 −b πx 1 πx πx πx

 d1+1     d1+1  πx 1 πx d2 −1 d2 ∼  bπx   b  =  πx  =: g2 d3+1 d3+1 πx 1 πx with multiplicity q − 1. If a = b = 0,

 d1+1  πx d2+1 gξw =  bπx  =: g3 d3 πx with multiplicity 1. 2 For w = [0 : 0 : 1] ∈ P (Fq)

 d1     d1  πx 1 πx d2 d2+1 gξw =  bπx   bπx  =  bπx  = g1 d3 d3+1 πx πx πx with multiplicity 1. 2 If w = [0 : 1 : a] ∈ P (Fq) and a 6= 0,

 d1     d1+1 d1  πx πx a πx aπx d2 d2 gξw =  bπx   1  =  bπx  d3 d3+1 πx πx πx

 d1+1 d1     d1  πx aπx −a aπx d2 d2+1 d2 ∼  bπx   πx 1  =  πx πx  d3+1 d3+1 πx 1 πx

 −1 d2−d1   d2+1 d2   d2+1  1 −a πx πx πx πx d1 d1 ∼  1   aπx  =  aπx  d3+1 d3+1 1 πx πx

50  d1  πx d2+1 ∼  πx  = g1 d3+1 πx with multiplicity q − 1. For a = 0,

 d1     d1+1  πx πx πx d2 d2 gξw =  πx   1  =  bπx  = g2 d3 d3+1 πx πx πx with multiplicity 1. Therefore the graph of Φx,2 is given by:

(d1 + 1, d1, d3 + 1) q2 + q q2 + q + 1

(d1, d1, d1) (d1 + 1, d1 + 1, d1) (d1, d1, d3) 1

(d1 + 1, d1 + 1, d3)

(d1 > d3)

(d1, d2 + 1, d2 + 1) (d1 + 1, d2 + 1, d3) q2 1

(d1, d2, d2) q2 q + 1 (d1, d2, d3) (d1, d2 + 1, d3 + 1) (d1 + 1, d2 + 1, d2) q

(d1 + 1, d2, d3 + 1)

(d1 > d2) (d1 > d2 > d3)

We proceed with the investigation of places of larger degree. Let us ﬁx a place x of degree one and let y be a place of degree d ≥ 1. For determining the edges of − ± + Gy,r, we have to ﬁnd the standard representative of gξw in G(R ) \ G(R )/G(R ) for d n g = πx , d ∈ (Z )+ and for every w ∈ Gr(n − r, n)(κ(y)) where ξw is as in Theorem 2.2.7.

51 The problem here is that ξw has nontrivial entries in a place different from x and, a priori we can not use the reduction to a standard representative in the double − ± + coset G(R ) \ G(R )/G(R ). Thus we have to find an equivalence class for ξw which depends only on the x-component. Let S be the set of (n × n)-matrices (aij)n×n defined as follows. Given λ = (j1, . . . , jn−r) ∈ J(n − r, n), (aij)n×n is an upper triangular matrix defined as follows:

d • aii = πx if i 6= jk for k = 1, . . . , n − r and aii = 1 if i is equals to some jk

Pd−1 l l d • aij := l=0 aijπx ∈ Fq[πx] if i < j, ajj = 1 and aii = πx

• aij = 0 otherwise.

d Pd−1 l l l Observe that we have #κ(y) = q possibilities for the sum l=0 aijπx with aij ∈ Fq.

Proposition 2.5.4. Keeping the above notation, the Φy,r−neighbours of [g] are the classes [gδ] with δ ∈ S.

Proof. We have to show that there is a bijection

1:1 Ψ: {ξw|w ∈ Gr(n − r, n)(κ(y))} −→ S such that [gξw] = [gΨ(ξw)] in G(F ) \ G(A)/K. First of all, we can suppose πy has a nontrivial valuation only in y and x. Indeed, let X D(πy) := vz(πy)z z∈|P1| be the divisor associated with πy, write

0 0 X D(πy) = D + D with D := y − dx and D := dx + vz(πy)z. z6=y

0 0 Since, deg D = 0 and |Cl (F )| = 1, we have D(πy) − D = D(f) for some f ∈ F. 0 −1 Moreover vy(f) = 0 since D(f) = D . Thus f ∈ OF,y \{0} and πyf is a uniformizer for y with −1 0 0 D(πyf ) = D(πy) − D(f) = D + D − D = y − dx. −1 Replacing πy by πyf , we can assume that πy has a nontrivial valuation only in y and x. Let δ ∈ G(F ) denote the inverse of (ξw)y =: h. For all places z 6= x, y, the canonical embedding G(F ) ,→ G(Fz)

52 sends δ to a matrix δz ∈ Kz since vz(πy) = 0. Let k ∈ K such that (k)z = δz for −1 z 6= x, y, (k)x = In and (k)y = In. Let δw = δξwk ∈ G(A). Observe that only the −1 x−component of δw is nontrivial, with (δw)x = h . We have

−1 −1 [gξw] = [δgξwk ] = [gδξwk ] = [gδx] where the first equality holds because δ ∈ G(F ) and k−1 ∈ K, the second because g and δ are diagonal matrices and the last due to the definition of δw. To finish the proof, we need to show that δw ∈ S and S = {δw}. The only −1 nontrivial component of δw is (δw)x = (ξw)y . Since vx(πy) = −d, we have vx(πy) = d −1 d d and πy = uπx for some u ∈ OF,x \{0}. Thus the diagonal of (δw)x has entries uπx at the positions that (ξw)y has entries πy. Using the reduction from Lemma 2.2.6, we have (δw)z = In for z 6= x and conclude that (δw)x is as desired. In the end, since each w ∈ Gr(n − r, n)(κ(y)) gives us a unique ξw and each ξw is associated with some δw ∈ S and #S = #{ξw}, Ψ is a bijection. Each δw is in a different class, i.e. the matrices as defined before occur as the x−components of an associated δw.

By the previous proposition, the graph Gy,r depends only on the degree of y. Since we have a representation for ξw as δw, whose only nontrivial component is the x-component, we can use the same reduction as before to ﬁnd the neighbours of a vertex in the graph of Φy,r. Note that if y is a place of degree one x, then Gy,r is the same graph as Gx,r.

Example 2.5.5. Let us calculate the graph Gy,1 for rank 2-bundles and deg(y) = 2. According the last proposition, we have to ﬁnd the standard representatives for

d1 d1 2 πx 1 πx πx a0 + a1πx d2 2 and d2 πx πx πx 1 where d1 ≥ d2 and a0, a1 ∈ Fq. In the ﬁrst case, we have

d1 d1 πx 1 πx d2 2 = d2+2 πx πx πx with multiplicity one. In the second case, we ﬁrst assume a0 6= 0 and let s = ∗ a0 + a1πx ∈ OF,x. Then

d1 2 d1+2 d1 πx πx a0 + a1πx πx sπx d2 = d2 πx 1 πx

−1 d2−d1 d2 −1 d2+2 1 −s πx πx −s πx ∼ d1 d1+2 = d1 d1+2 1 sπx πx sπx πx

53 d1 d1+2 −1 2 d1 d1 sπx πx 1 −s πx sπx πx ∼ −1 d2+2 = −1 d2+2 ∼ d2+2 −s πx 1 −s πx πx whose multiplicity is q(q − 1). If a0 = 0 and a1 6= 0, then

d1 2 d1+2 d1+1 πx πx a0 + a1πx πx a1πx d2 = d2 πx 1 πx

−1 d2−d1−1 d2 −1 d2+1 1 −a1 πx πx 0 −a1 πx ∼ d1+1 d1+2 = d1+1 d1+2 1 a1πx πx a1πx πx

d1+1 d1+2 −1 a1πx πx 1 −a1 πx ∼ −1 d2+1 −a1 πx 1

d1+1 d1+1 a1πx πx = −1 d2+1 ∼ d2+1 −a1 πx πx with multiplicity q − 1. If a1 = b1 = 0, then

d1 2 d1+2 πx πx a0 + a1πx πx d2 = d2 πx 1 πx with multiplicity 1. Therefore, the graph Gy,1 is as follows.

q − 1 q − 1 q − 1

(d, d) (d + 1, d + 1) (d + 2, d + 2) (d + 3, d + 3) q2 − q + 2 q2 − q + 2 q2 − q + 2

q2 − q + 1 q2 − q + 1

q − 1 q − 1

(d + 2, d) 1 (d + 3, d + 1)1 (d + 4, d + 2)

q2 − q + 1

q − 1

(d + 4, d) 1 (d + 5, d + 1)

(d + 6, d)

54 q2 q2 q2

(d + 1, d) (d + 2, d + 1) (d + 3, d + 2) (d + 4, d + 3) 1 1 1

q2 − q + 1 q2 − q + 1

q − 1 q − 1

(d + 3, d) 1 (d + 4, d + 1)1 (d + 5, d + 2)

q2 − q + 1

q − 1

(d + 5, d) 1 (d + 6, d + 1)

(d + 7, d)

Example 2.5.6. We conclude with the description of the graph Gy,2, for rank 3 bundles where y is a place of degree two. We do not write out all calculations, which are similar to the ones for Gy,1. By Proposition 2.5.4, we have to ﬁnd the standard representatives for gδw with

 d1  πx d2 g :=  πx  , and δw one of the matrices d3 πx

   2   2  1 πx a0 + a1πx πx a0 + a1πx 2 2  πx  ,  1  or  πx b0 + b1πx  , 2 2 πx πx 1 where d1 ≥ d2 ≥ d3 and ai, bi ∈ Fq for i = 0, 1. As before, we will write (d1, d2, d3) for the matrix g. In the ﬁrst case, we obtain

 d1    πx 1 d2 2 gδw =  πx   πx  ∼ (d1, d2 + 2, d3 + 2) d3 2 πx πx with multiplicity 1. In the second case, we have

 d1   2   d1+2 d1  πx πx a0 + a1πx πx (a0 + a1πx)πx d2 d2 gδw =  πx   1  =  πx  d3 2 d3+2 πx πx πx

55 and we have to analyse the following subcases. • a0 6= 0: then gδw ∼ (d1, d2 + 2d3 + 2) with multiplicity q(q − 1). • a0 = 0 and a1 = 0: then gδw ∼ (d1 + 2, d2, d3 + 2) with multiplicity 1. • a0 = 0 and a1 6= 0 : then gδw ∼ (d1 + 1, d2 + 1, d3 + 2) with multiplicity 1. In the third case, we have

 d1   2   d1+2 d1  πx πx a0 + a1πx πx (a0 + a1πx)πx d2 2 d2+2 d2 gδw =  πx   πx b0 + b1πx  =  πx (b0 + b1πx)πx  d3 d3 πx 1 πx and consider the following subcases:

2 2 • a0 6= 0 and b0 6= 0 : then gδw ∼ (d1, d2 + 2, d3 + 2) with multiplicity q (q − 1) .

• a0 = 0, a1 6= 0 and b0 6= 0 : then gδw ∼ (d1 + 1, d2 + 1, d3 + 2) with multiplicity q(q − 1)2.

• a0 = 0, a1 = 0 and b0 6= 0 : then gδw ∼ (d1 + 2, d2, d3 + 2) with multiplicity q(q − 1).

• a0 = 0, a1 6= 0 and b0 = 0, b1 6= 0 : then gδw ∼ (d1 + 1, d2 + 2, d3 + 1) with multiplicity (q − 1)2.

• a0 = 0, a1 6= 0 and b0 = 0, b1 = 0 : then gδw ∼ (d1 + 1, d2 + 2, d3 + 1) with multiplicity (q − 1).

• a0 = 0, a1 = 0 and b0 = 0, b1 6= 0 : then gδw ∼ (d1 + 2, d2 + 1, d3 + 1) with multiplicity (q − 1).

• a0 = 0, a1 = 0 and b0 = 0, b1 = 0 : then gδw ∼ (d1 + 2, d2 + 2, d3) with multiplicity 1.

• a0 6= 0, and b0 = 0, b1 = 0 : then gδw ∼ (d1, d2 + 2, d3 + 2) with multiplicity q(q − 1).

• a0 6= 0, and b0 = 0, b1 6= 0 : then gδw ∼ (d1, d2 + 2, d3 + 2) with multiplicity q(q − 1)2.

The graph Gy,2 can be illustrated as follows.

56 (d1 + 2, d2 + 2, d3) (d1 + 2, d2 + 1, d3 + 1)

1 q − 1

q3 − 2q2 + 2q − 1 q4 − q3 + q2 − q + 1

(d1 + 1, d2 + 1, d3 + 2) (d1, d2 + 2, d3 + 2) (d1, d2, d3) q2 − q + 1

q2 − q

(d1 + 1, d2 + 2, d3 + 1) (d1 + 2, d2, d3 + 2)

Observe that, the sum up of multiplicities of edges originating in [g] is q4 + q2 + 1 which is equal to #Gr(1, 3)(Fq2 ), as stated in Theorem 2.2.7.

57 Chapter 3

Hall algebras

The theory of Hall algebras was first discussed (in an elementary version) by Ernest Steinitz (1901) in his ”forgotten paper” [49]. In this note, Steinitz defines what we have called the Hall polynomials. Steinitz’s note is a summary of a lecture given at the annual meeting of the Deutsche Mathematiker-Vereinigung in Aachen in 1900. This note contains various results some conjectures, but no proofs, and it does not even give hints about the author’s approach. It remained forgotten until brought to light by Karsten Johnsen in 1982 in the survey [25]. Half a century after Steinitz’s lecture, Philip Hall ”rediscovery” the theory of Hall polynomials and Hall algebras in the survey [17]. However Hall did not publish anything more than a summary of his theory. The first complete work about Hall algebras, with definitions and proofs is due to Ian G. Macdonald in his book [33]. In what follows, a partition is any finite sequence λ = (λ1, . . . , λs) of positive integers in decreasing order. We denote by (1s) the partition (1,..., 1) with s positions equals to 1. The λi’s are called parts of λ. The number of parts is the length of λ, denoted by l(λ). The sum of the parts is the weight of λ, denoted by |λ|. The first object introduced in this theory are the well-known Hall polynomials. Hall’s original definition (1950) is as follows. Fix a prime number p. Let G be a finite abelian p-group, i.e. a direct sum of cyclic subgroups of orders pλ1 , . . . , pλs , where we may suppose that λ1 ≥ · · · ≥ λs. The sequence of exponents λ = (λ1, . . . , λs) is a partition, called the type of G, which describes G up to isomorphism. If µ and ν are λ partitions, let hµν(p) denote the number of subgroups H of G such that H has type λ µ and G H has type ν. Hall showed that hµν(p) is a polynomial function of p with integer coefficients and was able to determine its degree and leading coefficient. More generally, cf. the next example, in place of a finite abelian p-group we may consider modules of finite length over a discrete valuation ring R with finite residue λ λ field. In place of hµν(p) we have hµν(q) where q is the number of elements in the residue field. Example 3.0.7. Let p be a prime number, G be a finite abelian p-group. Then

58 s s p G = 0 for large s. Thus G may be regarded as a module over the ring Z p Z for all large s, hence as a finite module over the ring R = Zp of p-adic integers. Next, Hall used these polynomials to construct an algebra which reflects the lattice structure of the finite R-modules. Let HR be the free Z-module with bases (Iλ) λ indexed by the set of partitions. Define a multiplication in HR by using the hµν(q) as structure constants, i.e. X λ Iµ Iν := hµν(q)Iλ. λ

We will see in Section 3.1 that HR is a commutative, associative ring with identity, which is freely generated (as Z-algebra) by the generators I(1s) corresponding to the elementary R-modules. In the early of 1990s, the interest in the Hall algebras increased after Ringel formalized the notion of a Hall algebra associated to a ﬁnitary category (cf. Section 3.2) and associated this theory to quantum groups in the series of papers [37], [38] and [40]. In this chapter, we aim to give a short introduction to the theory of Hall algebras. The main references for this chapter are [33] and [44].

3.1 The classical Hall algebras

Throughout this section R will denote a discrete valuation ring, m its maximal ideal. Let us require k = R m be a finite field. We shall be concerned with finite length R-modules M, that is to say, modules M which posses a finite composition series. Equivalently finitely-generated R-modules such that msM = 0 for some s ≥ 0. Since k is finite, the finite R-modules are precisely those that have a finite number of elements. The Example 3.0.7 shows us that the concept of Hall algebras for finite R-modules in fact generalizes the concept of Hall algebras for finite p-groups as introduced by Philip Hall. Since R is a principal ideal domain, every finitely generated R-modules is a direct sum of cyclic R-modules. For a finite length R-module M, this means that M has a direct sum decomposition of the form

s M M = Rmλi i=1 where the λi are positive integers, which we may assume to be arranged in descending order: λ1 ≥ · · · ≥ λs > 0. In other words, λ = (λ1, . . . , λs) is a partition. We call λ the type of M.

59 0 0 0 The conjugate of a partition λ = (λ1, . . . , λs) is the partition λ = (λ1, . . . , λs) 0 0 0 whose parts are given by λi = #{j | λj ≥ i}. In particular, λ1 = l(λ), λ1 = l(λ ) and λ00 = λ. For two partitions λ, µ of n ∈ Z (i.e. |λ| = |µ| = n), we shall write λ ≥ µ if

λ1 + ··· + λi ≥ µ1 + ··· + µi for all i ≥ 1.

Lemma 3.1.1 ([33] (1.11)). Let λ, µ partition of n ∈ Z. Then λ ≥ µ ⇐⇒ µ0 ≥ λ0.

Lemma 3.1.2. Fix a partition λ and let M be a ﬁnite R-module of type λ. Let i−1 i µi = dimk(m M m M). Then µ = (µ1, . . . , µr) is the conjugate partition of λ. Proof. Since M has type λ, we may write

s M M = Rmλi . i=1

λi Let xj be the generator of the R m in the above description. Let π be the generator i−1 of m. Thus m M is generated by those of the πi−1xj which do not vanish, i.e. those for which λj ≥ i. Hence µj is equal to the number of indices j such that λj ≥ i. 0 Therefore, µi = λi. By Lemma 3.1.2, λ is determined uniquely by the module M. Clearly two finite R-modules are isomorphic if and only if thay have the same type, and every partition P λ occurs as a type. If λ = (λ1, . . . , λs) is the type of M, then |λ| := λi is the length l(M) of M i.e. the length of a composition series of M. The length is an additive function. If N is a submodule of M, the cotype of N in M is defined to be the type of MN. A finite length R-module M is cyclic (i.e. generated by one element ) if and only if its type is a partition (n) consisting of a single part n = l(M). We say M elementary if mM = 0, hence M is elementary if and only if the type of M is (1r). If M is a r elementary of type (1 ), then M is a vector space over k, and l(M) = dimk M = r.

Duality. Let π be a generator of the maximal ideal m. If m ≤ n, then multiplication n−m m n by π is an injective R-homomorphism of R m into R m . Consider the direct limit E := lim Rmn. −→

60 Then E is an injective R-module containing k, and it is the smallest injective R- module which contains k as a submodule. If M is any ﬁnite R-module, the dual of M is deﬁned to be

Mc := HomR(M,E).

Lemma 3.1.3. Let M be a ﬁnite length R-module. Then the dual Mc of M is ﬁnite length and isomorphic to M. In particular, it has the same type as M.

Proof. Observe that taking dual modules commutes with direct sums, hence it is enough to proof M ∼= Mc when M is ﬁnite length and cyclic. Let x be the generator of M, this is the case because a homomorphism from M to R is given by fa : rx 7→ ra for ﬁxed a ∈ R and every r ∈ R. Hence the isomorphism Mc → M is given by fa 7→ ax.

Since E is injective, an exact sequence 0 −→ N −→ M −→ MN −→ 0 gives rise to an exact sequence 0 ←− Nb ←− Mc ←− M\N ←− 0,

and M\N is the annihilator Ann(N) of N in M,c i.e. the set of all z ∈ Mc such that zN = 0. The natural mapping M → Mc is an isomorphism for all finite length R-modules M, and identifies N with Ann(Ann(N)). Hence we conclude: Proposition 3.1.4. The map N ↔ Ann(N) is a one-one correspondence between the submodules of M and M,c respectively. Moreover, this correspondence maps the set of all submodules N of M of type ν and cotype µ onto the set of all submodules Ann(N) of Mc of type µ and cotype ν. Next we define the Hall algebra of R. Fix M be a R-module of type λ. Let µ(1), . . . , µ(s) be partitions. Define

λ hµ(1)...µ(s) to be the number of chains of submodules of M

M = M0 ⊃ M1 ⊃ · · · ⊃ Ms = 0

(i) such that Mi−1 Mi has type µ , for 1 ≤ i ≤ s. In particular, for µ, ν partitions, λ hµν is the number of submodules N of M which has type ν and cotype µ. Since l(M) = l(M N) + l(N), we obtain the following lemma.

61 λ Lemma 3.1.5. hµν = 0 unless |λ| = |µ| + |ν|.

λ The Hall’s idea was to use the numbers hµν as the multiplication constants of a ring, which in deﬁned as follows. Let HR be the free Z-module with basis (Iλ) indexed by all partitions λ. Deﬁne a product in HR by the rule

X λ IµIν := hµνIλ. λ

By Lemma 3.1.5, the above sum on the right has only ﬁnitely many non-zero terms.

Theorem 3.1.6 (Steinitz, Hall, Macdonald). The following hold: (i) HR is a commutative and associative ring with an identity element. (ii) HR is generated (as a Z-algebra) by the elements I(1s) for all s ≥ 1, and they are algebraically independent over Z.

Proof. The identity element is I0. Associativity follows from the fact that the coef- λ ﬁcient of Iλ in both Iµ(IνIρ) and (Iµ(Iν)Iρ is just hµνρ. Commutativity follows from λ λ Proposition 3.1.4, which shows that hµν = hνµ. This proves part (i). For (ii) we proceed as follows. For any partition λ, consider the product

Π 0 := I 0 ··· I 0 λ (1λ1 ) (1λs )

0 0 0 where λ = (λ1, . . . , λs) is the conjugate partition of λ. The product Πλ0 will be a linear combination of the Iµ, say X Πλ0 = aλµIµ µ in which the coeﬃcient aλµ is by deﬁnition equal to the number of chains

M = M0 ⊃ M1 ⊃ · · · ⊃ Ms = 0 (3.1.1)

λ0 in a ﬁxed ﬁnite R-module M of type µ such that Mi−1 Mi is of type (1 i ), i.e. 0 elementary of length λi, for 1 < i < s. If such a chain 3.1.1 exists, that is, aλµ 6= 0, i then we must have mMi−1 ⊂ Mi and therefore m M ⊂ Mi for 1 < i < s. Hence

i l(M m M) ≥ l(M Mi), which by virtue of Lemma 3.1.2 gives the inequality

0 0 0 0 µ1 + ··· + µi ≥ λ1 + ··· + λi

62 for 1 ≤ i ≤ s. Hence µ0 ≥ λ0 and therefore µ ≤ λ by Lemma 3.1.1. Moreover, the same reasoning shows that if µ = λ, there is only one possible choice for the chain i 3.1.1, namely Mi = m M. Consequently we have aλµ = 0 unless µ ≤ λ, and aλλ = 1. In other words, the matrix (aλµ) is strictly upper unitriangular (see [33, Chapter I (1.11)]), and thus our ﬁrst equation X Πλ0 = aλµIµ µ can be solved by expressing the Iµ as integral linear combinations of the Πλ0 . Hence the Πλ0 form a Z-basis of HR, which proves part (ii).

λ Deﬁnition 3.1.7. The ring HR is the Hall algebra of R. The numbers hµ(1)...µ(s) are called Hall numbers.

Next we state the existence of the Hall polynomials. These polynomials are universal in the sense that they are independent of the choice of R, i.e. the choice of the (ﬁnite) ground ﬁeld k.

Theorem 3.1.8 (Hall, Macdonald, Schiﬀmann). For any triple of partitions λ, µ, ν λ satisfying |ν| + |µ| = |λ| there exists a unique polynomial hµν(t) ∈ Z[t] such that for any prime power q and any discrete valuation ring R whose residual ﬁeld has q λ λ elements, we have hµν = hµν(q). Proof. For a classical proof see [33, Chap. II, Section 4 or AZ.1]. See [44, Prop. 2.7] for an independent proof.

Macdonald’s book [33, Chap. II] contains a version of this theorem that includes λ as estimation for the degree of hµν(t) and its leading coefficient. The above theorem allows us to consider a universal version H of the Hall algebra ±1 HR, which is defined over the ring C[t ]. This universal Hall algebra is defined to be the algebra M ±1 H := C[t ]Iλ λ where λ runs trough the set of partitions and the multiplication is defined by

X λ IµIν = hµν(t)Iλ. λ

As in the classical version, the universal Hall algebra H is commutative and is a free ±1 polynomial ring over C[t ] in the generators (I(1r))r≥1.

63 Link with the Macdonald’s ring of symmetric functions. By the previous discussion, the classical Hall algebra HR or its universal version H, are N-graded polynomial rings with one generator in every degree. There is another such ring, which plays an important role in many branches of mathematics, and particularly in com- binatorics and representation theory: the Macdonald’s ring of symmetric functions. These symmetric function come into the picture in the following way. Consider for each n ≥ 1 the ring of symmetric polynomials in n independent variables. It is known that this ring is a polynomial ring Λn := Z[e1, . . . , en] generated by the elementary symmetric functions X ej := xi1 ··· xij

i1<···

3.2 The Hall algebras of a ﬁnitary category

As remarked earlier, the Hall algebra of a finitary category was formalized by Ringel in the early of 1990s. Ringel’s work laid the foundation for new links between Hall’s work and algebraic geometry, number theory and quantum algebras. Henceforth, major steps in these directions have been taken by Xiao [51], Kapranov [26], Schiffmann [43] and others. In this section, we introduce the notion of the Hall algebra of a finitary category in as much generality as possible.

64 Definition 3.2.1. A small abelian category A is called finitary if the following two conditions are satisfied for any two objects M,N ∈ Ob(A):

(i) #Hom(M,N) < ∞ (Hom-ﬁnite);

(ii) #Ext1(M,N) < ∞.

In the most important example of finitary category that we are interested in this thesis, A is linear over some finite field Fq, this yields

1 dimFq Hom(M,N) < ∞ dimFq Ext (M,N) < ∞ for any pair of objects M,N ∈ A.

Example 3.2.2. Simple examples of such categories are provided by the categories Rep Q~ Fq of finite-dimensional Fq-representations of a quiver, or more generally by the categories RepA of finite-dimensional representations of a finite-dimensional Fq- algebra A. The example that we are concerned with in this thesis are the categories CohX of coherent sheaves on some projective scheme of finite type over Fq (in this case, the finiteness properties of the definition of finitary categories holds by the famous Theorem 3.3.1 due to Serre).

We denote by K(A) the Grothendieck group of an abelian category A. If A is a finite length category i.e. any object of A has a finite composition sequence with simple factors, then K(A) is freely generated by the classes of the simple objects. We turn to the definition of the Hall algebra of a finitary category. Let A be a finitary category. Let us make the additional assumptions that its global dimension gldimA is finite and that Exti(M,N) < ∞ for any two objects M,N ∈ Ob(A) and for all i (here we implicitly assume that the groups Exti are well-defined). For any two objects M,N ∈ ObA we define

1 ∞ ! 2 i Y i (−1) hM,Nim := #Ext (M,N) . i=0

Since gldimA < ∞, Exti(M,N) = 0 for i 0 and the above product is ﬁnite. Using the long exact sequences in homology associated with the functor Hom, we can show that hM,Nim does not depend on a choice of a square root and only depends on the classes of M and N in the Grothendieck group. Hence we may deﬁne a form

h , im : K(A) × K(A) −→ C which is called the multiplicative Euler form.

65 Let k be a ﬁeld. When A is k-linear, then we usually consider the additive version of the Euler form instead, which is deﬁned by

X i i hM,Ni := (−1) dimk Ext (M,N). i Hence we have associated with any finitary k-linear category a lattice (K(A, h , i), i.e. a Z-module equipped with a Z-valued symmetric bilinear form. Let Iso(A) = Ob(A) ∼ be the set of isomorphism classes of objects of A. We thus define HA the Hall algebra of A to be the vector space M HA := C[M], M∈Iso(A) linearly spanned by symbols [M], where M runs through Iso(A). We define the mul- R tiplication in HA as follows. Given any three objects M,N,R of A, let SM,N denote the set of short exact sequences 0 −→ N −→ R −→ M −→ 0.

R Observe that SM,N is a finite set by definition of finitary category 3.2.1. Denote by aM R R the cardinality #Aut(M) for any object M of A, and consider hMN := #SM,N aM aN . We define the product in HA by the rule

X R [M][N] := hM,Nim hMN [R]. (3.2.1) R

Proposition 3.2.3 (Ringel [38]). The above product defines on HA the structure of an associative algebra. The unit is given by [0] where 0 is the zero object of A. We will proof of the above proposition after reinterpreting the definition of the Hall algebras. In this reinterpretation, HA is viewed as the set of finitely supported functions on Iso(A),

HA = {f : Iso(A) → C | supp(f) is ﬁnite } by identifying the symbol [M] with the characteristic function 1M . The product is rewritten as X (f g)(R) := hR/Q, Qimf(R/Q)g(Q). (3.2.2) Q⊆R Indeed, by bilinearity it is enough to check that the two products coincide when f = 1M and g = 1N . This is a consequence of the following lemma. Lemma 3.2.4. For any three objects M,N,R of A we have R ∼ ∼ hMN = #{L ⊂ R|L = N and R/L = M}. (3.2.3)

66 R Proof. The group Aut(M) × Aut(N) acts freely on SM,N , and the quotient is canonically identiﬁed with the right side of 3.2.3.

Remark 3.2.5. In the case that A is Fq-linear, we consider the following deﬁnitions for the multiplication in HA

−hM,Ni X R [M][N] := v hMN [R] R and X (f g)(R) := v−hR/Q,Qif(R/Q)g(Q), Q⊆R where v is a square root of q−1. Proof of Proposition 3.2.3. We will use the definition 3.2.2 instead 3.2.1 for the proof. Assume that f = 1M and g = 1N . Then the right side of 3.2.2 is a function supported on the set Ext1(N,M). Since Ext1(N,M) is finite, there are only finitely many such extensions, hence 1M 1N does indeed belong to HA. By bilinearity it follows that fg ∈ HA for any f, g ∈ HA. The associativity is as follows. Let f, g, h ∈ HA and M ∈ ObA, thus X f(gh) (M) = hM/N, Nimf(M/N)(gh)(N) N⊂M X = hM/N, NimhN/L, Limf(M/N)g(N/L)h(L) N⊂M,L⊂N X = hM/N, LimhM/N, N/LimhN/L, Limf(M/N)g(N/L)h(L) L⊂N⊂M by the bilinearity of the multiplicative Euler form. On the other hand, X (fg)h (M) = hM/R, Rim(fg)(M/R)h(R) R⊂M X = hM/R, Rimh(M/R)/S, Simf((M/R)/S)g(S)h(R). R⊂M,S⊂M/R Next observe that there is a bijection S 7→ S0 between the sets {S | S ⊆ M/R} and {S0 | R ⊆ S0 ⊆ M} satisfying S ∼= S0/R and (M/R)/S ∼= M/S0. From that we conclude

X 0 0 0 0 0 0 (fg)h (M) = hM/S ,RimhS /R, RimhM/S ,S /Rimf(M/S )g(S /R)h(R), R⊂S0⊂M which is none other than f(gh) (M). The statement concerning the unit is obvious.

67 Remark 3.2.6. Let us state some interesting observations concerning the Hall algebra HA.

(i) By definition, the product [M][N] is a linear combination of elements [R], where R is an extension of M by N. Hence HA encodes the structure of the set of short exact sequences in A. It is possible to define the Hall algebra for any exact category with the finiteness conditions of Definition 3.2.1, see [23].

(ii) The Hall algebra HA is naturally graded by the Grothendieck group K(A) of A, M M HA = HA[α] where HA[α] := C[M] α∈K(A) M=α where M denotes the class of M in the Grothendieck group K(A).

(iii) Assume that A is a semisimple category and let S = {Si}i∈I be the set of simple objects. If i 6= j, then

[Si][Sj] = [Si ⊕ Sj] = [Sj][Si]

1/2 [Si][Si] = (#End(Si)) (#End(Si) + 1)[Si ⊕ Si]

and in fact HA is a free commutative polynomial algebra in the generators {[Si]}i∈I .

(iv) In the case of a semisimple category A, as above, the algebra HA is commutative. However, this is a rather rare phenomenon: in general the set of extensions of M by N diﬀers of the set of extensions of N by M, in which case HA is not commutative.

(v) As in the classical case (see section 3.1), the product of more than two elements also has an interpretation in terms of number of ﬁltrations:

X Y ∼ [M1] ··· [Mr] = hMi,Mjim#{Lr ⊂ · · · ⊂ L1 = R | Li/Li+1 = Mi} [R] R i

where M1,...,Mr are objects in A.

R Deﬁnition 3.2.7. The integers hMN , used in the deﬁnition of the multiplication in HA, are called Hall numbers.

We summarize some properties of the Hall numbers in the following proposition.

Proposition 3.2.8. Let A be a Fq-linear ﬁnitary category with gldim(A) = 1. If M,N,P,Q ∈ Iso(A), then the following holds.

68 R (i) There are only ﬁnitely many isomorphism classes R ∈ Iso(A) such that hMN 6= 0.

(ii) If each object of A has a finite filtration with simple sub-quotients (Jordan- 2 M Hölder series), there exists only finitely pairs (R,S) ∈ Iso(A) such that hRS 6= 0.

N N (iii) hM0 = δMN and h0M = δNM (Kronecker symbols).

P R Q P Q R (iv) R∈Iso(A) hMN hRP = R∈Iso(A) hMRhNP .

dim Hom(M,N) P (v) q hMN aM aN /aP is an integer.

P R −hM,Ni (vi) R∈Iso(A) hMN aM aN aR = q . (vii) If M and N are indecomposable objects and P is a decomposable object, then P P q − 1 divides hMN − hNM . Proof. Part (i) holds because Ext1(−, −) is a finite set. With the assumptions, (ii) follows from the fact that the map Iso(A) → K(A) has finite fibers. By Proposition 3.2.3 the zero object 0 of A is the unity of HA, which implies (iii). Part (iv) is [37, Proposition 1]. The assertions (v) and (vi) are consequences of Proposition 1.3.4 in [41] (see also our classical reference [33] pag. 221). Finally, (vii) follows from the proof of [39, Proposition 1].

3.3 The Hall algebra of a curve

Next we turn our attention to the particular case of the category Coh(X) of coherent sheaves on smooth projective curve X over a finite field Fq. The category Coh(X) is never of finite type, i.e. there always are infinitely many indecomposable objects. Motived by some problems in number theory (e.g. Langlands program), Kapranov studied in his paper [26] the Hall algebra HX := HCoh(X). After Kapranov, many others authors have worked on the theory of Hall algebra of a curve. In the end of 1990s, Baumann and Kassel have published a paper [7] concerning the Hall algebra of the projective line. In the last years, Schiffmann has published several works concerning the Hall algebras of elliptic curves and curves of higher genus, cf. [10] (joint with Burban), [45] and [27] (joint with Kapranov and Vasserot).

n Theorem 3.3.1 (Serre). Let k be a ﬁeld. Let X ⊆ Pk be a projective variety. The category Coh(X) is a Hom-ﬁnite, abelian and noetherian category. Moreover, if X is smooth, then

i dimk Ext (−, −) < ∞ for every i ≥ 0 and gldimCoh(X) = dim X.

69 Proof. See [16, Corollary 3.2.3] or the original paper [46].

According to Theorem 3.3.1, Coh(X) is a finitary hereditary category if X is a smooth projective curve defined over a finite field k = Fq. Moreover, Coh(X) is Krull- Schmidt since it is Hom-finite and abelian. Therefore we can define the Hall algebra for Coh(X). Fix a smooth projective curve X of genus g over Fq. Since Coh(X) is Fq-linear, we will use the additive version of the Euler form to define the structure constants of the multiplication. The Riemann-Roch theorem yields

hF, Gi = (1 − g) rk(F) rk(G) + rk(F) deg(G) − rk(G) deg(F) (3.3.1)

We thus deﬁne the Hall algebra HX of X to be the Hall algebra of the category −1 Coh(X). Fix a square root v of q . Following 3.2.1, HX is the C−vector space M HX := C[F] [F] where [F] runs through the isomorphism classes of objects in Coh(X). The multiplication in HX is given by

−hF,Gi X H [F][G] := v hF,G[H]. [H]

H where hF,G are the Hall numbers deﬁned in the previous section. The Hall algebra HX has a natural grading in the subset

0 Z := (n, d) ∈ K0(X) n > 0 or n = 0 and d ≥ 0

0 L of the numerical Grothendieck group K0(X), given by HX [v] := F=v C[F], for v ∈ Z. For v = (n, d) ∈ Z, γ(v) denotes the gcd(n, d). tor L We will denote by HX := d≥0 HX [0, d] the subalgebra of torsion sheaves and vec L vec by HX := (n,d)∈Z,n>0 HX [n, d] the subalgebra of vector bundles. We denote by π vec the projection map HX → HX .

Remark 3.3.2. Observe that the categories Tor(X) and Torx also satisfy the conditions of Deﬁnition 3.2.1 and we may deﬁne HTor(X) and HTorx for x a closed point in X. In the last case, we can identify the Hall algebra HTorx with the classical Hall algebra and thus with the Macdonald’s ring of symmetric functions. We return to this discussion in chapter 4.

We end this section with some easy observations.

Proposition 3.3.3. The following holds in the Hall algebra HX .

70 (i) Let T be a torsion and E be a vector bundle on X. Then [E][T ] = [E ⊕ T ] in HX .

0 (ii) Let F ∈ Cµ and G ∈ Cµ0 . If µ > µ + deg(ωX ), then [G][F] = [F ⊕ G].

(iii) Let T1, T2 be torsion sheaves on X. Suppose that the set of support of T1 is disjoint of the set of support of T2, then [T1][T2] = [T1 ⊕ T2] in HX .

Proof. Item (i) follows from 1.4.2. By Lemma 1.4.5 we have (ii). Assertion (iii) is a consequence of the structure theorem of ﬁnitely generated torsion modules over a principal ideal domain.

In [7, Theorem 13], the authors exhibit several formulas for the Hall numbers in the Hall algebra HP1 of the projective line over Fq. Further, they follow Kapranov and prove some structure results for the Hall algebra of the projective line. For the Hall algebra of an elliptic curve, the main reference for this thesis are the works due to Schiﬀmann and others (cf. [10] and [45]).

71 Chapter 4

Graphs of Hecke operators for elliptic curves

Our goal in this chapter is to describe the graphs of Hecke operators for an elliptic function ﬁeld. For that we shall consider the geometric point of view on graphs of Hecke operators as introduced in 2.3. The main idea here is to bring up a relation between graphs of Hecke operators and Hall algebras and use it to develop an algorithm that allows us to calculate these graphs for an elliptic curve. We start this chapter by a introductory section where we recall (and improve) some facts and notation of the previous chapters and introduce new facts and notations that will be necessary for the development of the chapter.

4.1 Introduction

Notation. Throughout this chapter, X denotes an elliptic curve defined over Fq; that is, X is a smooth and geometrically irreducible projective curve of genus one together with the choice of a rational point. Note that, by Hasse-Weil’s√ inequality (we refer to [48] for a proof), we have |#X(Fq) − (q + 1)| ≤ 2 q, hence any smooth projective curve of genus one has such a point, say x0 ∈ X(Fq). We denote #X(Fqd ) by Nd. Let F be the function field of X. Recall that we denote by |X| the set of closed points of X or, equivalently, the set of places of F . For x ∈ |X|, we denote by Fx the completion of F at x, by Ox its integers, by πx ∈ Ox a uniformizer (we can suppose ∼ πx ∈ F ) and by qx the cardinality of the residue field κ(x) := Ox/(πx) = Fqx . Let |x| be the degree of x which is defined by the extension field degree [κ(x): Fq]; in other |x| words, qx = q . For an extension of finite fields Fqn of Fq we will denote by Xn the fiber product n X ×Spec(Fq) Spec(Fq ). Using our choice of a rational point x0 ∈ X(Fq), the Riemann-

72 Roch theorem yields a bijective map

1:1 0 X(Fqn ) = Xn(Fqn ) −→ Pic (Xn)

x 7−→ OXn (x − x0)

0 Therefore we can transport the group structure from Pic (Xn) to Xn(Fqn ) such that x0 is the neutral element. Moreover, the inclusions X(Fqm ) ⊆ X(Fqn ) for m|n are compatible with the group structure. To avoid confusion with the addition of divisors, we will denote the sum of x, y ∈ X(Fqn ) with respect to the group law by x ⊕ y ∈ X(Fqn ). Let Y be a scheme over Fq. We call the morphism FrY : Y → Y induced by the q sheaf homomorphism OY → OY : a 7→ a the Frobenius endomorphism of Y. This morphism is the identity in the underlying topological spaces and the raising at the q-th power at level of functions ([28] 3.2.4). The Frobenius endomorphism acts on the set X(Fqn ) of Fqn -rational points of X by composition FrX ◦ x : Spec(Fqn ) → X, where x ∈ X(Fqn ). Moreover, this action is compatible with the group structure of X(Fqn ). If m|n are positive integers, Frm then we have an obvious identification X(Fqm ) = X(Fqn ) X . In particular X(Fqn ) = Frn ∼ X(Fq) X . We have an identification of sets |X| = X(Fq) FrX where the quotient means that two Fq-rational points of X are identified if they have the same orbit under ∼ n the Frobenius action. Similarly, we have |Xn| = X(Fq) FrX . These two equalities allow us to define an action of FrX on the closed points of Xn. We denote this action by frobn : |Xn| → |Xn|. In particular |Xn| frobn = |X|. For a finite abelian group G we denote by Gb its group of characters. From the ∼ 0 isomorphism X(Fqn ) = Pic (Xn) that we fixed before, we obtain the following commutative diagram Fr∗ 0 X,n 0 Pic (Xn) / Pic (Xn) o o FrX X(Fqn ) / X(Fqn ). ∗ \0 The Frobenius FrX,n acts by duality on each group Pic (Xn) and we will denote this 0 action simply by FrX,n. We will denote by Pic^(Xn) (or in some cases only Pn) the 0 quotient Pic\(Xn) FrX,n. Let m, n be positive integers such that m|n. We define the relative norm map

n 0 0 Normm : Pic (Xn) −→ Pic (Xm) by n/m−1 n O mi Normm(L) := (FrX,n) (L) i=0

73 The fact that this map is well deﬁned follows from Galois descent i.e. a line bundle on X that it is isomorphic to its Frobenius conjugates, descends to a line bundle on X. For more details see Proposition 3.4 in [3]. By dualizing we obtain relative norm maps between the character groups for which we will use the same notation n \0 \0 Normm : Pic (Xn) −→ Pic (Xm).

Recall that we ﬁxed a rational point x0 on X and therefore for any integers d 0 n ≥ 1 and d ∈ Z we can identify Pic (Xn) with Pic (Xn) by subtraction dx0. This 0 allows us to extend any character ρ of Pic (Xn) to a character of Pic(Xn) by putting 0 ρ(OXn (x0)) = 1. Unless otherwise speciﬁed we will view the characters of Pic (Xn) as characters of Pic(Xn). ` ^0 Denote Xe := n Pic (Xn). By abuse of language, we call the elements of Xe characters even if they are actually orbits of characters. For a character ρ ∈ Xe, we say 0 that it is a character of degree n if ρ ∈ Pic^(Xn).

Coherent sheaves on elliptic curve As before, Coh(X) denotes the category of coherent sheaves on X. In complement of Section 1.4, let us outline, following Atiyah [5], the classification of coherent sheaves on elliptic curves (in [5] it is assumed that the ground field is algebraically closed, but the proof applies to an arbitrary field). Recall that we denote by rk(F) (resp. deg(F), resp. µ(F)) the rank (resp. degree, resp. slope) of a coherent sheaf F and that a sheaf F is semistable (resp. stable) if for any proper nontrivial subsheaf G ⊆ F we have µ(G) ≤ µ(F) (resp.µ(G) < µ(F)). The full subcategory Cµ of Coh(X) consisting of all semistable sheaves of a fixed slope µ ∈ Q ∪ {∞} is abelian, artinian and closed under extension (see Proposition 1.4.7). Moreover, if F, G are semistable with µ(F ) < µ(G), then Hom(G, F) = Ext(F, G) = 0 (cf. Lemma 1.4.5 ) Any sheaf F possesses a unique filtration (the Harder-Narasimhan filtration, or HN-filtration see Chapter 1 Lemma 1.4.3, or [18]) 0 = F r+1 ⊂ F r ⊂ · · · ⊂ F 1 = F i i+1 for which F F is semistable of slope µi and µ1 < ··· < µr. Moreover, recall that C∞ is the category of torsion sheaves and hence equivalent to the product category Q x Torx where x runs through the set of closed points of X and Torx denotes the category of torsion sheaves supported at x. Theorem 4.1.1 ([5] Theorem 7). The following hold. (i) The HN-filtration of any coherent sheaf splits (noncanonically). In particular, any indecomposable coherent sheaf is semistable. (ii) The set of stable sheaves of slope µ is the class of simple objects of Cµ. (iii) The choice of any rational point x0 ∈ X(Fq), induces an exact equivalence of abelian categories ν,µ : Cµ → Cν, for any µ, ν ∈ Q ∪ {∞}.

74 The assertions (i) and (ii) are stated in Proposition 1.4.6, we just put all together here. The proof of Atiyah also provides an algorithm to compute the equivalences in Theorem 4.1.1 (iii). His proof is for an algebraically closed field (of any characteristic). In [10], Burban and Schiffmann prove these equivalences for the finite field case. We briefly recall their treatment; for complete details see [10], section 1 (paragraph 1.2) and Appendix A. The Grothendieck group K0(X) of Coh(X) is isomorphic to Z ⊕ Pic(X) and the isomorphism is given by F 7→ rk(F), det(F) . Moreover, if we compose this morphism with the one sending a line bundle to its degree, then we get a group homomorphism 0 2 ϕ : K0(X) −→ K0(X) := Z F 7−→ rk(F), deg(F)

0 The group K0(X) is called the numerical Grothendieck group and for a sheaf F, we denote by F its image in this group. The Euler bilinear form h, i : K0(X)⊗K0(X) → Z is deﬁned by the formula F ⊗ G 7→ dim Hom(F, G) − dim Ext(F, G). By the Riemann-Roch theorem, one has hF, Gi = rk(F) deg(G) − rk(G) deg(F), in particular the Euler form is skew-symmetric. The kernel of ϕ is the radical of the Euler form, which is given by

radh−, −i := {F ∈ K0(X)| rk(F) = deg(F) = 0}. Hence we obtain an isomorphism ∼ 2 ϕ = (rk, deg) : K0(X) radh−, −i −→ Z , 0 and the Euler form is well deﬁned on classes in K0(X). The group of exact auto-equivalences Aut(Db(Coh(X))) of the bounded derived category of coherent sheaves on X acts on K0(X) by automorphism, since K0(X) is an invariant of the derived category. This action preserves the Euler form and its radical. Hence, we get a group homomorphism b π : Aut(D (Coh(X))) −→ SL2(Z) which sends an auto-equivalence F to the upper map in the following commutative diagram π(F) 2 / 2 . Z ZO =∼ ϕ−1 =∼ ϕ =∼ K0(X) radh−, −i / K0(X) radh−, −i F∗

75 The map π(F) belongs to SL2(Z) since the auto-equivalence F respects the Euler form and hence the orientation of Z2. Let F be indecomposable in Coh(X) with F = ϕ(F) = (n, d) ∈ Z2. Let g := gcd(n, d) be the greatest common divisor. If d 6= 0, then there is a matrix f ∈ SL2(Z) 0 −1 with f(n, d) = (0, g). If d = 0, then let f be the matrix 1 0 in SL2(Z) which b ﬂips the coordinate axes. Since π : Aut(D (Coh(X))) −→ SL2(Z) is surjective, we can lift f to an auto-equivalence F ∈ Aut(Db(Coh(X))). Since F is an equivalence of categories the image F(F) is indecomposable again. The object F(F) is then a coherent sheaf of rank 0. Therefore, T = F(F) is a torsion sheaf and F = F−1(T ). The above discussion shows us how Atiyah’s theorem works for the classiﬁcation ∼ of indecomposable vector bundles on X, and more generally the equivalences Cµ = Cν, for any µ, ν ∈ Q ∪ {∞}.

Reminder on graphs of Hecke operators We recall the geometric definition of the graphs Gx,r of an unramified Hecke operator as introduced in Section 2.3. These graphs encode the action of the unramified Hecke operators Φx,r on the space of automorphic forms for GLn over a global function field. We say that two exact sequences of sheaves

0 0 0 −→ F1 −→ F −→ F2 −→ 0 and 0 −→ F1 −→ F −→ F2 −→ 0

Deﬁnition 4.1.2. Let x ∈ |X|. For a vector bundle E ∈ BunnX we deﬁne

0 0 Vx,r(E) := {(E, E , m)|m = mx,r(E, E ) 6= 0},

0 0 0 and we call E a Φx,r-neighbour of E if mx,r(E, E ) 6= 0, and mx,r(E, E ) its multiplicity.

76 We deﬁne the graph Gx,r by a VertGx,r = BunnX and EdgeGx,r = Vx,r(E).

E∈BunnX

If Vx,r(E) = {(E, E1, m1),..., (E, Er, mr)}, we make the following drawing conventions to illustrate the graph Gx,r: vertices are represented by labelled dots, and an edge (E, E 0, m) together with its origin E and its terminus E 0 is drawn as m E E 0

The Φx,r-neighbourhood of E is thus illustrated as

m1 E1

4.2 Hall algebras and graphs of Hecke operators

Let Y be a smooth projective and geometric irreducible curve over a finite field Fq and HY its Hall algebra. See Section 3.3 for the definition of the Hall algebra of a curve. The following lemma is the main connection between the theory of graphs of Hecke operators and the theory of Hall algebras of a smooth and projective curve.

0 Lemma 4.2.1. Let Y be a smooth projective curve over Fq. If E, E ∈ BunnY, then

E 0 h r 0 = m (E, E ) Kx,E x,r

00 ⊕r 00 ∼ 0 Proof. Let A := {(0 −→ E −→ E −→ Kx −→ 0) E = E } be the set of repre- 0 sentatives classes of short exact sequences as in 4.1. Note that #A = mx,r(E, E ). Deﬁne 00 00 ∼ 0 00 ∼ r B := E ⊆ E E = E and E/E = Kx . 0 r Since Aut(E ) × Aut(Kx) acts freely on set of exact sequences

0 r 0 −→ E −→ E −→ Kx −→ 0

77 E h r 0 = #B. A → B we have Kx,E We claim that the map given by 00 r 00 0 −→ E −→ E −→ Kx −→ 0 7→ E

00 r 000 is a bijection. Indeed, if 0 −→ E −→ E −→ Kx −→ 0 and 0 −→ E −→ E −→ r 00 ∼ 0 ∼ 000 Kx −→ 0 have the same image via the above map, then E = E = E and

0 / E 00 / E < =∼ 0 / E 000 commutes. This implies that these two short exact sequences are the same element in A, thus we have a injection. The surjectivity is trivial. Therefore #A = #B. vec vec As Section 3.3, we denote by π the projection map HX → HX . Remark 4.2.2. It follows from the deﬁnition of the Hall algebra, the above lemma and the previous paragraph that the graph of the Hecke operator Gx,r is completely de- −nr|x| vec r −nr|x| vec r termined by v π (Kx E) where E runs through BunnY. The product v π (Kx E) in the Hall algebra give us all the edges in Gx,r that arrive in the vertex given by E. Example 4.2.3. Y = 1 The following is an easy example for PFq that exempliﬁes the connection between the Hall algebra for Y and the graph of a certain Hecke operator. In Theorem 13 [7] Baumann and Kassel prove that r r−1 r r Kx OP1 (d) = OP1 (d + |x|) ⊕ Kx + qx OP1 (d) ⊕ Kx , which gives us 1

OP1 (d + |x|) OP1 (d) for r = 1. Compare this with section 2.5.

Lemma 4.2.4. Let x ∈ |Y |. If E ∈ BunnY , then

vec r vec r π (Kx E) = π [Kx, E], where the commutator is taken in the Hall algebra HY .

r Proof. This follows from the fact that none of the elements in Ext(E, Kx) belongs to vec HY . Lemma 4.2.5 ([12], Lemma 2.10). For r = 1 in the previous lemma,

vec π (Kx E) = [Kx, E], where the commutator is taken in the Hall algebra HY .

78 We conclude this section with two applications of our results in chapter 2 to HY . The ﬁrst one is the following corollary and the second one is stated in the next remark. n n−1 n−2 r+1 X E (qx − 1)(qx − 1)(qx − 1) ··· (qx − 1) Corollary 4.2.6. h r 0 = Kx,E n−r n−r−1 n−r−2 0 (qx − 1)(qx − 1)(qx − 1) ··· (qx − 1) E ∈BunnX Proof. This follows from Theorem 2.2.7 where we prove that for a ﬁx x ∈ |X|

X 0 mx,r(E, E ) = #Gr(n − r, n)(κ(x)), 0 E ∈BunnY and the Lemma 4.2.1.

vec ⊕r Remark 4.2.7. The products π (Kx E) are explicitly determined for every E ∈ Bun 1 , 1 ≤ r ≤ n x 1 . n, nPFq and a closed point in PFq Namely, ﬁxed an positive integer in the chapter 2 we develop an algorithm to describe Gx,r for every x closed point in 1 1 ≤ r ≤ n. πvec(K⊕rE) PFq and By Remark 4.2.2 we are able to determine x .

4.3 Graphs of Hecke operators for elliptic curves

We return our attention to an elliptic curve X deﬁned over Fq. As we have seen, vec r describing the graph Gx,r for X is equivalent to calculating the products π (Kx E) in the Hall algebra of X (Remark 4.2.2) where E runs through BunnX. The strategy is to write these products in terms of elements in the twisted spherical Hall algebras and to use the structure of these algebras to compute the products. We shall need the usual notions of ν-integers. If ν 6= ±1, then we set νs − ν−s [s] := . ν ν − ν−1

2 −1 We shall mostly use [s] := [s]v where v = q . Let us recall some properties of the classical Hall algebra, i.e. the Hall algebra of finite modules over a discrete valuation ring (cf. Section 3.1 ). For a deeper discussion of classical Hall algebras, we refer, as before, to [33], chapters I, II and III. Recall that a partition is any finite sequence λ = (λ1, . . . , λm) of non-negative integers in decreasing order: λ1 ≥ · · · ≥ λm. The non-zero λi are called the parts of λ. The number of parts is the length of λ, denote by l(λ); and the sum of the Pm d1 d2 dm parts i=1 λi is the weight of λ, denote by |λ|. We denote by λ = (1 2 . . . m ) the partition that has exactly di parts equal to i. −1 For a finite field k, we denote by u a square root of (#k) . Denote by Ak the category of finite modules over the discrete valuation ring R := k[[t]]. For every s ∈ N there exists a unique indecomposable module of length s (up to isomorphism), s denoted by I(s), which is the quotient R t R. For a partition λ = (λ1, . . . , λm), we

79 write by Iλ := I(λ1) ⊕ · · · ⊕ I(λm). The collection {Iλ}λ where λ runs over all partitions is a complete collection of representatives for the isomorphism classes of objects of Ak. Let us denote by Λ := Z[e1, e2,...] the Macdonald’s ring of symmetric function, where X er = xi1 ··· xir

i1<···

±1 Λt = C[t ] ⊗ Z[e1, e2,...].

Furthermore, let us denote by pr the power-sum symmetric function given by

X r pr = xi .

For a partition λ = (λ1, . . . , λm), we denote by Pλ the Hall-Littlewood symmetric functions (see [33] III.2 for the deﬁnition). We write pλ := pλ1 ··· pλm and eλ := eλ1 ··· eλm . m(m−1) Proposition 4.3.1 ([33]). The assignment I(1m) 7→ u em extends to an algebra 2 isomorphism Ψk : HAk → Λt|t=u . Moreover, −1 P Ql −2i (i) Ψk (pm) = |λ|=m nu(l(λ) − 1)Iλ where nu(l) = i=1(1 − u ), and

2n(λ) P (ii) Ψk(I(λ)) = u Pλ where n(λ) = (i − 1)λi. Here, l(λ) is the length of λ and |λ| its weight.

Proof. The proof of the isomorphism and statement (ii) can be found in [33] III.3 (3.4). The statement (i) is [33], III.7 Ex. 2.

Let x be a closed point of X. Consider the category Torx of torsion sheaves on X ∼ supported at x. We have an equivalence of categories Torx = Aκ(x), (see Section 1.4) which provides us with an isomorphism of algebras Ψκ(x) : HTorx → Λt|t=v2|x| , where v2 = q−1. For m ∈ N and x ∈ |X|, we deﬁne the elements T(0,m),x ∈ HX by ( 0 if |x| 6 | m T(0,m),x := [m] −1 |x|Ψ (p m ) |x| | m. m κ(x) |x| if

Using Proposition 4.3.1, we can rewrite T(0,m),x explicitly as follows

[m]|x| X T = n (l(λ) − 1)K(λ), (0,m),x m ux x |λ|=m/|x|

80 for |x| | m. (µ) For any µ ∈ Q ∪ {∞}, we consider the subspace HX ⊂ HX that is spanned (µ) by classes {[F]|F ∈ Cµ}. Since the category Cµ is stable under extensions, HX is a subalgebra of HX . The exact equivalence µ,ν from Theorem 4.1.1 gives rise to an (ν) ∼ (µ) algebra isomorphism µ,ν : HX → HX . µ ≤ µ N~ H(µ) For 1 2, let µ1≤µ≤µ2 X stand for the (restricted) tensor product of spaces (µ) HX with µ1 ≤ µ ≤ µ2, ordered from left to right in increasing order, that is, for (νi) the vector spanned by elements of the form aν1 ⊗ · · · ⊗ aνr with aνi ∈ HX and µ1 ≤ ν1 < ··· < νr ≤ µ2. Let C[µ1, µ2] be the full subcategory of sheaves whose HN-decomposition contains only slopes µ ∈ [µ1, µ2]. This category is exact and, in particular, stable under extensions.

Lemma 4.3.2 ([10] Lemma 2.5). For any µ1 ≤ µ2 the Hall algebra of the exact category C[µ , µ ] is a subalgebra of H isomorphic to N~ H(µ) (via the multi- 1 2 X µ1≤µ≤µ2 X plication map).

(∞) Note that T(0,m),x ∈ HX . For every v ∈ Z with slope µ, we define the element Tv,x by µ,∞(T(0,m),x). Namely, there is f ∈ SL2(Z) such that f(0, m) = v and thus Tv,x := f · T(0,m),x (cf. paragraph 4.1). With the definitions of paragraph of Notations (Section 4.1) in mind, we define:

^0 Deﬁnition 4.3.3. For a character ρe ∈ Pic (Xn) and a closed point x ∈ X, we deﬁne

n−1 d−1 1 X 1 X ρ(x) := ρ((Fr )i(O (x0))) = ρ((Fr )i(O (x0))) e n X,n Xn d X,n Xn i=0 i=0

0 where x ∈ Xn is a closed point that sits above x and d = gcd(|x|, n).

Deﬁnition 4.3.4. For a character ρe ∈ Xe and for a point v ∈ Z we deﬁne the element X ρe Tv := ρe(x)Tv,x. x∈|X|

Remark 4.3.5. Since the above deﬁnition is an average over all possible choices for 0 the representative of ρe and the point x , it is well deﬁned. Proposition 4.3.6. Let v be a point in Z with d := γ(v). If y ∈ |X| is such that |y| | d, then X −1 ρe Tv,y = |y|Nd ρe(−y)Tv . ^0 ρe∈Pic (Xd)

81 Proof. Recall that for a ﬁnite abelian group G, ( X 0 if g 6= 1G ρ(g) = #G if g = 1G. ρ∈Gb

By deﬁnition X X ρe Tv = ρe(x)Tv,x = ρe(x)Tv,x. x∈|X| x∈|X| |x| | d

0 Summing up all orbits in Pic^(Xd), we obtain X X X ρe ρe(−y)Tv = ρe(x − y)Tv,x x∈|X| ρ∈Pic^0(X ) ρ∈Pic^0(X ) e d e d |x| | d |x|−1 X X 1 X = ρ((Fr )i(O (x0 − y0 − x )))T |x| X,d Xd 0 v,x x∈|X| i=0 ρ∈Pic^0(X ) |x| | d e d X 1 X = ρ(O (x0 − y0 − x ))T |x| Xd 0 v,x x∈|X| ρ∈Pic\0(X ) |x| | d d −1 = |y| NdTv,y

0 0 0 0 where x , y ∈ Xd with x above x and y above y.

⊕r In the following, we aim to write Kx and E (for E ∈ BunnX) in terms of the ρe elements Tv as the above deﬁnition. The next proposition tell us that this is possible.

ρ 0 Proposition 4.3.7 ([12] Proposition 3.4). The set T e v ∈ Z, ρ ∈ Pic^(Xd), d = v e γ(v) , generates the Hall algebra HX .

0 Deﬁnition 4.3.8. We call a character ρ ∈ Pic\(Xn) primitive of degree n if its orbit under the Frobenius FrX,n is of maximal cardinal, i.e. if it is of cardinal n. Equivalently, 0 ρ is primitive if there does not exist a character χ ∈ Pic\(Xm), m < n, m|n, such n that ρ = Normm(χ) ([12] Lemma 3.8).

0 Remark 4.3.9. Applying the above deﬁnition we see that ρ ∈ Pic\(Xn) is primitive 0 or there is a primitive character χ ∈ Pic\(Xm) with m|n, m < n such that ρ = n Normm(χ) (c.f. [12] Corollary 3.9).

82 Let Pn be the set of primitive characters of degree n modulo the action of Frobe- ` nius and P := n≥1 Pn. Next, we are able to deﬁne the twisted spherical Hall algebras. This subalgebras of HX will play an important role in the algorithm (section 4.4), to calculate the graphs of Hecke operators.

Deﬁnition 4.3.10. Let n ≥ 1 and ρe ∈ Pn be a primitive character. We deﬁne the ρe algebra UX , called the twisted spherical Hall algebra of X and ρ,e as the subalgebra of HX generated by n nγ(v) o Normn (ρe) Tnv v ∈ Z .

^0 For n = 1 and the trivial character ρe = 1 ∈ Pic (X), the above deﬁnition spe- cializes to the spherical Hall algebra UX introduced by Burban and Schiﬀmann in section 4 of [10] and considered implicitly by Kapranov in sections 3.8 and 5 of [26]. We also will need the combinatorial description of the twisted spherical Hall algebra as discovered in section 5 of [10] and generalized in section 3 of [12]. For v, w ∈ Z which are not proportional we denote by v,w := sign(det(v, w)) ∈ {±1} and by ∆v,w the triangle formed by the vectors o, v, v + w where o = (0, 0). We recall Pick’s formula ([36]), which will be useful for us. For any pair of linear independents points v, w ∈ Z2,

2 | det(v, w)| = γ(v) + γ(w) + γ(v + w) − 2 + 2 #(∆v,w ∩ Z ). (4.3.1)

∗ − 1 Deﬁnition 4.3.11. Fix σ, σ ∈ C with σ, σ 6∈ {±1} and set ν := (σσ) 2 and

i/2 −i/2 i/2 −i/2 ci(σ, σ) := (σ − σ )(σ − σ )[i]ν i

n Let Eσ,σ be the C-algebra generated by {tv | v ∈ Z} modulo the following relations:

0 1. If o, v, v are collinear then [tv, tv0 ] = 0.

2. If v, w are such that γ(v) = 1 and that ∆v,w has no interior lattice point, then θ [t , t ] = c v+w w v v,w nγ(w) n(ν−1 − ν)

where the elements θz, z ∈ Z are deﬁned by the following generating series: ! X i −1 X i θiz0 s = exp n(ν − ν) tiz0 s i≥0 i≥1

for any z0 ∈ Z such that γ(z0) = 1.

83 −1 Note that θz = n(ν − ν)tz whenever γ(z) = 1.

n Following section 4 of [10], we give a geometric interpretation for the algebra Eσ,σ. By a path in Z we understand a sequence p = (v1,..., vm) of non-zero elements of Z, which we represent graphically as the polygonal line in Z that joins the points o, v1, v1 + v2,..., v1 + ··· + vm. Let vwc ∈ [0, 2π) denote the angle between the segments ov and ow. We will call a path p = (v1,..., vm) convex if v[1v2 ≤ v[1v3 ≤ 0 · · · ≤ v[1vm < 2π. Put L0 := (0, −1) and let Conv be the collection of all convex paths p = (v1,..., vm) satisfying v[1L0 ≥ · · · ≥ v[mL0. Two convex paths p = 0 (x1,..., xr) and q = (y1,..., ys) in Conv are said to be equivalent if {x1,..., xr} = {y1,..., ys} i.e. p is obtained by permuting several segments of q of the same slope. For example the path p = ((2, −2), (1, 0), (2, 0), (1, 1)) is equivalent to the path q = ((2, −2), (2, 0), (1, 0), (1, 1)).

∼ p q

We denote by Conv the set of equivalence classes of paths in Conv0 and we will call the elements of Conv simply paths. We introduce the positive paths Conv+ and − the negatives paths Conv as the paths p = (v1,..., vm) ∈ Conv such that v[mL0 ≥ π and v[1L0 < π, respectively. By concatenating paths we obtain an identiﬁcation Conv ≡ Conv+ × Conv−.

p1 p p 2 o 3

p4 p1 is not convex − + p2 ∈ Conv , p3 ∈ Conv p4 is convex, but p4 6∈ Conv

84 Fix an integer n ≥ 1. With a path p = (v1,..., vm) ∈ Conv, we associate the n element tp ∈ Eσ,σ deﬁned by

tp := tv1 ··· tvm , n which is a well deﬁned element of Eσ,σ due to relation (1). n L Lemma 4.3.12 ([10] Lemma 5.6). The algebra Eσ,σ is equal to p∈Conv+ Ctp. Next we associate with each coherent sheaf a convex path in Z. By Theorem 4.1.1, the HN-ﬁltration splits, hence for every vector bundle E ∈ BunnX we can write

E = E1 ⊕ · · · ⊕ Em where Ei ∈ Cµi , Ei = (ni, di) and µ1 < ··· < µm. We associate with E the convex path + p(E) = (v1,..., vm) ∈ Conv where vi := (ni, di). Let P(x, r, E) the polygon determined by the region below the convex path given by E and above the path given by ((0, −r|x|), v1,..., vm).

p(E) o

(0, −r|x|) P(x, r, E)

0 0 Theorem 4.3.13. Let E, E ∈ BunnX and x ∈ |X|. Suppose that mx,r(E, E ) 6= 0. Then p(E 0) is contained in the polygon P(x, r, E). Proof. Let E, E 0 be rank n vector bundles on X such that 0 ⊕r 0 −→ E −→ E −→ Kx −→ 0, 0 i.e. mx,r(E, E ) 6= 0. Denote by E = E1 ⊕ · · · ⊕ Es, where Ei := vi and µ(v1) < ··· < 0 0 0 0 0 µ(vs), the HN-decomposition of E. Let E = E1 ⊕ · · · ⊕ Et, where Ei := vi and 0 0 0 µ(v1) < ··· < µ(vt), be HN-decomposition of E . Observe that the statement of the theorem is equivalent to say that p(E) is con- 0 0 0 tained in the polygon delimited by p(E ) and ((0, r|x|), v1,..., vt). 0 0 0 First suppose that E is semistable, i.e. E = E1. Since E must appears in the ⊕r 0 product Kx E in the Hall algebra HX , it follows from Lemma 4.3.2 that p(E) is 0 ∆ 0 o v a convex path in the triangle (v1,(0,r|x|)) whose vertices are the origin , 1 and (0, r|x|). The general case follows by concatenation of paths. Namely, we may write E 0 as P 0 0 i

85 In the particular case of n = 2 and E = L1 ⊕ L2 with deg L2 − deg L1 > |x|, we obtain a complete list of possible neighbours of E in Gx,1, for every x ∈ |X|.

Corollary 4.3.14. Fix x ∈ |X| and n = 2. Let L1, L2 be line bundles on X with deg(L2) > deg(L1) + |x|. Let us denote E = L1 ⊕ L2, then Vx,1(E) = (E, L1(−x) ⊕ L2, m1), (E, L1 ⊕ L2(−x), m2) where m1 = mx,1 E, L1(−x) ⊕ L2 and m2 = mx,1 E, L1 ⊕ L2(−x) .

Proof. By the last theorem there are only decomposable vector bundles on Vx,1(E). All decomposable vector bundles on Vx,1(E) have this form, c.f. Proposition 2.3.5.

Later (Proposition 4.5.2) we will be able to show that m1 = 1 and m2 = qx in the above corollary. This recovers the special case for n = 2 in [31]. Next we state the Theorem 5.2 of [12], which yields the relationship between the ρe k algebras UX and Eσ,σ deﬁned before. Recall that Ni denotes the number of rational −1/2 points of X over Fqi and that v = q . By the Hasse-Weil theorem (see e.g. [19], Appendix C), there exist conjugate algebraic numbers σ, σ, satisfying σσ = q such that i i i Ni = q + 1 − (σ + σ ) for any i ≥ 1. These numbers σ, σ are the eigenvalues of the Frobenius automorphism 1 acting on H (X, Ql). Note that

i v [i]#X( i ) c (σ, σ) = Fq , i i cf. Deﬁnition 4.3.11. For σ, σ ﬁxed as before, we denote ci(σ, σ) simply by ci.

Theorem 4.3.15 ([12] Theorem 5.2). Let σ, σ as in the Hasse-Weil theorem. For any two integers n, m ≥ 1 and any two diﬀerent primitive characters ρe ∈ Pn and σe ∈ Pm, we have

ρe n (i) The twisted spherical Hall algebra UX is isomorphic to Eσ,σ. This isomorphism is given by nγ(v) Normn (ρe) tv 7−→ Tnv .

ρ e σe (ii) The algebras UX and UX commute with each other.

(iii) The Hall algebra HX decomposes into a commutative restricted tensor product

∼ O 0 ρe HX = UX ρe∈P

86 Notation. By Atiyah’s classiﬁcation of vector bundles over X, see Theorem 4.1.1, an irreducible vector bundle E is semistable. Let µ be its slope. Then its image via ∞,µ is an irreducible torsion sheaf on X. Hence E is completely determined by (i) its image E = (n, d) in the numerical Grothendieck group Z2; (ii) a closed point x ∈ |X| which is the support of ∞,µ(E); and (iii) a weight ` that determines the (`) (n,d) unique irreducible torsion sheaf with support in x as Kx . We denote E by E(x,`) .

4.4 The algorithm

In the following, we explain the main theorems that provide the basis for our algo- E rithm to calculate the structure constants h ⊕r 0 . Kx E

Theorem 4.4.1 ([10]). Let F = F1 ⊕ · · · ⊕ Fs be a coherent sheaf on X. If Fi ∈ Cµi for i = 1, . . . , n, and µ1 < ··· < µs, then

P hFi,Fj i [F] = v i

Proof. This theorem follows from Lemma 2.4 and (2.6) in [10].

Theorem 4.4.2. If F ∈ Cµ, then

n n i ρ eijk Y X X X ρeij1 j F = ··· a(ρij1 ,..., ρijk ) Tvij ··· Tvij . e e j 1 kj i=1 j=1 ρij ρij e 1 e kj

0 for some a(ρij ,..., ρij ) ∈ where ρij runs through Pic ^(Xγ(v )) for i = 1, . . . , n e 1 e kj C e k ijk Pn Pkj and j = 1, . . . , ni. Moreover, µ(vijk ) = µ and i=1 k=1 vijk = F. Proof. By Theorem 4.1.1, F corresponds to a torsion sheaf T . Since Tor(X) is the Q direct product of blocks x∈|X| Torx where Torx is the category of torsion sheaves supported at x, we may write

(λ1) (λn) T = Kx1 ⊕ · · · ⊕ Kxn , where λ1, . . . , λn are partitions and xi 6= xj if i 6= j. Since the Ext of two torsion sheaves with disjoint support is trivial, we have

(λ1) (λn) T = Kx1 ···Kxn

87 (λi) in the Hall algebra of X. Via Proposition 4.3.1, Kxi corresponds to the Hall- P (x, q−1). Littlewood symmetric function λi xi From [33], I.2.12, the power-sums gen- [q±1] erate the Macdonald’s ring of symmetric functions (as C xi -algebra). Thus we can write n Xi P (x, q−1) = a p λi xi ij λij j=1 for certain aij ∈ C and partitions λij with |λij| = |λi|. Hence

ni (λi) X Kx = bij T(0,mij ),xi ··· T(0,mij ),xi i 1 kj j=1

Pkj where k=1 mijk = |xi||λi|, for j = 1, . . . , ni. Thus n n Y Xi T = bij T(0,mij ),xi ··· T(0,mij ),xi , 1 kj i=1 j=1 with bij ∈ C. Using Atiyah’s theorem again, we get n n Y Xi F = bij Tvij ,xi ··· Tvij ,xi 1 kj i=1 j=1

Pn Pkj where γ(vijk ) = mijk , µ(vijk ) = µ and i=1 k=1 vijk = F. The desired follows from Proposition 4.3.6. The next result is a particular case of the previous theorem, which will be of later use. Corollary 4.4.3. Let x ∈ |X| and r ≥ 1 an integer. Then

n ρ ⊕r X X X ρeij1 eijk Kx = ··· bij T ··· T (0,mi1 ) (0,mik ) i=1 ρ ρ ei1 eik

0 for some bij ∈ where ρi runs through the elements in Pic^(Xm ) for i = 1, . . . , n. C e k ik ⊕r Proof. Since Kx ∈ C∞, this follows from previous theorem.

Example 4.4.4. Let x be a degree d closed point in X. By deﬁnition T(0,d),x = [d]Kx −1 and thus Kx = [d] T(0,d),x. From Proposition 4.3.6,

−1 X ρe T(0,d),x = dNd ρe(−x)T(0,d). ρe∈Pd Therefore, K = [d]−1dN −1 P ρ(−x)T ρe . x d ρe∈Pd e (0,d)

88 ⊕r Remark 4.4.5. Via Proposition 4.3.1, Kx corresponds to the elementary symmetric functions er and these functions have an explicitly description in terms of power- sums given by Newton’s formula, see [33] I.(2.11’). These formulas are useful for the calculation of our graphs. We state these formulas in the following Step-4 of our algorithm.

Theorem 4.4.6. Let E ∈ BunnX, then

ρ ρ e1jk esjk X ρe1j1 j ρesj1 j E = aijk Tv1j ··· Tv1j ··· Tvsj ··· Tvsj , 1 kj 1 kj ρ eijk

0 for some aij ∈ where ρij runs through the elements in Pic ^(Xγ(v )) and the path k C e k ijk given by v ,..., v ,..., v ,..., v is a convex path which deﬁnes the same 1j1 1jkj sj1 sjkj polygonal line as p(E) for all j = 1, . . . , m.

Proof. Since the Harder-Narasinham ﬁltration splits (c.f. Theorem 4.1.1), we may write E = E1 ⊕ · · · ⊕ Es with Ei ∈ Cµi for i = 1, . . . , s, and µ1 < ··· < µs. By Theorem 4.4.1, P hEi,Ej i E = v i

Theorem 4.4.7. Let x ∈ |X|, r ≥ 1 an integer and E ∈ BunnX, then

m X ρi ρi K⊕r E = a T e 1 ··· T e ` x i vi1 vi` i=1

0 for some ai ∈ where ρi runs through the elements in Pic ^(Xγ(v )), and where C e j ij vij ∈ Z are such that vi1 ,..., vi` deﬁnes a convex path in ∆(rk(E),deg(E)),(0,r|x|) for all i = 1, . . . , n.

Proof. Considering the HN-decomposition E = E1 ⊕ · · · ⊕ Es with Ei ∈ Cµi and µ < ··· < µ , K⊕r, E ∈ C[µ , ∞]. N~ H(µ) 1 s we obtain that x 1 Since µ1≤µ≤∞ X is a subalgebra H , K⊕r E N~ H(µ) of X the product x still belong to µ1≤µ≤∞ X and thus we may write

m0 X K⊕r E = a0 a ··· a x i µi1 µit i=1

(µ ) a0 ∈ a ∈ H ij µ ≤ µ < ··· < µ ≤ ∞ i = 1, . . . , m0. with i C, µij X and 1 i1 it for all Hence

m00 X K⊕r E = a00 E ···E x i µi1 µit i=1

89 a00 ∈ E ∈ C . µ ≤ µ < ··· < µ ≤ ∞ i = 1, . . . , m00 with i C and µij µij Since 1 i1 it for all and E ···E ∈ Ext(K⊕r, E), E ,..., E µi1 µit x it follows from Lemma 4.3.2 that µi1 µit deﬁnes a convex path in ∆(rk(E),deg(E)),(0,r|x|). E E Since each µij is semistable we can use Theorem 4.4.1 to write µij in terms of the generators of the twisted spherical Hall algebras. The theorem follows from rearranging the terms.

The algorithm Fix r, n ≥ 1 integers and x ∈ |X|. Let E ∈ BunnX. As explained in Remark 4.2.2, it suﬃces for the determinate of the graph Gx,r to calculate the vec ⊕r vec ⊕r products π Kx E = π [Kx , E] for every E ∈ BunnX. Our aim is to write the ⊕r product Kx E as in Theorem 4.4.7 and then use the deﬁnition of the elements in the twisted spherical Hall algebras and the structure of the Hall algebra HX to calculate this product explicitly. In the following steps, we outline how to perform such calculations.

Step 1 The Harder-Narashimhan decomposition tells us that every vector bundle E can be written as follows E = E1 ⊕ · · · ⊕ Es where each Ei ∈ Cµi for i = 1, . . . , s, and µ1 < ··· < µs.

Step 2 We write each Ei in terms of elements in the twisted spherical Hall algebras. Namely, via Atiyah’s theorem (4.1.1) Ei corresponds to a torsion sheaf of degree gcd(rk(Ei), deg(Ei)). Such a torsion sheaf is the direct sum of torsions sheaves with disjoint support. Let Tx be a torsion sheaf with support in x. By Proposition 4.3.1, Tx corresponds to the Hall-Littlewood symmetric function Pλ for some partition λ.

Writing Pλ as linear combination of pλi for some partitions λ1, . . . , λm, we are able to write the Ei as sums of products of elements in the twisted spherical Hall algebras as stated in Theorem 4.4.2.

Step 3 By Theorem 4.4.1,

P hEi,Ej i E = v i

By the previous step, we may write E as sums of products of elements in the spherical Hall algebras, cf. Theorem 4.4.6.

90 ⊕r Step 4 By Corollary 4.4.3 the previous step applies also to Kx . By the deﬁnition of ρe Tv , we only have to known how to write the elementary symmetric functions in terms of power-sums (cf. Proposition 4.3.1). These are the well-known Newton formulas:

m X i−1 mem = (−1) piem−i. i=1

⊕r Therefore, we may write Kx as sums of products of elements in the twisted spherical Hall algebras with slope ∞.

⊕r Step 5 Since we know how to write E and Kx as elements in the twisted spherical ⊕r Hall algebras, the product Kx E can be written as sums of products of elements in the twisted spherical Hall algebras. By Corollary 4.4.3 and Theorem 4.4.6, we obtain an expression of the form

m ρ ρ ⊕r X ρei1 eik ρei1 ei` Kx E = ai T ··· T Tvi ··· Tvi (0,mi1 ) (0,mik ) 1 ` i=1 The problem here is that these products of elements in the spherical Hall algebras ρ e σe are not in increasing order of slopes. Since the algebras UX and UX commute for diﬀerent primitive characters, we are left with the problem to write these products ρe in increasing order of slopes in a twisted spherical Hall algebra UX for a ﬁxed ρe ∈ Pn.

ρe Step 6 In order to write products in UX in increasing order of slopes, we use the n isomorphisms from Theorem 4.3.15 to reduce the problem to calculations in Eσ,σ. n Lemma 4.3.12 tells us that we can write any element in Eσ,σ as a linear combination + of elements tp where p ranges through convex paths in Conv . If x, y ∈ Z are such that there are not interior lattice points in ∆x,y, (as in the proof of Lemma 4.3.12) then either

γ(x) = γ(y) = γ(x + y) = 2 or γ(x) = 1 or γ(y) = 1.

In th ﬁrst case, we may assume up to the SL2(Z)-action that x = (0, 2) and y = (2, 0), applying the Deﬁnition 4.3.11 item (2) yields

2 c2 tx, ty = c t(1,1) + c2 − 2 t(2,2), c1 where c ∈ C. For the last two cases, the relation (2) in Deﬁnition 4.3.11 directly yields that we can write the bracket tx, ty as linear combination of tp where p runs through the convex paths in ∆x,y.

91 On the other hand, if Z∩∆x,y is not empty, we can every write the bracket tx, ty in terms of brackets tx0 , ty0 such that ∆x0,y0 has no interior lattice points. To do that, we can subdivide ∆x,y into smaller triangles by successively choosing a lattice point zi for i = 0, . . . , s such that ∆zi−1,zi does not have an interior lattice point, where z0 = x and zs = y. Therefore, using the previous paragraph we may every write tx, ty as linear combination of tp where p runs through the set of convex paths in ∆x,y.

vec ⊕r vec ⊕r Step 7 By Lemma 4.2.4, π Kx E = π Kx , E . Aiming to use the relations n ⊕r of Eσ,σ as in the previous step, we consider Kx , E . For x, x1,..., x` ∈ Z with µ(x) = ∞ and µ(x1) ≤ · · · ≤ µ(x`) the formula

` X tx, tx1 ··· tx` = tx1 ··· txi−1 tx, txi txi+1 ··· tx` i=1 combined with tztw = tz, tw + twtz for µ(z) > µ(w) and with the fact (previous step) that we can calculate tz, tw using the relation (2) of Deﬁnition 4.3.11 results in ﬁnitely many steps in tx, tx1 ··· tx` = c1tp1 + ··· + cstps

+ for p1,..., ps ∈ Conv . Therefore,

m X ρi ρi K⊕r, E = a T e 1 ··· T e ` x i vi1 vi` i=1

in increasing order of slopes i.e. µ(vi1 ) < ··· < µ(vi` ), for i = 1, . . . , m. By construction (or Theorem 4.4.7) pi = (vi1 ,..., vi` ) runs through the convex paths in ∆(rk(E),deg(E)),(0,r|x|).

ρ Step 8 T eij P ρ(x) Next we replace the vij by their deﬁnition. Using that ρ is zero un- x = x K⊕r E T ··· T , less 0, we can write the product x as linear combination of vi1 ,xi1 vin ,xin where pi = (vi1 ,..., vin ) runs through the convex paths in the triangle deﬁned by ⊕r Kx and E.

Step 9 Finally, since Tvi,xi Tvj ,xj = Tvi,xi ⊕ Tvj ,xj if either µ(vi) < µ(vj) or µ(vi) =

µ(vj) and xi 6= xj, our problem reduces to solve the case Tvi,xTvj ,x with µ(vi) = µ(vj).

92 Step 10 In order to solve the previous step consider Tvi,x,Tvj ,x ∈ HTorx with

µ(vi) = µ(vj). By Proposition 4.3.1 once again we can express the product Tvi,xTvj ,x ±1 as a product of power-sums. Since the Hall-Littlewood functions form a C[ux ]-basis for the Macdonald ring of symmetric functions we can write the product of these two power-sums as a linear combination of Hall-Littlewood functions. Since the Hall- (λ) Littlewood functions Pλ correspond to Kx , we have an explicit description for the vec ⊕r product Tvi,xTvj ,x, and therefore an explicit description for the product π Kx E .

⊕r Remark 4.4.8. In fact, our algorithm does not only calculate the products Kx E, but also a more general class of products in the Hall algebra of an elliptic curve. For instance the products of any torsion sheaf by a vector bundle.

4.5 Calculating structure constants

In this section we will apply our algorithm to calculate some structure constants for the graphs Gx,r. Our results cover some cases of particular interest and are meant to exemplify the way the algorithm works. An explicit description of the graphs Gx,r for all n, all x and all r seems out of reach. But we cover the case of |x| = 1, n = 2, r = 1 in completion in next section. (n,d) Theorem 4.5.1. Fix an integer n ≥ 1. Let x ∈ |X| with |x| = 1. Let E = E(y,1) with 0 0 ∼ (n,d+1) n|d be a stable vector bundle on X. Then mx,1 E , E 6= 0 if and only if E = E(z,1) n 0 0 0 where z = x + Norm1 (y ), where y ∈ |Xn| sits above y. In this case mx,1 E , E = 1. Graphically, 1

(n,d+1) (n,d) E(z,1) E(y,1)

(n,d) are all incoming arrows at E(y,1) in Gx,1.

(n,d) Proof. Via Atiyah’s classiﬁcation, E = E(y,1) corresponds to Ky where |y| = n. We vec aim to calculate π (Ky E) following the algorithm from section 4.4. \0 First note that T(0,n),y = [n]Ky. For ρ ∈ Pic (Xn) consider

ρe X 0 T(0,n) = ρe(x )T(0,n),x0 x0∈|X| |x0| | n and by Proposition 4.3.6

−1 X ρe T(0,n),y = nNn ρe(−y)T(0,n). ρe∈Pn

93 Hence −1 −1 X ρe Ky = [n] nNn ρe(−y)T(0,n) ρe∈Pn and −1 −1 X ρe E = [n] nNn ρe(−y)T(n,d). ρe∈Pn Therefore

vec π (Ky E) = Ky, E h i −1 X σ −1 −1 X ρe = N1 σ(−x)T(0,1), [n] nNn ρe(−y)T(n,d) σ∈Pic\0(X) ρe∈Pn

−1 −1 −1 X σ ρe = N1 [n] nNn σ(−x)ρ(−y) T(0,1),T(n,d) σ∈P1 ρe∈Pn −1 −1 −1 X n 0 σ = N1 [n] nNn cn σ(−x − Norm1 (y )) T(n,d+1) σ∈P1 n ρ=Norm1 (σ) −1 n X X 0 n 0 = N1 v σ(x − x − Norm1 (y ))T(n,d+1),x0 σ∈P1 x0∈|X| n = v T(n,d+1),z n (n,d+1) = v E(z,1) ,

n 0 0 where z = x + Norm1 (y ) with y ∈ |Xn| sits above y.

Theorem 4.5.2. Let x be a closed point in X and L1,..., Ln line bundles on X. If 0 E = L1 ⊕ · · · ⊕ Ln with deg(Li) > deg(Li−1) + |x| for i = 1, . . . , n, then

n 0 [ 0 k−1 Vx,1(E ) = E , L1 ⊕ · · · ⊕ Lk(−x) ⊕ · · · ⊕ Ln, qx . k=1

Proof. Following the algorithm in section 4.4, the proposition will follow from calcu- vec lating π (KxE) = [Kx, E]. We have

(1,d1) (1,dn) P h(1,d ),(1,d )i (1,d1) (1,dn) E = E ⊕ · · · ⊕ E = v i

Denote by d the degree of x. By deﬁnition, T(0,d),x = [d]Kx. By Proposition 4.3.6, X −1 −1 σe Kx = [d] dNd σe(−x)T(0,d). ^0 σe∈Pic (Xd)

94 E (1,di) K |x | = 1 By Atiyah’s classiﬁcation (Theorem 4.1.1) (xi,1) corresponds to xi where i i = 1, . . . , n. T = K i = 1, . . . , n for Since (0,1),xi xi , we have for that X K = N −1 ρ (−x )T ρei xi 1 ei i (0,1) ^0 ρei∈Pic (X) by Proposition 4.3.6. Thus

(1,di) −1 X ρei E = N ρi(−xi)T , i = 1, . . . , n. (xi,1) 1 e (1,di) for ^0 ρei∈Pic (X) Therefore h X vec −1 −1 σe π (KxE) = [d] dNd σe(−x)T(0,d), σe∈Pd n P X X ρ ρ i i

ρ T σe ,T ei = 0 σ = Normdρ . σ = Normdρ Observe that (0,d) (1,di) unless 1 i When 1 i, we have by Theorem 4.3.15 that T σe ,T ρei = c T ρei . (0,d) (1,di) d (1,di+d)

95 This yields P vec −1 −1 i

4.6 The case of rank 2

We shall use our algorithm to describe the graphs of unramiﬁed Hecke operators for n = 2, r = 1 and a closed point x of degree one.

96 We start with an explicit description for the vertices of Gx,1 for n = 2. In [32] 1 and [31], Lorscheid describe explicitly the set PBun2X of isomorphism classes of P - bundles over X ([19], Ex. II.7.10). We derive a description of Bun2X from this by determining the of representatives for each classes in PBun2X. We call a vector bundles E indecomposable if for every decomposition E = E1 ⊕ E2 into two subbundles E1, E2 one factor is trivial and the other is isomorphic to E. The Krull-Schmidt theorem (c.f. [4] Theorem 2) holds for the category of vector bundles over X, i.e. every vector bundle on X has a unique decomposition into a direct sum of indecomposable subbundles, up to permutation of factors.

An extension of scalars Fqi F/F , or geometrically π : Xi := X×Spec Fq Spec Fi → X, deﬁnes the inverse image or the constant extension of vector bundles

∗ π : BunnX −→ BunnXi. E 7−→ π∗E

The isomorphism classes of rank n bundles that after extension of constants to Fqi ∗ 1 become isomorphic to π E are classiﬁed by H (Gal(Fqi /Fq), Aut(E ⊗Fqi )), c.f. [3] Sec- tion 1. Since the algebraic group Aut(E ⊗ Fqi ) is an open subvariety of the connected algebraic group End(E⊗Fqi ), it is itself a connected algebraic group. As a consequence 1 of Lang’s theorem ([?] Corollary to Theorem 1), we have H (Gal(Fqi /Fq), Aut(E ⊗ ∗ Fqi )) = 1. We deduce that π is injective. In particular, one can consider the constant extension to the geometric curve X = X ×SpecFq SpecFq, where Fq is an algebraic closure of Fq. Then two vector bundles are isomorphic if and only if they are geometrically isomorphic, i.e. their constant extensions to X are isomorphic. We can therefore think BunnX as a subset of BunnXi and BunnX. On the other hand, π : Xi → X deﬁnes the direct image or the trace of vector bundles

π∗ : BunnXi −→ BunniX.

E 7−→ π∗E

∗ ⊕i ∗ L τ For E ∈ BunnX we have that π∗π E = E and for E ∈ BunnXi that π π∗E = E τ τ where τ ranges over Gal(Fqi /Fq) and E is deﬁned by the stalks Ex = Eτ −1(x). We call a vector bundle geometrically indecomposable if its extension to X is indecomposable. In [3], Theorem 1.8, it is shown that every indecomposable vector bundle over X is the trace of a geometrically indecomposable bundle over some constant extension Xi of X. We are interested in the case i = n = 2.The set Bun2X is the disjoint union of the set of classes of decomposable rank 2 bundles and the set of classes of indecomposable dec ind gi bundles. We denote theses sets by Bun2 X and Bun2 X, respectively. Let Bun2 X ⊂ ind Bun2 X be the subset of classes of geometrically indecomposable vector bundles. tr ind gi Since the rank is 2, the complement Bun2 X = Bun2 X − Bunn X consists of classes

97 0 0 0 00 00 of traces π∗L where L ∈ PicX2. Moreover, π∗L = π∗L if, and only if, L ∈ 0 0 τ 0 {L , (L ) }, where τ generates Gal(Fq2 |Fq) and π∗L decomposes if, and only if, L0 ∈ PicX (cf. Proposition 6.4 of [32]). Thus we have a disjoint union

dec tr gi Bun2X = Bun2 X q Bun2 X q Bun2 X. Namely, dec Bun2 X = L1 ⊕ L2 L1, L2 ∈ PicX and tr 0 0 0 0 Bun2 X = L ⊗ π∗L L ∈ PicX, L ∈ Pic X2 − Pic X . gi Next we determine Bun2 X. Since g = 1 and ωX = OX , the Riemann-Roch theorem reduces to −1 dimFq Γ(L) − dimFq Γ(L ) = deg L. Since Γ(L) is non-zero if and only if L is associated to an eﬀective divisor ([19] Proposition II.7.7(a)), we obtain:  ∼ 0 if deg L ≤ 0 and L 6= OX ,  ∼ dimFq Γ(L) = 1 if L = Ox, and  deg L if deg L > 0.

∗ The units Fq operate by multiplication on the Fq−vector space 0 ∼ 0 ∨ Ext(L, L ) = Hom(L, L ωX )

0 ∨ ∗ The multiplication of a morphism L → L ωX by an a ∈ Fq is nothing else but −1 0 ∨ multiplying the generic stalk Lη by a and all stalks (L ωX )x at a closed point x 0 ∨ by a. This induces automorphisms on both L and L ωX , respectively. Thus, two 1 0 ∗ elements of Ext (L, L ) that are Fq-multiplies of each other deﬁne the same bundle on Bun2X. We get a well-deﬁned map

0 PExt(L, L ) −→ Bun2X where the projective space PExt(L, L0) is deﬁned as the empty set when Ext(L, L0) is trivial. ∼ ∼ Let L ∈ PicX. Serre duality asserts that Ext(L, L) = Hom(L, L) = Γ(OX ) is one- dimensional. Thus PExt(L, L) contains only one element. By the above discussion, this determine a rank 2-bundle E(L). For a place x of degree one, the Fq-vector space ∼ ∼ Ext(L, L(x)) = Hom(L, L(x)) = Γ(OX (x)) is also one-dimensional and deﬁnes a rank 2 bundle Ex(L). Proposition 4.6.1. gi Bun2 X = Ex(L) x ∈ |X|, |x| = 1 and L ∈ PicX ∪ E(L) L ∈ PicX .

98 Proof. The proof follows from Proposition 7.1.4 of [30] by removing the action of PicX.

Our next task is to prove the following theorem which describes the graph Gx,1 for a degree one place x and n = 2.

Theorem 4.6.2. Let x be a closed point of X of degree one. Then the edges of Gx,1 for n = 2 are given by the following list. n o (1) Vx,1(E) = (E, L1(−x) ⊕ L2, 1), (E, L1 ⊕ L2(−x), q)

for E = L1 ⊕ L2 and either deg(L2) − deg(L1) > 1 or deg(L2) − deg(L1) = 1 ∨ and L2 ⊗ L1 6' OX (x).

n o (2) Vx,1(E) = (E, L1(−x) ⊕ L2, 1), (E, L1 ⊕ L2(−x), 1), (E, E(L1), q − 1)

for E = L1 ⊕ L2, deg(L2) − deg(L1) = 1 and L2(−x) 'L1.

n o (3) Vx,1(E) = (E, L1(−x) ⊕ L2, 1), (E, L1 ⊕ L2(−x), 1), (E, Ex0 (L2(−x)), q − 1)

for E = L1 ⊕ L2, deg(L1) = deg(L2) and L1 6' L2.

n o (4) Vx,1(E) = (E, L(−x) ⊕ L, q + 1) for E = L ⊕ L, L ∈ PicX.

n o (5) Vx,1(E(L)) = (E(L), Ex0 (L), q), (E(L), L ⊕ L(−x), 1) 0 2 0 for L ∈ PicX and x = Norm1(y ) − x where y ∈ X(Fq) is the rational point 0 associated to L and y ∈ |X2| sits above y.

n 0 o (6) Vx,1(Ex0 (L)) = (Ex0 (L), L ⊗ π∗(L ), 1), (Ex0 (L), L1 ⊕ L2, 1) S n o (Ex0 (L), E(L), q − 1) 0 0 0 where x ∈ X(Fq), L ∈ PicX, for L = OX2 (y − x0) ∈ Pic X2, 2 0 Norm1(y) = x − x and Li ∈ PicX associated to xi (i = 1, 2) 0 with x1 + x2 = x − x.

0 n 0 o (7) Vx,1(L ⊗ π∗(L )) = (L ⊗ π∗(L ), Ex0 (L), q + 1) 0 0 0 where L ∈ Pic X2, L ∈ PicX and x = y − x.

The rest of this section is dedicated to the proof of Theorem 4.6.2. Part (1) follows from Proposition 4.5.2 and Lemma 4.6.4. Proposition 4.5.2, Lemma 4.6.6 and Lemma 4.6.5 proof part (2). Part (3) follows from lemmas 4.6.3 and 4.6.7. Part (4) follows

99 from Lemma 4.6.8. Lemma 4.6.7 and Lemma 4.6.8 prove (5). Part (6) follows from Proposition 4.5.1, Lemma 4.6.4 and Lemma 4.6.5. Finally, (7) follows from Lemma 4.6.7. Lemma 4.6.3. Let E = E (1,d−1) ⊕ E (1,d) with x 6= x . Then (x1,1) (x2,1) 1 2

(1,d) (1,d) (1,d) (1,d) mx,1 E ⊕ E , E = 1 and mx,1 E 0 ⊕ E 0 , E = q (y1,1) (y2,1) (y1,1) (y2,1)

0 0 where y1 = x + x1, y2 = x2, y1 = x1 and y2 = x + x2. Proof. H X E = v E (1,d−1)E (1,d) . In the Hall algebra X of , we have (x1,1) (x2,1) Via Atiyah’s clas- E (1,d−1) K E (1,d) K . siﬁcation, (x1,1) corresponds to x1 and (x2,1) corresponds to x2 By Proposition −1 P ρei 4.3.6, K = T(0,1),x = N ρi(−xi)T , for i = 1, 2. Thus, xi i 1 ρei∈P1 e (0,1)

(1,d−1) −1 X ρe1 (1,d) −1 X ρe2 E = N ρ1(−x1)T , E = N ρ2(−x2)T (x1,1) 1 e (1,d−1) (x2,1) 1 e (1,d) ρe1∈P1 ρe2∈P1 and −2 X ρe1 ρe2 E = vN1 ρe1(−x1)ρe2(−x2) T(1,d−1)T(1,d). ρe1,ρe2∈P1 Therefore, h i vec −1 X ρe −2 X ρe1 ρe2 π (Kx E) = N1 ρe(−x)T(0,1), vN1 ρe1(−x1)ρe2(−x2) T(1,d−1)T(1,d) ρe∈P1 ρe1,ρe2∈P1 h i −3 X ρe ρe1 ρe2 = vN1 ρe(−x)ρe1(−x1)ρe2(−x2) T(0,1),T(1,d−1)T(1,d) ρ,e ρe1,ρe2∈P1 h i −3 X ρe ρe ρe2 = vN1 ρe(−x − x1)ρe2(−x2) T(0,1),T(1,d−1) T(1,d) ρe6=ρe2 h i X ρe1 ρe ρe + ρe(−x − x2)ρe1(−x1)T(1,d−1) T(0,1),T(1,d) ρe6=ρe1 h i X ρe ρe ρe + ρe(−x − x1 − x2) T(0,1),T(1,d−1)T(1,d) . ρe

By Pick’s formula, ∆(1,d−1),(0,1) ∩ Z = ∆(1,d),(0,1) ∩ Z = ∅. It follows from Deﬁnition 4.3.11 and Theorem 4.3.15 that h i h i ρe ρe ρe ρe ρe ρe T(0,1),T(1,d−1) = c1T(1,d), T(0,1),T(1,d) = c1T(1,d+1) and h i ρe ρe ρe ρe ρe ρe ρe T(0,1),T(1,d−1)T(1,d) = c1 T(1,d)T(1,d) + T(1,d−1)T(1,d+1) .

100 Thus vec −3 X ρe ρe2 π (Kx E) = vc1N1 ρe(−x − x1)ρe2(−x2)T(1,d)T(1,d) ρ,e ρe2∈P1 X ρe1 ρe + ρe(−x − x2)ρe1(−x1)T(1,d−1)T(1,d+1) ρ,e ρe1∈P1 −3 2 (1,d) (1,d) −2 2 (1,d−1) (1,d+1) = vc1N1 N1 E ⊕ E + v N1 E 0 ⊕ E 0 (y1,1) (y2,1) (y1,1) (y2,1) 2 (1,d) (1,d) (1,d−1) (1,d+1) = v E ⊕ E + E 0 ⊕ E 0 (y1,1) (y2,1) (y1,1) (y2,1)

0 0 where y1 = x + x1, y2 = x2, y1 = x1 and y2 = x + x2. The multiplicities of the edges follows from Remark 4.2.2, namely by multiplication by v−2. Lemma 4.6.4. Let E = E (1,d) ⊕ E (1,d) and x 6= x . Then (x1,1) (x2,1) 1 2

(2,2d+1) (1,d/2) (1,d/2+1) m E 0 , E = 1, m E ⊕ E , E = q and x,1 (x ,1) x,1 (y1,1) (y2,1)

(1,d/2) (1,d/2+1) mx,1 E 0 ⊕ E 0 , E = q (y1,1) (y2,1) where

0 0 0 x = x + x1 + x2, y1 = x2, y2 = x + x1, y1 = x1 and y2 = x + x2.

−2 P ρe1 ρe2 Proof. Since x1 6= x2, E = N ρ1(−x1)ρ(−x2)T T . Thus 1 ρe1,ρe2∈P1 e e (1,d) (1,d) h i vec −3 X ρe ρe1 ρe2 π (Kx E) = N1 ρe(−x)ρe1(−x1)ρe2(−x2) T(0,1),T(1,d)T(1,d) ρ,e ρe1ρe2∈P1 h i ρe ρe ρe 2 ρe ρe ρe From T(0,1),T(1,d)T(1,d) = c1 T(2,d+1) + 2c1 T(1,d)T(1,d+1) and h i ρe ρe ρe T(0,1),T(1,d) = c1T(1,d+1), it follows that

vec −3 X ρe2 ρe π (Kx E) = c1N1 ρe(−x − x1)ρe2(−x2)T(1,d)T(1,d+1) ρe6=ρe2 −3 X ρe1 ρe + c1N1 ρe(−x − x2)ρe1(−x1)T(1,d)T(1,d+1) ρe6=ρe1 −3 X ρe ρe ρe + c1N1 ρe(−x − x1 − x2) c1T(2,2d+1) + 2T(1,d)T(1,d+1) ρe (1,d/2) (1,d/2+1) (1,d/2) (1,d/2+1) 2 (2,2d+1) = E ⊕ E + E 0 ⊕ E 0 + v E 0 (y1,1) (y2,1) (y1,1) (y2,1) (x ,1)

0 0 0 where x = x + x1 + x2, y1 = x2, y2 = x + x1 y1 = x1, and y2 = x + x2.

101 (2,2d) 0 Lemma 4.6.5. Let E = E(x0,2) with |x | = 1. Then

(1,d) (1,d+1) (2,2d+1) m E ⊕ E , E = q − 1 and m E 0 , E = 1 x,1 (y1,1) (y2,1) x,1 (y ,1)

0 0 0 2 00 00 0 where y1 = x , y2 = x + x and y = x + Norm1(x ), with x ∈ |X2| sits above x . (2) Proof. Observe that E corresponds to Kx0 via Atiyah’s Theorem 4.1.1, and by deﬁ- (2) 2 −2 ⊕2 nition Kx0 = [2] T(0,2),x0 − (1 − v )Kx . By Newton’s formula,

2 ⊕2 v 2 2 K 0 = T 0 − T 0 x 2 (0,1),x [2] (0,2),x and thus 2 E = `1 T(2,2d),x0 − `2 T(1,d),x0 2 −2 v2 −2 v2 where `1 := [2] + (1 − v ) [2] and `2 := (1 − v ) 2 . Hence h i vec −1 −1 X 0 ρe χe π (Kx E) = `1N2 N1 ρe(−x)χe(−x ) T(0,1),T(2,2d) ρe∈P1 χe∈P2 h i −3 X 0 0 ρe ρe1 ρe2 − `2N1 ρe(−x)ρe1(−x )ρe2(−x ) T(0,1),T(1,d)T(1,d) . ρ,e ρe1,ρe2∈P1 h i 2 ρe χe If χe 6= Norm1ρe, then T(0,1),T(2,2d) = 0 by Theorem 4.3.15. Otherwise h i ρe χe ρe T(0,1),T(2,2d) = c2T(2,2d+1).

By the above observation and a calculation similar to that in the proof of Lemma 4.6.3, we have

vec 2 [2] (2,2d+1) (1,d) (1,d+1) 2 (2,2d+1) π (Kx E) = `1v E 0 − 2`2 E ⊕ E − `2v E 0 2 (y ,1) (y1,1) (y2,1) (y ,1) 2 (2,2d+1) 2 (1,d) (1,d+1) = v E 0 + (q − 1)v E ⊕ E (y ,1) (y1,1) (y2,1)

0 0 0 2 00 00 0 where y1 = x , y2 = x + x and y = x + Norm1(x ), with x ∈ |X2| sits above x . (1,d) (1,d) (1,d) (1,d+1) Lemma 4.6.6. Let E = E 0 ⊕ E 0 , then m E ⊕ E , E = 1 where (x ,1) (x ,1) x,1 (y1,1) (y2,1) 0 y1 = x, y2 = x + x . Proof. By Atiyah’s classiﬁcation and Newton’s formulas, we have

2 v X 2 2 X E = N −2 ρ (−x0)T ρe1 − N −1 χ(−x0)T χe . 2 1 e1 (1,d) [2] 2 e (2,2d) ρe1∈P1 χe∈P2

102 Thus,

v2 X h i πvec(K E) = N −3 ρ(−x)ρ (−2x0) T ρe ,T ρe1 T ρe1 x 2 1 e e1 (0,1) (1,d) (1,d) ρ,e ρe1∈P1 h i 2 −3 X 0 0 ρe ρe1 ρe2 + v N1 ρe(−x)ρe1(−x )ρe2(−x ) T(0,1),T(1,d)T(1,d) ρ,e ρe1,ρe2∈P1 ρe16=ρe2 v2 X h i − N −1N −1 ρ(−x)χ(−x0) T ρe ,T χe . [2] 1 2 e e (0,1) (2,2d) ρe∈P1 χe∈P2 h i h i ρe χe 2 ρe χe We have T(0,1),T(2,2d) = 0 unless χe = Norm1(ρe), in such case T(0,1),T(2,2d) = ρe c2T(2,2d+1) (see Deﬁnition 4.3.11 and Theorem 4.3.15). Therefore

πvec(K E) = v2 E (1,d) ⊕ E (1,d+1) x (y1,1) (y2,1)

0 where y1 = x, y2 = x + x .

(2,d) Lemma 4.6.7. Let E = E(x0,1) with gcd(2, d) = 1. Then

(2,d+1) (2,d+1) mx,1(E(y0,2) , E) = q, mx,1 E(y,1) , E = q + 1,

and m E (1,(d+1)/2) ⊕ E (1,(d+1)/2), E = q − 1 x,1 (y1,1) (y2,1) 2 0 where |y| = 2, Norm1(z) = x + x with z ∈ |X2| sits above y. And y1 6= y2, y1 + y2 = x + x0 and |y0| = 1, 2y0 = x + x0.

−1 P 0 ρe1 Proof. In the Hall algebra HX , E = N ρ1(−x )T , thus 1 ρe1∈P1 e (2,d) h i vec −1 X ρe −1 X 0 ρe1 π (Kx E) = N1 ρe(−x)T(0,1),N1 ρe1(−x )T(2,d) ρe∈P1 ρe1∈P1 h i −2 X 0 ρe ρe1 = N1 ρe(−x)ρe1(−x ) T(0,1),T(2,d) ρ,e ρe1∈P1 h i −2 X 0 ρe ρe = N1 ρe(−x − x ) T(0,1),T(2,d) ρe∈P1 By Theorem 4.3.15, Pick’s formula and Deﬁnition 4.3.11

h i θ(2,d+1) T ρe ,T ρe = c , (0,1) (2,d) 1 v−1 − v

103 Norm2ρ −1 2 2 −1 1 e (v −v) ρe where θ(2,d+1) = (v − v) T(2,d+1) + 2 T(1,(d+1)/2) . Hence

vec −2 X 2 00 0 π (Kx E) = c1 N1 ρe(Norm1(y ) − x − x )T(2,d+1),y0 y0∈|X| ρe∈P1 (v−1 − v) X + ρ(y + y − x − x0)T T . 2 e 1 2 (1,(d+1)/2),y1 (1,(d+1)/2),y2 y1,y2∈|X| ρe∈P1

00 0 where y ∈ |X2| sits above y . (2,d+1) 0 Let us calculate which term separated. Since T(2,d+1),y0 = [2]E(y0,1) if |y | = 2 and [2] −2 (1,(d+1)/2) (1,(d+1)/2) (2,d+1) 0 T(2,d+1),y0 = 2 (1 − v ) E(y0,1) ⊕ E(y0,1) + E(y0,2) if |y | = 1, the ﬁrst term yields 2 X (2,d+1) X v + 1 (1, d+1 ) (1, d+1 ) (2,d+1) (v2 + 1)E + (1 − v−2)E 2 ⊕ E 2 + E , (y,1) 2 (y0,1) (y0,1) (y0,2) y∈|X|,|y|=2 y0∈|X|, |y0|=1 Norm2(z)=x+x0 2 0 0 1 Nomr1(z )=x+x

0 0 0 where z, z ∈ |X2|, z sits above y and z sits above y . About the second term, (v−1 − v) X c N −2 ρ(y + y − x − x0)T T 1 1 2 e 1 2 (1,(d+1)/2),y1 (1,(d+1)/2),y2 y1,y2∈X ρe∈P1 d+1 d+1 2 d+1 d+1 2 X (1, 2 ) (1, 2 ) 1 − v X (1, 2 ) (1, 2 ) = (1 − v ) E ⊕ E + E 0 E 0 . (y1,1) (y2,1) 2 (y ,1) (y ,1) y16=y2 y0∈X 0 0 0 y1+y2=x+x 2y =x+x

d+1 d+1 (1, 2 ) (1, 2 ) Rest us calculate the product E 0 E 0 . Via the equivalence C d+1 ≡ C∞ and (y ,1) (y ,1) 2 ∼ 2|x| the isomorphism HTory0 = Λv , the previous product corresponds to the product e1e1 of elementary symmetric function in Λv2|x| . By Newton’s formula

−2 e1e1 = 2e2 + p2 = (v + 1)e2 + P(2)

−2 ⊕2 (2) and Ky0 Ky0 = (v + 1)Ky0 + Ky0 . Thus,

d+1 d+1 d+1 d+1 (1, 2 ) (1, 2 ) −2 (1, 2 ) (1, 2 ) (2,d+1) E(y0,1) E(y0,1) = (v + 1) E(y0,1) ⊕ E(y0,1) + E(y0,2) . Therefore the second term yields

d+1 d+1 −2 2 d+1 d+1 2 2 X (1, 2 ) (1, 2 ) v − v X (1, 2 ) (1, 2 ) 1 − v X (2,d+1) (1−v ) E ⊕E + E 0 ⊕E 0 + E 0 . (y1,1) (y2,1) 2 (y ,1) (y ,1) 2 (y ,2) y16=y2 y0∈|X| y0∈|X| 0 y1+y2=x+x 2y0=x+x0 2y0=x+x0 Putting all together follows the proof.

104 (1,d−1) (1,d) Lemma 4.6.8. Let E = E(x0,1) ⊕ E(x0,1). Then

(1,d) (1,d) (2,2d) mx,1 E(x0,1) ⊕ E(x0,1), E = q + 1, mx,1 E(x0,2) , E = 1

(1,d−1) (1,d+1) and mx,1 E(x0,1) ⊕ E(x0,1) , E = q. Proof. By a similar calculation as in the proof of Lemma 4.6.3, we have

vec −1 π (Kx E) = vc1N1 T(1,d),x0 T(1,d),x0 + T(1,d−1),x0 T(1,d+1),x0 since we are considering x as the neutral element of X(Fq). By Proposition 4.3.1, T(1,d),x0 corresponds to the elementary symmetric function e1, thus the product T(1,d),x0 T(1,d),x0 corresponds to the product e1e1, which can be written as e1e1 = 2e2 +p2 by Newton’s formula. Hence

−2 (1,d) (1,d) (2,2d) T(1,d),x0 T(1,d),x0 = (v + 1) E(x0,1) ⊕ E(x0,1) + E(x0,2) and therefore

vec 2 (1,d) (1,d) 2 (2,2d) (1,d−1) (1,d+1) π (Kx E) = (1 + v ) E(x0,1) ⊕ E(x0,1) + v E(x0,2) + E(x0,1) ⊕ E(x0,1) .

The lemma follows after multiplying with v−2.

This concludes the proof of Theorem 4.6.2. We end this section with an explicit example.

Example 4.6.9. The easiest example are given by elliptic curves with only one rational point. There are up to a isomorphism three such elliptic curves (see for example [47] 2.4.4 and Ex. 3 of 2.4): X2 over F2 defined by the Weierstrass equation 2 3 2 3 y + y = x + x + 1,X3 over F3 defined by the Weierstrass equation y = x + 2x + 2 2 3 and X4 over F4 defined by the Weierstrass equation y +y = x +α where F4 = F2(α). n Observe that in this example Xn does not denote X ×Spec Fq Spec Fq . Since the class number is 1,

dec Bun2 Xq = {OXq (n) ⊕ OXq (m) | n, m ∈ Z, n ≥ m} and gi Bun2 Xq = {Ex(OXq (n)), E(OXq (n)) | n ∈ Z} 0 ∼ 0 ∼ for q ∈ {2, 3, 4}. One calculates that Pic X2 ⊗ F4 = Z/5Z, Pic X3 ⊗ F9 = Z/7Z 0 ∼ and Pic X4 ⊗ F16 = Z/9Z, thus

tr 0 0 Bun2 Xq = {OXq (n)π∗(L1), ··· , OXq (n)π∗(Lq) | n ∈ Z}

105 0 0 0 where Pic Xq ⊗ Fq2 = {L0,..., L2q}. By Theorem 4.6.2 we obtain the following ﬁgure.

3 2 1 2 1 3 3 1 1 1 1 1 2 1 3 2 1 2 1 3 3 1 1 1 1 1 2 1 3 2 1 2 1 3 3 1 1 1 2 1 1 1

traces geometrically indecomposable sum of two lines bundles

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