Generalized Jacobians

Andrew Kobin Fall 2016 – Spring 2017 Contents Contents

Contents

0 Introduction 1

1 Classical Jacobians 2 1.1 Elliptic Functions ...... 2 1.2 Elliptic Curves ...... 8 1.3 The Classical Jacobian ...... 14 1.4 Jacobians of Higher Genus Curves ...... 19

2 Curves 21 2.1 Divisors ...... 22 2.2 Sheaves and Cohomology ...... 23 2.3 R´epartitions...... 27 2.4 Meromorphic Diﬀerentials ...... 29 2.5 Serre Duality and Riemann-Roch ...... 32

3 Maps From Curves 34 3.1 Moduli and Local Symbols ...... 34 3.2 The Main Theorem ...... 38 3.3 A Word About Symmetric Products ...... 44

4 Curves with Singularities 46 4.1 Normalization ...... 46 4.2 Singular Riemann-Roch ...... 48 4.3 Diﬀerentials on Singular Curves ...... 49

5 Constructing Jacobians 52 5.1 The Jacobian of a Curve ...... 52 5.2 Generalized Jacobians ...... 55 5.3 The Canonical Map ...... 57 5.4 Invariant Diﬀerential Forms ...... 61 5.5 The Structure of Generalized Jacobians ...... 62

6 Class Field Theory for Curves 68 6.1 Coverings and Isogenies ...... 69 6.2 Class Field Theory ...... 70

i 0 Introduction

0 Introduction

These notes come from the 2016-2017 Galois-Grothendieck Seminar at the University of Virginia. The general theme of the year’s seminar was “generalized Jacobians”: their construction and application for algebraic curves over an arbitrary field. Let C be a curve over a field k. We will almost always assume C is irreducible, projective and smooth. For the first chapter, k will be the complex numbers C. Our goal is to construct a new object J, the Jacobian of C, which is a projective algebraic variety over k of dimension equal to the genus of C. We will see that J is also a group, and in fact is an algebraic group. This means that since J is a projective (hence complete) algebraic group, it is abelian. Such an algebraic group is called an abelian variety. When the genus of C is greater than 0, there will be a canonical embedding C,→ J. The class field theory of k(C) = K will also be of interest. In general, for a field K, class field theory describes the abelian extensions L/K, i.e. finite Galois extensions with Gal(L/K) abelian, in terms of C itself. In the case of K = k(C) the function field of a curve, such an extension will be of the form L = k(C0) for a curve C0. Then the field embedding K ⊆ L corresponds to a ramified cover C0 → C. It turns out that if C0 → C is unramified, then it corresponds to a finite index subgroup of the surface group

* g + Y Σg = x1, y1, . . . , xg, yg : [xi, yi] = 1 i=1 where g is the genus of C. Even better, these covers of curves can be realized by covers of algebraic varieties. When C0 → C is a ramiﬁed cover, the usual notion of Jacobian will be replaced by a generalized Jacobian, which is isomorphic to a direct product of the Jacobian of C and another algebraic group. We will see in the ﬁrst chapter that when k = C, the theory has powerful connections to the theory of complex manifolds. In particular, the Jacobian J can be realized as a complex g ab ∼ 2g torus C /Λ where Λ is a lattice, and we will have π1(C) = π1(J) = Z .

1 1 Classical Jacobians

1 Classical Jacobians

In this chapter we review the classical theory of Jacobians for complex algebraic curves, starting with the construction and basic properties of elliptic functions, their connection to elliptic curves and their Jacobians, and then describing the construction in arbitrary dimension.

1.1 Elliptic Functions

Let Λ ⊆ C be a lattice, i.e. a free abelian subgroup of rank 2. Then Λ can be written

ω1 Λ = Zω1 + Zω2 for some ω1, ω2 ∈ C such that 6∈ R. ω2

Deﬁnition. A function f : C → C ∪ {∞} is doubly periodic with lattice of periods Λ if f(z + `) = f(z) for all ` ∈ Λ and z ∈ C. Deﬁnition. An elliptic function is a function f : C → C ∪ {∞} that is meromorphic and doubly periodic. It is not obvious that doubly periodic functions even exist! We will prove this shortly.

Deﬁnition. Let Λ ⊆ C be a lattice. The set

Π = Π(ω1, ω2) = {t1ω1 + t2ω2 | 0 ≤ ti < 1}

is called the fundamental parallelogram, or fundamental domain, of Λ. We say a subset Φ ⊆ C is fundamental for Λ if the quotient map C → C/Λ restricts to a bijection on Φ.

ω1

ω2 Π

Lemma 1.1.1. For any choice of basis [ω1, ω2] of Λ, Π(ω1, ω2) is fundamental for Λ. Lemma 1.1.2. Let Λ be a lattice. Then

2 1.1 Elliptic Functions 1 Classical Jacobians

(a) If Π is the fundamental domain of Λ, then for any α ∈ C, Πα := Π+α is fundamental for Λ. [ (b) If Φ is fundamental for Λ, then C = Φ + `. `∈Λ Corollary 1.1.3. Suppose f is an elliptic function with lattice of periods Λ and Φ fundamental for Λ. Then f(C) = f(Φ). Proposition 1.1.4. A holomorphic elliptic function is constant.

Proof. Let f be such an elliptic function and let Φ be the fundamental domain for its lattice of periods. Then Π is compact and hence f(Π) is as well. In particular, f(C) = f(Π) ⊆ f(Π) is bounded, so by Liouville’s theorem, f is constant. The prominence of tools from complex analysis (e.g. Liouville’s theorem in the above proof) is obvious in the study of elliptic functions. Another important result for computations is the residue theorem:

Theorem (Residue Theorem). For any meromorphic function f on a region R ⊆ C, with isolated singularities z1, . . . , zk ∈ R. Then if ∆ = ∂R,

k Z X f(z) dz = 2πi Res(f; zi). ∆ i=1

Proposition 1.1.5. Let f be an elliptic function. If α ∈ C is a complex number such that ∂Πα does not contain any of the poles of f, then the sum of the residues of f inside ∂Πα equals 0.

Proof. Fix a basis [ω1, ω2] of Λ and set ∆ = ∂Πα. By the residue theorem, it’s enough to R show ∆ f(z) dz = 0. We parametrize the boundary of Π as follows:

γ1 = α + tω1

γ2 = α + ω1 + tω2

γ3 = α + (1 − t)ω1 + ω2

γ4 = α + (1 − t)ω2.

γ3

γ4 Πα γ2

γ1 α

3 1.1 Elliptic Functions 1 Classical Jacobians

We show that R f(z) dz+R f(z) dz = 0 and leave the proof that R f(z) dz+R f(z) dz = 0 γ1 γ3 γ2 γ2 for exercise. Consider Z Z Z 1 Z 1 f(z) dz + f(z) dz = f(α + tω1)(ω1 dt) + f(α + (1 − t)ω1 + ω2)(−ω1 dt) γ1 γ3 0 0 Z 1 Z 0 = ω1 f(α + tω1) dt + ω1 f(α + sω1) ds since f is elliptic 0 1 Z 1 Z 1 = ω1 f(α + tω1) dt − f(α + sω1) ds = 0. 0 0 Hence the sum of the residues equals 0.

Corollary 1.1.6. Any elliptic function has either a pole of order at least 2 or two poles on the fundamental domain of its lattice of periods.

Proposition 1.1.7. Suppose f is an elliptic function with fundamental domain Π and α ∈ C n such that ∆ = ∂Πα does not contain any zeroes or poles of f. Let {aj}j=1 be a ﬁnite set of Pn zeroes and poles in Πα, with mj the order of the pole aj. Then j=1 mj = 0.

m Proof. For a pole z0, we can write f(z) = (z − z0) g(z) for some holomorphic function g(z), with g(z0) 6= 0. Then

f 0(z) g0(z) = (z − z )−1 m + (z − z ) . f(z) 0 0 g(z)

f 0 Hence Res f ; z0 = m. Then the statement follows from Proposition 1.1.5. Proposition 1.1.7 has an analogue in algebraic geometry: if f is a rational function on P an algebraic curve C, the formal sum (f) = mjaj, where the aj and mj are deﬁned as P above, is called the principal divisor associated to f and its degree is deg(f) = mj. Then one can prove that deg(f) = 0. Continuing in the complex setting, let f be an elliptic function and let a1, . . . , ar be the poles and zeroes of f in the fundamental domain of Λ. Write ordai f = mi if ai is a pole Pr of order −mi or if ai is a zero of multiplicity mi. The sum ord(f) = i=1 mi is called the order of f. Then Corollary 1.1.6 says that there are no elliptic functions of order 1. We will show that the ﬁeld of elliptic functions with period lattice Λ is generated by an order 2 and an order 3 function.

Let f be elliptic and z0 ∈ C with ordz0 f = m. Then for any ` ∈ Λ, ordz0+` f = m as well. Indeed, if z0 is a zero then

(m−1) 0 = f(z0) = f(z0) = ... = f (z0)

(k) but f (z) is also elliptic for all k ≥ 1. If z0 is a pole of f, the same result can be obtained 1 using f instead of f. If Φ1 and Φ2 are any two fundamental domains for Λ, then for all a1 ∈ Φ1, there is a unique a2 ∈ Φ2 such that a2 = a1 + ` for some ` ∈ Λ. Thus Propositions 1.1.5 and 1.1.7 hold

4 1.1 Elliptic Functions 1 Classical Jacobians

for any fundamental domain of Λ, so it follows that ord(f) is well-deﬁned on the quotient C/Λ. Now given any meromorphic function f(z) on C, we would like to construct an elliptic function F (z) with lattice Λ. Put X F (z) = f(z + `). `∈Λ There are obvious problems of convergence and (in a related sense) the order of summation. 1 It turns out we can do this construction with f(z) = zm , m ≥ 3 though. First, we need the following result from complex analysis, which can be proven using Cauchy’s integral formula and Morera’s theorem.

Lemma 1.1.8. Let U ⊆ C be an open set and suppose (fn) is a sequence of holomorphic functions on U such that fn → f uniformly on every compact subset of U. Then f is 0 0 holomorphic on U and fn → f uniformly on every compact subset of U.

Proposition 1.1.9. Let Λ be a lattice with basis [ω1, ω2]. Then the sum X 1 |ω|s ω∈Λr{0} converges for all s > 2.

Proof. Extend the fundamental domain by translation by the vectors ω1, ω2 and ω1 + ω2, and call the boundary of the resulting region ∆:

Λ Λ ∆

Λ Λ

Then ∆ is compact, so there exists c > 0 such that |z| ≥ c for all z ∈ ∆. We claim that for all m, n ∈ Z, |mω1 + nω2| ≥ c · max{|m|, |n|}. The cases when m = 0 or n = 0 are trivial, so without loss of generality assume m ≥ n > 0. Then n |mω + nω | = |m| ω + ω ≥ |m|c. 1 2 1 m 2 Hence the claim holds. Set M = max{|m|, |n|} and arrange the sum in question so that the 1 |ω|s are added in order of increasing M values. Then the sum can be estimated by

∞ ∞ X 1 X 8M X 1 ≤ ∼ . |ω|s csM s M s−1 ω∈Λr{0} M=1 M=1 This converges for s > 2 by p-series.

5 1.1 Elliptic Functions 1 Classical Jacobians

Proposition 1.1.10. Let n ≥ 3 and deﬁne X 1 F (z) = . n (z − ω)n ω∈Λ

Then Fn(z) is holomorphic on C r Λ and has poles of order n at the points of Λ. Moreover, Fn is doubly periodic and hence elliptic.

Proof. Fix r > 0 and let Br = Br(0) be the open complex r-ball centered at the origin in C. Let Λr = Λ ∩ Br be the lattice points contained in the closed r-ball. Then the function X 1 F (z) = n,r (z − ω)n ω∈ΛrΛr 1 C is holomorphic on Br. To see this, one has |z−ω|n ≤ |ω|n for some constant C and for all C z ∈ Br, ω ∈ Λ r Λr. Then |ω|n converges by Proposition 1.1.9, so by the Weierstrass M-test, 1 |z−ω|n converges uniformly and hence Fn,r(z) is holomorphic. It follows from the definition that Fn has a pole of order n at each ω ∈ Λ. Finally, for ` ∈ Λ, we have X 1 X 1 F (z + `) = = = F (z) n (z + ` − ω)n (z − η)n n ω∈Λ η∈Λ since the series is absolutely convergent and we can rearrange the terms. This shows that elliptic functions exist and more specifically that for each n ≥ 3, there is at least one elliptic function of order n. Unfortunately the previous proof won’t work to construct an elliptic function of order 3. However, Weierstrass discovered the following elliptic function. Definition. The Weierstrass ℘-function for a lattice Λ is defined by 1 X 1 1 ℘(z) = + − . z2 (z − w)2 ω2 ω∈Λr{0} Theorem 1.1.11. For any lattice Λ, ℘(z) is an elliptic function with poles of order 2 at the 0 points of Λ and no other poles. Moreover, ℘(−z) = ℘(z) and ℘ (z) = −2F3(z). Proof. (Sketch) To show ℘(z) is meromorphic, one estimates the summands by

1 1 D − ≤ (z − ω)2 ω2 |ω|3

for some constant D and all z ∈ Br, ω ∈ Λ r Λr as in the previous proof. 0 Next, ℘(z) can be diﬀerentiated term-by-term to obtain the expression ℘ (z) = −2F3(z). And proving that ℘(z) is odd is straightforward: 1 X 1 1 ℘(−z) = + − (−z)2 (−z − ω)2 ω2 ω∈Λr{0} 1 X 1 1 = + − = ℘(z) z2 (z − (−ω))2 (−ω)2 −ω∈Λr{0}

6 1.1 Elliptic Functions 1 Classical Jacobians

after switching the order of summation. Finally, proving ℘(z) is doubly periodic is diﬃcult since we don’t necessarily have absolute convergence. However, one can reduce to proving ℘(z + ω1) = ℘(z) = ℘(z + ω2). Then using the formula for ℘0(z), we have d [℘(z + ω ) − ℘(z)] = −2F (z + ω ) + 2F (z) dz 1 3 1 3 = −2F3(z) + 2F3(z) = 0

since F3(z) is elliptic by Proposition 1.1.10. Hence ℘(z + ω1) − ℘(z) = c is constant. Evalu- ω1 ω1 ω1 ating at z = − 2 , we see that c = ℘ 2 − ℘ − 2 = 0 since ℘(z) is odd. Hence c = 0, so it follows that ℘(z) is doubly periodic and therefore elliptic.

Lemma 1.1.12. Let ℘(z) be the Weierstrass ℘-function for a lattice Λ ⊆ C and let Π be the fundamental domain of Λ. Then

(1) For any u ∈ C, the function ℘(z) − u has either two simple roots or one double root in Π.

0 ω1 ω2 ω1+ω2 (2) The zeroes of ℘ (z) in Π are simple and they only occur at 2 , 2 and 2 .

ω1 ω2 ω1+ω2 (3) The numbers u1 = ℘ 2 , u2 = ℘ 2 and u3 = ℘ 2 are precisely those u for which ℘(z) − u has a double root. Proof. (1) follows from Corollary 1.1.6. 0 ω1 ω2 ω1+ω2 (2) By Theorem 1.1.11, deg ℘ (z) = 3 so it suﬃces to show that 2 , 2 and 2 are all ω1 roots. For z = 2 , we have ω ω ω ω ℘0 1 = −℘0 − 1 = −℘0 1 − ω = −℘0 1 2 2 2 1 2

0 0 ω1 since ℘ (z) is elliptic. Thus ℘ 2 = 0. The others are similar. (3) The double roots occur exactly when ℘0(u) = 0, so use (2). We now prove that any elliptic function can be written in terms of ℘(z) and ℘0(z).

Theorem 1.1.13. Fix a lattice Λ ⊆ C and let E(Λ) be the ﬁeld of all elliptic functions with lattice of periods Λ. Then E(Λ) = C(℘, ℘0). Proof. Take f(z) ∈ E(Λ). Then f(−z) ∈ E(Λ) as well and thus we can write f(z) as the sum of an even and an odd elliptic function: f(z) + f(−z) f(z) − f(−z) f(z) = f (z) + f (z) = + . even odd 2 2 We will prove that every even elliptic function is rational in ℘(z), but this will imply the fodd(z) theorem, since then feven(z) = ϕ(℘(z)) and ℘0(z) = ψ(℘(z)) for some ϕ, ψ ∈ C(℘(z)) and we can then write f(z) = ϕ(℘(z)) + ℘0(z)ψ(℘(z)). Assume f(z) is an even elliptic function. It’s enough to construct ϕ(℘(z)) such that f(z) ϕ(℘(z)) only has (potential) zeroes and poles at z = 0 in the fundamental parallelogram

7 1.2 Elliptic Curves 1 Classical Jacobians

f(z) for Λ, since then by Corollary 1.1.6, ϕ(℘(z)) is holomorphic and then by Proposition 1.1.4 it is constant. Suppose f(a) = 0 for a some zero of order m. Consider ℘(z) = u. If ω1 ω2 ω1+ω2 u 6= ℘ 2 , ℘ 2 , ℘ 2 then ℘(z) = u has precisely two solutions in the fundamental parallelogram, z = a and z = a∗ where  ω1 + ω2 − a if a ∈ Int(Π) ∗  a = ω1 − a if a is parallel to ω1  ω2 − a if a is parallel to ω2.

∗ (Notice that since f is even, f(a) = 0 implies f(a ) = 0 as well.) Moreover, if orda f = 0 then ∗ ω1 ω2 ω1+ω2 ∗ orda f = m. Note that a = a holds precisely when a is in the set Θ := 0, 2 , 2 , 2 . Let Z (resp. P ) be the set of zeroes (resp. poles) of f(z) in Π. Then the assignment a 7→ a∗ is in fact an involution on Z and P , so we can write

0 0 00 00 Z = Z1 ∪ · · · ∪ Zr ∪ Z1 ∪ · · · ∪ Zs 0 0 00 00 P = P1 ∪ · · · ∪ Pu ∪ P1 ∪ · · · ∪ Pv 0 0 00 00 where the Zi and Pi are the 2-element orbits of the involution and the Zj and Pj are the 0 0 0 00 00 1-element orbits. Of course then s, v ≤ 3. For a ∈ Z , set ord 0 f = m and for a ∈ Z , i i ai i j j 00 0 0 0 00 00 set ord 00 f = m , which is even. Likewise, for b ∈ P , set ord 0 f = n and for b ∈ P , set ai i i i bi i j j 00 ord 00 f = n which is even. Then we deﬁne ϕ(℘(z)) by bi i

r 0 s 00 Q 0 mi Q 00 mj /2 i=1(℘(z) − ℘(ai)) j=1(℘(z) − ℘(aj )) ϕ(℘(z)) = u 0 v . Q 0 ni Q 00 nj i=1(℘(z) − ℘(bi)) j=1(℘(z) − ℘(bj )) Then ϕ(℘(z)) has only potential zeroes/poles at z = 0 in the fundamental parallelogram, so we are done.

1.2 Elliptic Curves

Let Λ ⊆ C be a lattice. There is a canonical way to associate to the complex torus C/Λ ∼ an elliptic curve E such that C/Λ = E(C). We would also like to reverse this process, i.e. ∼ given an elliptic curve E, deﬁne a lattice Λ ⊆ C such that C/Λ = E(C). This procedure generalizes for a curve C of genus g > 1 and produces its Jacobian, C,→ Cg/Λ = J(C). We need the following lemma from complex analysis.

Lemma 1.2.1. Suppose f0, f1, f2,... is a sequence of analytic functions on the ball Br(z0) with Taylor expansions ∞ X (n) k fn(z) = ak (z − z0) . k=0 P∞ Then if F (z) = n=0 fn(z) converges uniformly on Bρ(z0) for all ρ < r, each series Ak = P∞ (n) n=0 ak converges and F (z) has Taylor expansion ∞ X k F (z) = Ak(z − z0 ). k=0

8 1.2 Elliptic Curves 1 Classical Jacobians

Let ℘(z) be the Weierstrass ℘-function for Λ. Then ℘0(z)2 is an even elliptic function, so 0 2 by Theorem 1.1.13, ℘ (z) ∈ C(℘). On a small enough neighborhood around z0 = 0, 1 X 1 1 ℘(z) − = − z2 (z − ω)2 ω2 ω∈Λr{0}

is analytic. Moreover, for each ω ∈ Λ r {0} we have 1 1 2z 3z2 = + + + ... (z − ω)2 ω2 ω3 ω4 1 1 2z 3z2 =⇒ − = + + ... (z − ω)2 ω2 ω2 ω4 which is uniformly convergent. Hence Lemma 1.2.1 shows that

∞ ∞ 1 X X k + 1 X ℘(z) − = zk = (k + 1)G zk z2 ωk+2 k+2 ω∈Λr{0} k=1 k=1 where G = G (Λ) := P 1 . These G are examples of modular forms. m m ω∈Λr{0} ωm m Deﬁnition. The series G (Λ) = P 1 is called the Eisenstein series for Λ of m ω∈Λr{0} ωm weight m. From the above work, we obtain the following formulas: 1 ℘(z) = + 3G z2 + 5G z4 + 7G z6 + ... z2 4 6 8 1 ℘(z)2 = + 6G + ... z4 4 1 9G ℘(z)3 = + 4 + 15G + ... z6 z2 6 2 ℘0(z) = − + 6G z + ... z3 4 4 24G ℘0(z)2 = − 4 − 80G − ... z6 z2 6 This implies: Proposition 1.2.2. The functions ℘ and ℘0 satisfy the following relation:

0 2 3 ℘ (z) = 4℘(z) − g2℘(z) − g3

where g2 = 60G4 and g3 = 140G6.

3 Consider the polynomial p(x) = 4x − g2x − g3, where the gn are deﬁned for the lattice Λ ⊆ C.

ω1 ω2 Proposition 1.2.3. p(x) = 4(x − u1)(x − u2)(x − u3) where u1 = ℘ 2 , u2 = ℘ 2 and ω1+ω2 u3 = ℘ 2 are distinct roots.

9 1.2 Elliptic Curves 1 Classical Jacobians

0 2 3 Thus (x, y) = (℘(z), ℘ (z)) determine an equation y = 4x −g2x−g3 which is the deﬁning 2 equation for an elliptic curve E0 over C. Let E = E0 ∪ {[0, 1, 0]} ⊆ P be the projective closure of E0. The point [0, 1, 0] is sometimes denoted ∞. Theorem 1.2.4. The map

ϕ : C/Λ −→ E(C) ( [℘(z), ℘0(z), 1], z 6∈ Λ z + Λ 7−→ ϕ(z + Λ) = [0, 1, 0], z ∈ Λ is a bijective, biholomorphic map.

Proof. Assume z1, z2 ∈ C are such that z1 + Λ 6= z2 + Λ. Without loss of generality we may assume z1, z2 ∈ Π, the fundamental domain of Λ (otherwise, translate). If ℘(z1) = ℘(z2) and 0 0 ∗ ℘ (z1) = ℘ (z2), then with the notation of Theorem 1.1.13, we must have z2 = z1 6= z1 and ω1 ω2 ω1+ω2 0 0 0 0 thus z1, z2 6∈ Θ = 0, 2 , 2 , 2 . Since ℘ (z) is odd, we get ℘ (z1) = ℘ (z2) = −℘ (−z2) = 0 −℘ (z1), but this implies ℘(z1) = 0, contradicting z1 6∈ Θ. Therefore ϕ is one-to-one. 0 Next, we must show that for any (x0, y0) ∈ E(C), x0 = ℘(z) and y0 = ℘ (z) for some 0 z ∈ C. If ℘(z1) = x0, then it’s clear that ℘ (z1) = y0 or −y0. Now one shows as in the 0 previous paragraph that we must have ℘ (z1) = y0. 2 3 Now consider F (x, y) = y − p(x), where p(x) = 4x − g2x − g3. If (x0, y0) satisfies ∂F F (x0, y0) = 0 and y0 6= 0, then ∂y (x0, y0) 6= 0 and thus the assignment (x, y) 7→ x is a local chart about (x0, y0). Likewise, (x, y) 7→ y defines a local chart about (x0, y0) when x0 6= 0. Finally, we conclude by observing that a locally biholomorphic map is biholomorphic. In general, an elliptic curve can be defined by a Weierstrass equation E : y2 = f(x) = ax3 + bx2 + cx + d.

X Y This embeds into projective space via (x, y) 7→ [x, y, 1]. Setting x = Z and y = Z , we also obtain a homogeneous equation for the curve: E : ZY 2 = aX3 + bX2Z + cXZ2 + dZ3. The single point at inﬁnity, [0, 1, 0], can be studied by dehomogenizing via the coordinates Z X z˜ = Y andx ˜ = Y , which yield E :z ˜ = ax˜3 + bx˜2z˜ + ax˜z˜2 + dz˜3.

We have shown that a lattice Λ ⊆ C determines elliptic functions ℘(z) and ℘0(z) that satisfy 0 2 3 ℘ (z) = 4℘(z) − g2℘(z) − g3 and that this polynomial expression has no multiple roots. Therefore the mapping z 7→ (℘(z), ℘0(z)) determines a bijective correspondence C/Λr{0} → E(C) r {∞} which can be extended to all of C/Λ → E(C) (this is Theorem 1.2.4). There is a natural group structure on C/Λ induced from C, but what is not so obvious is that E(C) also possesses a group structure, the so-called “chord-and-tangent method”. Let E be an elliptic curve over an arbitrary field k, let O ∈ E(k) be the point at infinity 2 and fix two points P,Q ∈ E(k). In the plane Pk, there is a unique line containing P and Q; call it L. (If P = Q, then take L to be the tangent line to E at P .) By Bézout’s theorem, E ∩ L = {P, Q, R} for some third point R ∈ E(k), which may not be distinct from P and Q if multiplicity is counted. Let L0 be the line through R and O and call its third point R0.

10 1.2 Elliptic Curves 1 Classical Jacobians

Q P

P + Q

Addition of two points P,Q ∈ E(k) is deﬁned by P + Q = R0, where R0 is the unique point lying on the line through R and O. If R = O, we set R0 = O.

Proposition 1.2.5. Let E be an elliptic curve with O ∈ E(k). Then

(a) If L is a line in P2 such that E ∩ L = {P, Q, R}, then (P + Q) + R = O. (b) For all P ∈ E(k), P + O = P .

(d) For all P ∈ E(k), there exists a point −P ∈ E(k) satisfying P + (−P ) = O.

(e) For all P, Q, R ∈ E(k), (P + Q) + R = P + (Q + R).

Together, (b) – (e) say that chord-and-tangent addition of points defines an associative, commutative group law on E(k). The proofs of (a) – (d) are rather routine using the definition of this addition law, whereas verifying associativity is notoriously difficult. There are formulas for the coordinates of P + Q that make this possible though (see Silverman).

Theorem 1.2.6. The map ϕ : C/Λ → E(C) is an isomorphism of abelian groups. Proof. Consider the diagram ϕ × ϕ C/Λ × C/Λ E(C) × E(C)

α β

/Λ E( ) C ϕ C where α and β are the respective group operations. Since C/Λ × C/Λ is a topological group, it’s enough to show the diagram commutes on a dense subset of C/Λ × C/Λ. Consider

2 Xe = {(u1, u2) ∈ C | u1, u2, u1 ± u2, 2u1 + u2, u1 + 2u2 6∈ Λ}.

11 1.2 Elliptic Curves 1 Classical Jacobians

∼ 2 Then Xe = C so X = Xe mod Λ × Λ is dense in C/Λ × C/Λ. Take (u1 + Λ, u2 + Λ) ∈ X and set u3 = −(u1 + u2). Then u1 + u2 + u3 = 0 in C/Λ. Set P = ϕ(u1),Q = ϕ(u2) and R = ϕ(u3) ∈ E(C). By the assumptions on X, the points P, Q, R are distinct. We want to 0 show ϕ(u1 + u2) = ϕ(u1) + ϕ(u2) = P + Q. Since ℘(z) is even and ℘ (z) is odd, we see that ϕ(−z) = −ϕ(z) for all z ∈ C/Λ. Thus ϕ(u1 + u2) = −ϕ(−(u1 + u2)) = −R so we need to show P + Q + R = O, i.e. P, Q, R are colinear. Since u1 6= u2, the line PQ is not vertical, 0 so there exist a, b such that ℘ (ui) = a℘(ui) + b for i = 1, 2. Consider the elliptic function

f(z) = ℘0(z) − (a℘(z) + b).

Then on the fundamental domain Π, f only has a pole at 0, so ord0 f = −3. Also, u1 and u2 are distinct zeroes of f, so there is a third point ω ∈ Π such that deg(f) = u1+u2+ω−3·0 = 0, i.e. u1 + u2 + ω = 0. Solving for ω, we get ω = −(u1 + u2) = u3. It follows that R = ϕ(u3) is on the same line as P and Q, so we are done.

The compatibility of the group operations of C/Λ and E(C) is highly useful. For example, ﬁx N ∈ N and let E[N] = {P ∈ E(C) | [N]P = O}, where [N]P = P + ... + P . The points of E[N] are called the N-torsion points of E. For | {z } N N = 2, the points P such that P = −P are exactly the intersection points of E with the x-axis along with O = [0, 1, 0]:

In general, one can show that #E[N] = N 2. This is hard to see from the geometric ∼ picture, but working with the isomorphism E(C) = C/Λ from Theorem 1.2.6, we see that 1 since C/Λ = R/Z×R/Z as an abelian group, the N-torsion is given by (C/Λ)[N] = N Z/Z× 1 2 N Z/Z. This is a group of order N . A morphism in the category of elliptic curves is called an isogeny. Explicitly, ϕ : E1 → E2 is an isogeny between two elliptic curves if it is a (nonconstant) morphism of schemes that takes the basepoint O1 ∈ E1 to the basepoint O2 ∈ E2.

12 1.2 Elliptic Curves 1 Classical Jacobians

Proposition 1.2.7. Suppose Λ1, Λ2 ⊆ C are lattices and f : C/Λ1 → C/Λ2 is a holomorphic map. Then there exist a, b ∈ C such that aΛ1 ⊆ Λ2 and

f(z mod Λ1) = az + b mod Λ2.

Proof. As topological spaces, C/Λ1 and C/Λ2 are complex tori with the same universal covering space C, so any f : C/Λ1 → C/Λ2 lifts to F : C → C making the diagram commute: F C C

π1 π2

C/Λ1 C/Λ2 f

Since covers are local homeomorphisms, it follows that F is holomorphic as well. Thus for any z ∈ C, ` ∈ Λ1,

π2(F (z + `) − F (z)) = f(π1(z + `) − π1(z)) = f(π1(z) − π1(z)) = f(0) = 0.

So F (z + `) − F (z) ∈ Λ1 for any ` ∈ Λ1 and the function L(z) = F (z + `) − F (z) is constant. It follows that F 0(z + `) = F 0(z), so F 0 is holomorphic and elliptic, but this means by Proposition 1.1.4 that F 0(z) = a for some constant a. Hence F (z) = az + b as claimed.

Corollary 1.2.8. Any holomorphic map f : C/Λ1 → C/Λ2 is, up to translation, a group homomorphism. In particular, if f(0) = 0 then f is a homomorphism.

Corollary 1.2.9. For any elliptic curve E, the group of endomorphisms End(E) has rank at most 2.

Proof. Viewing E(C) = C/Λ for some Λ = Z + Zτ, we get End(E) = {f : E → E | f is an isogeny} = {f : C/Λ → C/Λ | f is holomorphic and f(0) = 0} by Corollary 1.2.8 = {z ∈ C | zΛ ⊆ Λ} = {z ∈ C | z(Z + Zτ) ⊆ (Z + Zτ)} ⊆ Z + Zτ. Hence rank End(E) ≤ 2. It turns out that there are two possible cases for the rank of End(E), breaking down as follows:

End(E) = Z.

End(E) is an order O in some imaginary quadratic number ﬁeld K/Q. In this case, E is said to have complex multiplication.

13 1.3 The Classical Jacobian 1 Classical Jacobians

1.3 The Classical Jacobian

For the isomorphism ϕ : C/Λ → E(C) in Theorem 1.2.6, let ψ = ϕ−1 : E(C) → C/Λ be the inverse map. To understand this map explicitly, we will show how to construct a torus for ∼ every elliptic curve, i.e. ﬁnd a lattice Λ ⊆ C such that C/Λ = E(C).

Lemma 1.3.1. Any lattice Λ ⊆ C can be written Z P Λ = dz : P ∈ Λ . 0

Notice that each differential form dz on C satisfies d(z + `) = dz for all ` ∈ Λ by Lemma 1.3.1. Thus dz descends to a differential form on C/Λ, which by abuse of notation we will also denote by dz. Formally, this is the pushforward of dz along the quotient π : C → C/Λ. This implies:

Lemma 1.3.2. Any lattice Λ ⊆ C can be written Z Λ = dz : γ is a closed curve in C/Λ passing through 0 . γ

For an elliptic curve E defined by the equation y2 = f(x), fix a holomorphic differential form ω on E(C). (In general, the space of holomorphic differential forms on a curve has dimension equal to the genus of the curve, so in the elliptic curve case, there is exactly one such ω, up to scaling.)

Deﬁnition. The lattice of periods for an elliptic curve E is Z Λ = ω : γ is a closed curve in E passing through P γ

where P ∈ E(C) is ﬁxed.

Example 1.3.3. Under the map ϕ : C/Λ → E(C), z 7→ (x, y) = (℘(z), ℘0(z)), we see that dx = ℘0(z) dz = y dz

dx dx 2 so ω = y is a differential form on E(C). In fact, ω = f 0(x) , where E is defined by y = f(x), is holomorphic because f 0(x) 6≡ 0. This differential form is also holomorphic at O = [0, 1, 0], so up to scaling, this is the unique holomorphic form on E.

Historically, mathematicians were interested in studying solutions to elliptic integrals, or integrals of the form Z dx √ . ax3 + bx + c When f(x) = ax3 + bx + c, the expression ω = √ dx is precisely the holomorphic ax3+bx+c diﬀerential form deﬁning the lattice of periods of the elliptic curve E : y2 = f(x).

14 1.3 The Classical Jacobian 1 Classical Jacobians

For a more functorial description, let VE = Γ(E, ΩE) be the space of all holomorphic diﬀerential forms on E. If γ is a curve in E(C), there is an associated linear functional ∗ ϕγ ∈ VE deﬁned by

ϕγ : VE −→ C Z ω 7−→ ω. γ

Fixing the basepoint O ∈ E(C), the lattice of periods for E can be written

Λ = {ϕγ : γ ∈ π1(E(C),O)}.

∗ In other words, this deﬁnes a map π1(E(C),O) → VE , γ 7→ ϕγ.

∗ Deﬁnition. The Jacobian of an elliptic curve E is the quotient J(E) = VE /Λ.

For each point P ∈ E(C), the coset ϕγ + Λ is an element of the Jacobian, where γ is a path from O to P . This deﬁnes an injective map i : E,→ J(E).

Proposition 1.3.4. Suppose σ : E1 → E2 is an isogeny between elliptic curves, so that σ(O1) = O2. Then there is a map τ : J(E1) → J(E2) making the following diagram commute: σ E1 E2

i1 i2

J(E ) J(E ) 1 τ 2

∗ ∗ Proof. The pullback gives a contravariant map σ : VE2 → VE1 , ω 7→ σ ω = ω ◦ σ. Taking ∗∗ ∗ ∗ ∗∗ ∗ the dual of this gives a linear map σ : VE1 → VE2 deﬁned by (σ ρ)(ω) = ρ(σ ω) for any ∗ ρ ∈ VE1 and ω ∈ VE2 . Taking ρ = ϕγ1 for a path γ1 in E1 gives Z Z ∗ ∗ ∗ ρ(σ ω) = ϕγ1 (σ ω) = σ ω = ω = ϕσ(γ1)ω. γ1 σ(γ1)

∗∗ Thus σ ϕγ1 = ϕσ(γ1). If γ1 is a closed curve through O1, then σ(γ1) is a closed curve passing

through O2 = σ(O1). Hence if ΛE1 , ΛE2 are the lattices of periods for E1,E2, respectively, ∗∗ ∗∗ we have σ (λE1 ) ⊆ ΛE2 . So σ factors through the quotients, deﬁning τ:

∗∗ ∗ ∗ τ = σ : VE1 /ΛE1 −→ VE2 /ΛE2 . It is immediate the diagram commutes.

Lemma 1.3.5. For any elliptic curve E, the inclusion i : E,→ J(E) induces an isomorphism

∗ i : π1(E,O) −→ π1(J(E), i(O)).

15 1.3 The Classical Jacobian 1 Classical Jacobians

Unfortunately, the construction of the Jacobian given so far is not algebraic so it would be hard to carry over to curves over an arbitrary ground ﬁeld. To construct Jacobians algebraically, we will prove Abel’s theorem:

Theorem 1.3.6 (Abel). Suppose Λ ⊆ C is a lattice with fundamental domain Π and take P P any set {ai} ⊂ Π such that there are integers mi ∈ Z satisfying mi = 0 and miai ∈ Λ. Then there exists an elliptic function f(z) whose set of zeroes and poles is {ai} and whose

orders of vanishing/poles are ordai f = mi.

Given a lattice Λ ⊆ C, we may assume Λ = Z + Zτ for some τ ∈ C with im τ > 0. Deﬁnition. The theta function for a lattice Λ is

∞ X 2 θ(z, τ) = eπi(n τ+2nz). n=−∞

2 2 One has |eπi(n τ+2nz)| = e−π(n im τ+2n im z) for any z ∈ C, which implies that the above series converges absolutely.

Proposition 1.3.7. Fix a theta function θ(z) = θ(z, τ). Then

(1) θ(z) = θ(−z).

(2) θ(z + 1) = θ(z).

(3) θ(z + τ) = e−πi(τ+2z)θ(z).

Properties (2) and (3) together say that θ(z) is what’s known as a semielliptic function. 1+τ For our purposes, this will be good enough. Notice that for z = 2 , we have 1 + τ 1 + τ θ = θ − + (1 + τ) 2 2 πi τ+2 − 1+τ 1 + τ = e ( ( 2 ))θ − 2 1 + τ 1 + τ = eπiθ − = −θ . 2 2

1+τ Thus z = 2 is a zero of θ(z). 1+τ Lemma 1.3.8. All zeroes of θ(z, t) are simple and are of the form 2 + ` for ` ∈ Λ. (x) 1+τ (x) (x) Lemma 1.3.9. For x ∈ C, set θ (z, τ) = θ z − 2 − x . Then θ (z) = θ (z, τ) satisﬁes:

(1) θ(x)(z + 1) = θ(x)(z).

(2) θ(x)(z + τ) = e−πi(2(z−x)−1)θ(x)(z).

We now prove Abel’s theorem (1.3.6).

16 1.3 The Classical Jacobian 1 Classical Jacobians

Proof. Given such a set {ai} ⊂ Π, let x1, . . . , xn be the list of all ai with mi > 0, listed with repetitions corresponding to the number mi. For example, if m1 = 2 then x1 = x2 = a1. Likewise, let y1, . . . , yn be the list of all ai with mi < 0, once again with repetitions. By the P hypothesis mi = 0, there are indeed an equal number of each. Set

Qn (xi) i=1 θ (z) f(z) = n . Q (yi) i=1 θ (z) Then by Lemma 1.3.9, f(z + 1) = f(z). On the other hand, the lemma also gives

Qn (xi) i=1 θ (z + τ) f(z + τ) = n Q (yi) i=1 θ (z) 2πi Pn x −Pn y = e ( i=1 i i=1 i)f(z) P = e2πi miai f(z) X = f(z) since miai = 0.

Therefore f(z) is elliptic. Note that θ(z) is a meromorphic function, so by complex analysis, the integral

1 Z θ0(z) dz 2πi ∂Π θ(z) counts the number of zeroes of θ(z) in the fundamental domain Π, up to multiplicity. To ensure no zeroes lying on ∂Π are missed, we may shift Π → Πα for an appropriate α ∈ C. Parametrize ∂Π as in Proposition 1.1.5. Then once again the integrals along γ2 and γ4 cancel since θ(z + 1) = θ(z). On the other hand,

θ(z + τ) = e−πi(τ+2z)θ(z) =⇒ θ0(z + τ) = e−πi(τ+2z)(−2πiθ(z) + θ0(z)) θ0(z + τ) θ0(z) =⇒ = −2πi + . θ(z + τ) θ(z)

This implies

Z θ0(z) Z θ0(z) Z θ0(z) Z θ0(z) Z θ0(z) dz = dz + dz + dz + dz ∂Π θ(z) γ1 θ(z) γ2 θ(z) γ3 θ(z) γ4 θ(z) Z θ0(z) Z θ0(z) Z θ0(z) Z θ0(z) = dz + dz + dz + dz γ1 θ(z) γ3 θ(z) γ2 θ(z) γ4 θ(z) Z θ0(z) Z θ0(z) = dz − dz + 2πi + 0 γ1 θ(z) γ1 θ(z) = 2πi.

1+τ It follows that θ(z) has exactly one zero in Π, and it must be z = 2 .

17 1.3 The Classical Jacobian 1 Classical Jacobians

Deﬁnition. For a curve E (need not be elliptic), deﬁne: P A divisor on E is a formal sum D = nP P over the points P ∈ E, with nP ∈ Z. The abelian group of all divisors is denoted Div(E). P P The degree of a divisor D = nP P ∈ Div(E) is deg(D) = nP . The set of all degree 0 divisors is denoted Div0(E).

For a meromorphic function f on E(C) = C/Λ, the principal divisor associated to P f is (f) = degP P where nP = ordP f. The group of all principal divisors is denoted PDiv(E). The Picard group of E is the quotient group Pic(E) = Div(E)/ PDiv(E). The degree zero part of the Picard group is written Pic0(E) = Div0(E)/ PDiv(E).

The inverse map ψ : E → C/Λ extends to the group of divisors on E: Ψ : Div(E) −→ C/Λ X X nP P 7−→ nP ψ(P ).

Definition. The map Ψ : Div(E) → C/Λ is called the Abel-Jacobi map. Recall that ψ : P 7→ R ω + Λ ∈ /Λ where ω is a fixed holomorphic differential form γP C 0 0 on E and γP is a path connecting O ∈ E(C) to P . If O is another basepoint and ψ is the corresponding map, we have ψ(P ) = ψ(O0) + ψ0(P ) for all P ∈ E. So it appears that Ψ is 0 P not well-defined. However, this issue vanishes when we restrict Ψ to Div (E): if D = nP P is a degree 0 divisor, then X Ψ(D) = nP ψ(P )

X 0 0 = nP (ψ(O ) + ψ (P ))

0 X X 0 = ψ(O ) nP + nP ψ (P )

X 0 0 = 0 + nP ψ (P ) = Ψ (D).

0 0 ∼ Corollary 1.3.10. The map Ψ : Div (E) → C/Λ induces an isomorphism Pic (E) = C/Λ. Proof. One can prove that Ψ is a surjective group homomorphism. Moreover, Abel’s theorem (1.3.6) implies that ker Ψ = PDiv(E).

0 Consider the map iO : E → Div (E) that sends P 7→ P −O. This ﬁts into a commutative diagram:

Div0(E) Ψ

iO C/Λ

ψO E

18 1.4 Jacobians of Higher Genus Curves 1 Classical Jacobians

On the level of the Picard group, this diagram looks like

Pic0(E) Ψ

iO C/Λ

ψO E

and every arrow is a bijection.

1.4 Jacobians of Higher Genus Curves

Let C be a complex curve of genus g ≥ 2 and let V = Γ(C, ΩC ) be the vector space of ∗ ∼ g holomorphic differential forms on C. Then dimC V = g, so V = C . As in the previous R ∗ section, for any path ω in C the assignment ϕγ : ω 7→ γ ω defines a functional ϕγ ∈ V . As for elliptic curves, we define:

Deﬁnition. The lattice of periods for C is

∗ Λ = {ϕγ ∈ V | γ is a closed curve in C}.

Lemma 1.4.1. Λ is a lattice in V ∗.

Deﬁnition. The Jacobian of C is the quotient space J(C) = V ∗/Λ.

As with elliptic curves, we have a map ψ : C → J(C) called the Abel-Jacobi map, which sends P 7→ ϕγP + Λ, where γP is a curve through P . Also, ψ extends to the divisor group of C as a map Ψ : Div(C) −→ J(C) which is canonical when restricted to Div0(C). The Abel-Jacobi theorem generalizes Theo- rem 1.3.6 and Corollary 1.3.10.

Theorem 1.4.2. Let C be a curve of genus g > 0 and let Ψ : Div0(C) → J(C) be the Abel-Jacobi map. Then

(1) (Abel) ker Ψ = PDiv(C).

(2) (Jacobi) Ψ is surjective.

Therefore Ψ induces an isomorphism Pic0(C) ∼= J(C).

0 Just as with elliptic curves, if we ﬁx a basepoint O ∈ C, the map iO : C → Div (C),P 7→ P − O determines a commutative diagram

19 1.4 Jacobians of Higher Genus Curves 1 Classical Jacobians

Pic0(C) Ψ

iO J(C)

ψO C

However, this time not every map is a bijection. In particular, dim C = 1 < g = dim J(C). To remedy this, let Cg be the g-fold product of C and consider the map

ψg : Cg −→ J(C)

(P1,...,Pg) 7−→ ψ(P1) + ... + ψ(Pg) where + denotes the group law on J(C).

Theorem 1.4.3 (Jacobi). ψg : Cg −→ J(C) is surjective.

There is still work to do to show that the natural map Cg → Pic0(C) is surjective. It turns out that J(C) is birationally equivalent to the symmetric power C(g) = Cg/ ∼, where (P1,...,Pg) ∼ (Pσ(1),...,Pσ(g)) for any permutation σ ∈ Sg. Jacobi proved that this birational equivalence is enough to endow Pic0(C) ∼= J(C) with the structure of an algebraic group.

Theorem 1.4.4. J(C) is an abelian variety.

20 2 Curves

2 Curves

Let k be an algebraically closed ﬁeld. Deﬁnition. An algebraic curve over k is an algebraic variety X of dimension 1 which is irreducible, nonsingular and complete.

Denote by k(X) the function field of X, so that tr degk k(X) = dim X = 1. We may then think of k(X) as a finite extension of k(t) = k(P1). Conversely, we have: Proposition 2.0.1. Every field extension K/k of transcendence degree 1 is the function field K = k(X) for some algebraic curve X over k. Moreover, the correspondence {algebraic curves} −→ {transcendence degree 1 function fields K/k} X 7−→ k(X) is one-to-one.

Proof. For K/k with tr degk K = 1, define X to be the set of DVRs of K/k with the cofinite topology, i.e. a set of DVRs U is open if and only if X r U is finite. Define a structure sheaf T on X by U 7→ O(U) := O∈U O. Then one checks that (X, O) is an algebraic curve and the assignment X 7→ k(X) is a bijection. Therefore the study of algebraic curves is precisely the study of transcendence degree 1 fields over k. For X such a curve and a point P ∈ X, define the local ring at P to be the set of “rational functions at P ”: f O = ∈ k(X): g(P ) 6= 0 . P g Proposition 2.0.2. For any point P ∈ X,

n f o (a) OP is a local noetherian ring with maximal ideal mP = g ∈ OP : f(P ) = 0 .

(b) OP is regular and dim OP = 1.

n (c) OP is a DVR of k(X) with valuation vP (f) = n where f = tP u for mP = (tP ) and × u ∈ OP .

We call a generator tP of the maximal ideal mP a local uniformizer at P , or simply a uniformizer. Deﬁnition. Let f ∈ k(X),P ∈ X and r > 0. We say f has a pole of order r at P if vP (f) = −r, and a zero of order r at P if vP (f) = r. Lemma 2.0.3. Let X be an algebraic curve. Then (1) A function f ∈ k(X) is regular at a point P if and only if f has no pole at P . (2) Every nonconstant rational function f ∈ k(X) has a pole. (3) Each f ∈ k(X) has ﬁnitely many zeroes and poles.

21 2.1 Divisors 2 Curves

2.1 Divisors

Definition. For a variety X, a closed, irreducible subvariety of X with codimension 1 is called an irreducible divisor. The divisor group of X, denoted Div(X), is defined to be the free abelian group on the set of irreducible divisors of X. That is, elements of Div(X) are P formal linear combinations Z nZ Z where nZ ∈ Z and the sum is over irreducible divisors Z ⊂ X. For a curve X, an irreducible divisor of X is a (closed) point P ∈ X, so the divisor group may be written ( ) X Div(X) = D = nP P : nP ∈ Z and nP 6= 0 for finitely many P . P ∈X P P Definition. The degree of a divisor D = nP P is deg(D) = nP .

× P For a nonzero rational function f ∈ k(X) , the formal sum (f) = vP (f)P ∈ Div(X) is called the principal divisor associated to f. Lemma 2.1.1. The set of principal divisors PDiv(X) = {(f) ∈ Div(X): f ∈ k(X)×} is a subgroup of Div(X). Deﬁnition. The Picard group of a curve X is the quotient group Pic(X) = Div(X)/ PDiv(X). We say two divisors D,D0 ∈ Div(X) are linearly equivalent, written D ∼ D0, if [D] = [D0] in Pic(X), that is, if D = D0 + (f) for some f ∈ k(X)×. P For D = nP P , we say D is eﬀective, written D ≥ 0, if nP ≥ 0 for each P ∈ X. Then there is a natural partial ordering on Div(X): write D ≥ D0 if D − D0 ≥ 0. Proposition 2.1.2. For all principal divisors (f) ∈ PDiv(X), deg((f)) = 0. Proof. First, if f is constant then (f) = 0 since X is complete. Otherwise, we may view f 1 −1 −1 1 as a function f : X → Pk. Then (f) = f (0) − f (∞) for the points 0, ∞ ∈ Pk, since

−1 X ∗ f (P ) = vQ(f tP )Q. Q∈X

But on the other hand, this means deg(f) = deg(f −1(0)−f −1(∞)) = [k(X): k(f)]−[k(X): k(f)] = 0.

This shows that the map deg : Div(X) → Z induces a well-defined degree map on Pic(X). Moreover, if Div0(X) denotes the set of divisors of degree 0, then PDiv(X) ⊆ Div0(X) so we may define the degree 0 Picard group Pic0(X) = Div0(X)/ PDiv(X). Definition. Let D be an effective divisor on X. The Riemann-Roch space for D is the k-vector space L(D) = {f ∈ k(X)× : D + (f) ≥ 0} ∪ {0}.

Write `(D) = dimk L(D).

22 2.2 Sheaves and Cohomology 2 Curves

Deﬁnition. For D ∈ Div(X), the projective space

0 0 0 ∼ |D| := {D ∈ Div(X): D ∼ D and D ≥ 0} = P(L(D)) is called the complete linear system of D. Any projective subspace of |D| is called a linear system. Note that `(D) = dim |D| + 1. Remark. The spaces L(D) are useful for understanding maps to projective space. If |D| ⊆ PN and δ ⊆ |D| is a linear system, deﬁne the base locus of δ: n X o B(δ) = P ∈ X : nP 6= 0forall nP P ∈ δ .

N Then a basis f0, . . . , fN for L(D) determines a rational map ϕδ = (f0, . . . , fN ): X 99K P which is regular on X r B(δ). Thus those δ for which B(δ) = ∅ (called basepoint free linear systems) are important: they determine morphisms X → PN . N Theorem 2.1.3. For a divisor D ∈ Div(X), the map ϕ|D| : X → P is an embedding if and only if `(D − P − Q) < `(D − P ) < `(D) for all P,Q ∈ X,P 6= Q.

2.2 Sheaves and Cohomology

In this section we briefly review the notation for sheaves on curves and their cohomology. A good reference for proofs of the results in this section is Serre’s Faisceaux Algébriques Cohérents (FAC). Let X be an algebraic curve over k and let Top(X) denote the small category on X, whose objects are open sets U ⊆ X and whose morphisms are inclusions V,→ U. Definition. A presheaf (of abelian groups) on X is a contravariant functor Top(X)op → AbGps. Let F be a presheaf on X. For every inclusion of open sets V,→ U in X, there is an induced map fU,V : F(U) → F(V ) which is called a restriction; we write s|V for the image fU,V (s) of any s ∈ F(U). Definition. A sheaf (of abelian groups) on X is a presheaf F : Top(X)op → AbGps satisfying the following conditions:

(a) For every open U ⊆ X and every cover {Ui} of U, if s ∈ F(U) such that s|Ui = 0 for all i, then s = 0.

(b) For every open U ⊆ X and every cover {Ui} of U, if si ∈ F(Ui) such that si|Ui∩Uj =

sj|Ui∩Uj for all i, j, then there exists a unique s ∈ F(U) such that s|Ui = si for all i. Remark. Equivalent to the notion of a sheaf on a space X is the notion of an ´espace ´etale, which is, loosely, an assignment x 7→ Fx to every x ∈ X an abelian group Fx, along with et ` a projection map π : F = x∈X Fx → X sending any f ∈ Fx to x, which satisfy the following conditions:

23 2.2 Sheaves and Cohomology 2 Curves

(a) For all f ∈ F et, there are open neighborhoods V ⊆ F et of f and U ⊆ X of π(f) such that π|V : V → U is a homeomorphism. (b) Let (F et × F et)0 = π∗(F et × F et) := {(f, g) ∈ F et × F et : π(f) = π(g)}. Then the maps (F et × F et)0 → F et, (f, g) 7→ f + g and F et → F et, f 7→ −f are continuous.

One may deﬁne sheaves with values in other categories, e.g. sheaves of rings, sheaves of modules, sheaves of vector spaces, analogously.

Example 2.2.1. Fix an abelian group G, considered with the discrete topology. Then the constant sheaf on X associated to G is the sheaf U 7→ FG(U) where FG(U) is the set of ∼ continuous functions U → G. Note that if U is connected, then FG(U) = G as abelian groups. Alternatively, a constant sheaf is given by its ´espace´etale x 7→ Fx = G for all x ∈ X. et ` Here, F = x∈X Fx is homeomorphic to X × G. Example 2.2.2. For an algebraic curve X, the assignment

f U 7→ O (U) = ∈ k(X)× : g(P ) 6= 0 for all P ∈ U X g

defines a sheaf OX on X, sometimes called the structure sheaf of X. In the éspaceétale construction, OX,P = OP , the local ring at P ∈ X. Note that for any open set U ⊆ X, T OX (U) = P ∈U OP .

Deﬁnition. For a point P ∈ X, the stalk at P of F is the direct limit FP := lim F(U) over −→ all U containing P .

Observe that for any P ∈ X, FP coincides with the abelian group associated to P in the ´espace´etaleof F.

Deﬁnition. For a sheaf F on X and an open set U ⊆ X, a section of F over U is an element s ∈ F(U).

Remark. Let Γ(U, F) denote the set of sections of the ´espace ´etaleover U, i.e. continuous et ∼ maps s : U → F for which π ◦ s = idU . Then Γ(U, F) = F(U) as abelian groups, so the terminology ‘section’ agrees with the topological one.

Deﬁnition. Let X be a curve and F, F 0, G sheaves on X.

G is a subsheaf of F if for every open U ⊆ X, G(U) is a subgroup of F(U).

Given a subsheaf G of F, the quotient sheaf is the sheaf F/G associated (by sheaﬁ- ﬁcation) to the presheaf U 7→ F(U)/G(U).

0 0 A morphism of sheaves ϕ : F → F consists of a morphism ϕU : F(U) → F (U) for every open U ⊆ X, such that for every inclusion of open sets V,→ U, the diagram

24 2.2 Sheaves and Cohomology 2 Curves

ϕU F(U) F 0(U)

(·)|V (·)|V

F(V ) F 0(V ) ϕV

commutes. Remark. In general, a morphism of sheaves ϕ : F → F 0 is controlled by the induced 0 morphisms on stalks ϕP : FP → FP . For example, ϕ is injective (resp. surjective) if and only if every ϕP is injective (resp. surjective). For a sheaf F on X, set H0(X, F) := F(X), called the global sections of F. Lemma 2.2.3. For any exact sequence of sheaves 0 → F 0 → F → F 00 → 0, there is an exact sequence of abelian groups

0 → H0(X, F 0) → H0(X, F) → H0(X, F 00).

This says that the functor F 7→ H0(X, F) is left exact. We will deﬁne sheaf cohomology as the right derived functors of H0(X, −). First, we need:

Proposition 2.2.4. For a variety X, the category of sheaves of OX -modules has enough injectives. Definition. Let X be a variety over k. The right derived functors of H0(X, −) are denoted Hi(X, −) and the ith cohomology of X with coefficients in F is the k-vector space i i i H (X, F). Set h (F) = dimk H (X, F). Remark. The sheaf cohomology groups Hi(X, F) correspond with the definition of cohomology coming from the Cechˇ complex C•(U, F) for covers U of X. When X is a curve, we have the following important result from FAC. Proposition 2.2.5. Let X be an algebraic curve and F a sheaf on X. Then Hi(X, F) = 0 for all i ≥ 2. P We now define a sheaf on X associated to a divisor D = nP P ∈ Div(X). For each open set U ⊆ X, set

× L(D)(U) = {f ∈ k(X) : vP (f) ≥ −nP for all P ∈ U}.

Then U 7→ L(D)(U) deﬁnes a sheaf of k-vector spaces on X, denoted L(D). In fact, L(D) is a subsheaf of the constant sheaf on X associated to k(X) (which by abuse of notation we will also write as k(X)). Note that H0(X, L(D)) = L(D), the Riemann-Roch space for D. Proposition 2.2.6. For any divisor D on X, Hi(X, L(D)) = 0 for i ≥ 2 and is a ﬁnite dimensional k-vector space for i = 0, 1.

25 2.2 Sheaves and Cohomology 2 Curves

Proof. First suppose D = 0. Then L(0) = OX is the sheaf of regular functions on X and since X is complete,

0 0 H (X, L(0)) = H (X, OX ) = OX (X) = k.

0 1 Hence h (L(0)) = 1. We will prove dimk H (X, L(0)) < ∞ later. For the rest, it’s enough to prove that the statements hold simultaneously for D and D0 = D + P , for any divisor D and any point P ∈ X. For any point Q ∈ X, we have L(D)Q ⊆ L(D + P )Q so we may regard L(D) as a subsheaf of L(D + P ). Consider the exact sequence of sheaves

0 → L(D) → L(D + P ) → F → 0 where F is the quotient sheaf. Then FQ = 0 for all Q 6= P , and for Q = P we get L(D + P )P 1 0 dimk FP = dimk = 1, so H (X, F) = 0 and H (X, F) = k. Thus the long exact L(D)P sequence in sheaf cohomology is

0 → L(D) → L(D0) → k → H1(X, L(D)) → H1(X, L(D0)) → 0.

This yields two short exact sequences:

0 → L(D) → L(D0) → A → 0 0 → B → H1(X, L(D)) → H1(X, L(D0)) → 0.

Applying the additive function dimk gives

0 dimk L(D) + dimk A = dimk L(D ) 1 0 1 dimk B + dimk H (X, L(D )) = dimk H (X, L(D))

so Hi(X, L(D)) and Hi(X, L(D0)) are ﬁnite dimensional simultaneously, for i = 0, 1.

Deﬁnition. The arithmetic genus of X is the dimension

1 1 g = dimk H (X, L(0)) = dimk H (X, OX ).

The following is a preliminary version of the Riemann-Roch theorem, which we will prove in full generality at the end of the chapter.

Theorem 2.2.7 (Riemann-Roch, First Version). For an algebraic curve X and a divisor D ∈ Div(X), `(D) − h1(L(D)) = 1 − g + deg(D).

Proof. First suppose D = 0. Then `(0) = dimk L(0) = 1 since (f) ≥ 0 is equivalent to f being regular on X, which is equivalent to f being constant since X is complete. Also, deg(0) = 0 so we get 1 − g on both sides of the equation. Now let P ∈ X. To prove the theorem in general, it will be enough to show the formula holds for any D and D + P simultaneously. On the left, we have `(D)−h1(L(D)) = h0(L(D))−h1(L(D)) = χ(X, L(D)), the Euler characteristic of the sheaf L(D) on X. Set χ(D) := χ(X, L(D)). Then for any P ∈

26 2.3 R´epartitions 2 Curves

X, we have deg(D+P ) = deg(D)+1, so in particular 1−g+deg(D+P ) = 1−g+deg(D)+1. Thus it’s enough to show χ(D + P ) = χ(D) + 1. As in the proof of Proposition 2.2.6, consider the exact sequence of sheaves

0 → L(D) → L(D + P ) → F → 0

where F is the quotient sheaf. Then as before, h1(X, F) = 0 and h0(X, F) = 1. Consider the long exact sequence in sheaf cohomology:

0 → L(D) → L(D + P ) → H0(X, F) → H1(X, L(D)) → H1(X, L(D + P )) → 0.

Applying dimk gives

`(D) − `(D + P ) + h0(F) − h1(L(D)) + h1(L(D + P )) = 0 =⇒ `(D) − `(D + P ) + 1 − h1(L(D)) + h1(L(D + P )) = 0,

which is equivalent to χ(D) + 1 = χ(D + P ). Hence the Riemann-Roch equality holds for all D. This gives us a reasonable tool for calculations, but to obtain the most general form of Riemann-Roch, we will need to interpret h1(L(D)). It turns out that there is a ‘canonical divisor’ K on X for which H1(X, L(D)) ∼= L(K − D) for all divisors D.

2.3 R´epartitions

To interpret h1(L(D)) in Theorem 2.2.7, we want to find a ‘canonical divisor’ for which h1(L(D)) = `(K − D) for all divisors D on X. We first interpret H1(X, L(D)) using répartitionsand then relate the dual of this space to the space of meromorphic differentials on our curve. Finally, this will allow us to define the canonical divisor.

Definition. A répartition on X is a family {rP }P ∈X with rP ∈ k(X) and such that rP 6∈ OP for finitely many P . The set of all répartitions on X is denoted R. Note that R is a k-algebra.

Remark. R may also be identified as the ring of adèles of k, as in adèlicnumber theory. P For a divisor D = nP P ∈ Div(X), define the subspace

R(D) = {{rP } : vP (rP ) ≥ nP } ⊆ R.

As with the Riemann-Roch spaces, we have R(D0) ⊇ R(D) when D0 ≥ D. Also, note that [ R = R(D). D∈Div(X)

As with divisors, there is a natural embedding k(X) ,→ R, f 7→ {f}.

Proposition 2.3.1. For any D ∈ Div(X), H1(X, L(D)) ∼= R/(R(D) + k(X)).

27 2.3 R´epartitions 2 Curves

Proof. Regard L(D) as a subsheaf of the constant sheaf k(X) on X. This determines a short exact sequence of sheaves 0 → L(D) → k(X) → Q → 0 where Q is the sheaﬁﬁcation of the presheaf k(X)/L(D). The long exact sequence in cohomology is

H0(X, k(X)) → H0(X, Q) → H1(X, L(D)) → H1(X, k(X)) → · · ·

Then we have H0(X, k(X)) ∼= k(X) since k(X) is a constant sheaf, and H1(X, k(X)) = 0, either by computing using Cechˇ cohomology or using the fact that k(X) is a ﬂasque sheaf. In any case, the long exact sequence becomes:

k(X) → H0(X, Q) → H1(X, L(D)) → 0.

Moreover, Q has finite support (it vanishes off of a finite set, namely the support of D), so it is a skyscraper sheaf and thus

0 ∼ M ∼ H (X, Q) = (k(X)/L(D))P = R/R(D) P ∈X

Now since k(X) is a ﬁeld, the natural k-algebra map k(X) → R/R(D), f 7→ {f} is injective, so by exactness we get H1(X, L(D)) ∼= R/(R(D) + k(X)) as desired. Set I(D) = R/(R(D) + k(X)) and let J(D) = I(D)∗ denote the algebraic dual. By duality, we have J(D0) ⊆ J(D) if D0 ≥ D. Set [ J = J(D). D∈Div(X)

For f ∈ k(X) and α ∈ J, deﬁne a map

fα : R −→ k r 7−→ α(fr).

Since α vanishes on k(X), fα vanishes on k(X) as well. Moreover, if α ∈ J(D) and f ∈ L(D0), then fα vanishes on R(D − D0) and hence fα ∈ J(D − D0) ⊆ J. This shows that J has the structure of a k(X)-vector space.

Proposition 2.3.2. dimk(X) J ≤ 1. Proof. Suppose α, α0 ∈ J are k(X)-linearly independent. We may ﬁnd a divisor D ∈ Div(X) 0 such that α, α ∈ J(D). Set d = deg(D). For each n ≥ 0, pick some divisor ∆n of degree 0 0 n. If f ∈ L(∆n) then by the above, fα, fα ∈ J(D − ∆n). Moreover, if fα + gα = 0 for g ∈ L(∆n), then we must have f = g = 0. This deﬁnes an injection

L(∆n) × L(∆n) −→ J(D − ∆n) (f, g) 7−→ fα + gα0.

28 2.4 Meromorphic Diﬀerentials 2 Curves

Thus dim J(D − ∆n) ≥ 2`(∆n), but by the ﬁrst form of Riemann-Roch (Theorem 2.2.7),

dim J(D − ∆n) = dim I(D − ∆n)

= − deg(D − ∆n) + g − 1 + `(D − ∆n)

= −(d − n) + g − 1 + `(D − ∆n).

In particular, when n > d, we have deg(D − ∆n) < 0 so `(D − ∆n) = 0. Hence the Riemann-Roch formula becomes

dim J(D − ∆n) = −(d − n) + g − 1 = n + (g − 1 − d).

On the other hand, by Riemann-Roch,

2`(∆n) ≥ 2(1 − g + deg(∆n)) = 2(1 − g + n) = 2n + (2 − 2g),

so dim J(D − ∆n) ≥ 2`(∆n) implies that n + (g − 1 − d) ≥ 2n + (2 − 2g). Taking n large enough gives a contradiction, so α, α0 must be linearly independent.

In fact, it will follow from our work in the next section that dimk(X) J = 1.

2.4 Meromorphic Diﬀerentials

Let X be a curve.

Deﬁnition. The space of meromorphic diﬀerential forms on X is the k-algebra generated by symbols df for each f ∈ k(X), subject to the relations dfg = gdf − fdg, written

Ω(X) = hdf | f ∈ k(X)i/(dfg − gdf − fdg).

The set Ω(X) has a natural k(X)-vector space structure.

Lemma 2.4.1. For any curve X, dimk(X) Ω(X) = 1.

For each P ∈ X, let Ω(X)P be the meromorphic differentials at P , i.e. the algebra generated by df for f ∈ OP , subject to the same relations as above. For any differential ω ∈ Ω(X)P , we can write ω = f dt for some local uniformizer t at P . The divisor associated to a differential ω ∈ Ω(X) is then defined as X (ω) = ordP (ω)P ∈ Div(X) P ∈X

where ordP (ω) = ordP (f) if ω = f dt at P .

Lemma 2.4.2. The order ordP (ω) does not depend on the choice of uniformizer t. Definition. For a divisor D ∈ Div(X), define the meromorphic differentials at D by

Ω(D) = {ω ∈ Ω(X):(ω) ≥ D}.

29 2.4 Meromorphic Diﬀerentials 2 Curves

There is another useful local invariant of a meromorphic differential, which we define now. For P ∈ X, let ObP = k[[T ]] be the topological completion of the local ring OP , which comes equipped with a map OP → ObP , t 7→ T for a local uniformizer t; this map is an embedding. Then the fraction field of this ring is FbP := Frac(ObP ) = k((T )) so via the embedding just defined, we can view functions f ∈ OP as formal Laurent series in T ,

∞ X i f = aiT . i=−n

In analogy with complex analysis, we deﬁne:

Deﬁnition. The residue of ω ∈ Ω(X) at P ∈ X is resP (ω) := a−1 if ω = f dt and P∞ i f = i=−n aiT .

Fix P ∈ X and let (O, m, v) be the DVR of the local ﬁeld FbP and let Dk(FbP ) be the ring of k-diﬀerentials on FbP . Set ∞ \ n Q = m dO ⊆ Dk(FbP ) n=1

0 and deﬁne D = Dk(FbP )/Q.

P∞ i Lemma 2.4.3. Let f = i=−m ait ∈ k((T )), where t is a uniformizer at P , and set P∞ i−1 0 ft = i=−m iait , the formal derivative of f with respect to t. Then df = ft dt in D and dt is a basis for D0.

N N+1 Proof. It is enough to show df − ft dt ∈ m dO for all N. Write f = f0 + t f1 for some N f0 ∈ k((T )) and f1 ∈ O. Then ft = (f0)t + (N + 1)t g for some g ∈ O, so

N N+1 N+1 N df − ft dt = (N + 1)t f1 dt + t df1 − t g dt ∈ m dO.

0 Hence df − ft dt = 0 in D . Finally, note that t 7→ ft is a nontrivial derivation on FbP which vanishes on Q, so D0 6= 0. This proves that D0 has dimension 1 and is generated by dt.

Proposition 2.4.4. Let P ∈ X and let t be a uniformizer at P . Then

(1) resP is k-linear.

(2) resP (ω) = 0 for any ω ∈ Odt.

(3) resP (dg) = 0 for any g ∈ FbP .

dg (4) resP g = vP (g) for any g ∈ FbP .

Proposition 2.4.5. The residue resP (ω) is well-deﬁned, i.e. does not depend on the choice of uniformizer t.

30 2.4 Meromorphic Diﬀerentials 2 Curves

PN −n Proof. By Lemma 2.4.3, write ω = ω0 + n=−N ant dt for some ω0 ∈ Ω(X) with vP (ω0) ≥ 0. Let the residue of ω with respect to t be rest(ω) = a−1. If u is another uniformizer PN −n at P , deﬁne resu(ω) the same way. Then resu(ω) = n=−N anresu(t dt) but by (4) of dt −n Proposition 2.4.4, resu t = vP (t) = 1 so it suﬃces to show resu(t dt) = 0 for n ≥ 2. Let 1 n−1 −n g = − n−1 t , so that for n ≥ 2, t dt = dg. Then by (3) of Proposition 2.4.4, resu(dg) = 0, −n so resu(t dt) = 0 as required. P Theorem 2.4.6 (Residue Formula). For any ω ∈ Ω(X), P ∈X resP (ω) = 0.

Proof. We ﬁrst establish this for X = P1. In this case, k(X) = k(t) so for any diﬀerential 1 ω ∈ Ω(P ), ω = f dt for some f ∈ k(t). Since resP is k-linear by Proposition 2.4.4, we may assume that f = tn or f = (t − a)−n for some n > 0 and a ∈ k. If f = tn, the only pole of n 1 ω = t dt is at P = ∞. At ∞, u = t is a uniformizer, so we get

−(n+2) ω = f dt = −u du =⇒ res∞(f dt) = 0

by Proposition 2.4.4. Hence P res (ω) = res (ω) = 0. The other case is similar. P ∈P1 P ∞ Now let X be any algebraic curve and take a rational function ϕ ∈ k(X). This may be viewed as a morphism of curves ϕ : X → P1, corresponding to the ﬁeld extension 1 k(P ) = k(t) ,→ k(X). Set E = k(t) and F = k(X). Then the trace TrF/E gives a map on meromorphic diﬀerentials

1 Tr : Ω(X) −→ Ω(P ) ω 7−→ TrF/E(f) dϕ

1 if ω = f dϕ for f ∈ k(X). Take P ∈ P and Q ∈ X such that ϕ(Q) = P , and let EbP and FbQ be the complete local ﬁelds at P and Q, respectively. Then the valuations at these points are related by vQ = evP where e is the ramiﬁcation index of F/E. Further, there is an isomorphism ∼ Y F ⊗E EbP = FbQ Q→P where the product is over all Q ∈ X for which ϕ(Q) = P . Under this isomorphism, for any P f ∈ F , TrF/E(f) = Q→P TrQ(f), which implies X resP (Tr(f) dϕ) = resP (TrQ(f)) dϕ, Q→P

but resQ(f dϕ) = resP (Tr(f) dϕ), so we get X X X X resQ(ω) = resQ(ω) = resP (Tr(ω)) Q∈X P ∈P1 Q→P P ∈P1

which is 0 by the P1 case proven above. Therefore the residue formula holds for X.

31 2.5 Serre Duality and Riemann-Roch 2 Curves

2.5 Serre Duality and Riemann-Roch

Residues allow us to deﬁne a pairing

Ω(X) × R −→ k X (ω, r) 7−→ hω, ri := resP (rP ω) P ∈X

if r = {rP }. Note that if r ∈ k(X) ⊆ R, then by the residue theorem (2.4.6), hω, ri = 0 for all ω ∈ Ω(X). Further, if D is a divisor, r = {rP } ∈ R(D) and ω ∈ Ω(D), then rP ω ∈ Ω(X)P for all P ∈ X, since vP (rP ) ≥ −vP (D) and vP (ω) ≥ vP (D). Hence the pairing passes to the quotient I(D) = R/(R(D) + k(X)):

h·, ·i : Ω(D) × I(D) −→ k.

Consider the map θ : Ω(D) → J(D) deﬁned by ω 7→ hω, ·i. Lemma 2.5.1. Let D be a divisor on X and ω ∈ Ω(X). If θ(ω) ∈ J(D), then ω ∈ Ω(D).

Proof. If ω 6∈ Ω(D), there is a point P ∈ X for which vP (ω) < vP (D). Set n = vP (ω) + 1 and construct a r´epartition r = {rQ} by ( t−n, if Q = P rQ = 0, if Q 6= P.

Then vP (rP ω) = −1, so resP (rP ω) 6= 0. Thus hω, ri 6= 0 since the pairing is 0 everywhere else. However, r ∈ R(D) implies hω, ri = 0, a contradiction. Theorem 2.5.2 (Serre Duality). For any divisor D ∈ Div(X), θ : Ω(D) → J(D) is an isomorphism. Proof. Suppose θ(ω) = 0. Then θ(ω) ∈ J(D) for every D, so by Lemma 2.5.1, ω ∈ Ω(D) for every D. However, \ Ω(D) = 0 D∈Div(X)

since X is complete, so ω = 0. This proves θ is injective. On the other hand, dimk(X) Ω(X) = 1 and dimk(X) J ≤ 1 by Proposition 2.3.2, so the map Ω(X) → J must be surjective. Suppose α ∈ J(D). Then there is some ω ∈ Ω(X) with θ(ω) = α, but by Lemma 2.5.1, this means ω ∈ Ω(D). Therefore θ is surjective and hence an isomorphism. Let D be a divisor on X. By Theorem 2.2.7, we already have

`(D) − i(D) = deg(D) − g + 1

0 1 where `(D) = dimk L(D) = dimk H (X, L(D)) and i(D) = dimk Ω(D) = dimk H (X, L(D)). Let ω, ω0 ∈ Ω(X). Since Ω(X) has dimension 1, we may write ω = fω0 for f ∈ k(X). Then (ω) = (ω0) + (f), so in particular every meromorphic diﬀerential form on X belongs to the same nontrivial linear equivalence class in Pic(X), call it K.

32 2.5 Serre Duality and Riemann-Roch 2 Curves

Deﬁnition. The class K ∈ Pic(X) such that [(ω)] = K for any ω ∈ Ω(X) is called the canonical divisor class of X.

Fix a diﬀerential form ω0 ∈ Ω(X), so that K = [(ω0)]. For any D ∈ Div(X) and ω ∈ Ω(X), (ω) ≥ D is equivalent to (f) ≥ D − (ω0), ω = fω0 for f ∈ k(X). This proves:

Lemma 2.5.3. ω ∈ Ω(D) if and only if f ∈ L(K − D), where ω = fω0. Theorem 2.5.4 (Riemann-Roch). For an algebraic curve X with canonical divisor K and any divisor D ∈ Div(X),

`(D) − `(K − D) = 1 − g + deg(D).

Proof. Lemma 2.5.3 shows that i(D) = `(K − D) so this is just a reinterpretation of Theo- rem 2.2.7.

33 3 Maps From Curves

3 Maps From Curves

3.1 Moduli and Local Symbols

In this chapter, let k be an algebraically closed field and X a curve over k. Fix a finite subset of points S ⊆ X. Definition. A modulus on X supported on S, written m, is an assignment of a positive integer nP > 0 to each point P ∈ S. Equivalently, a modulus m may be identified with an P effective divisor P ∈X nP P supported on S. Definition. For a function g ∈ k(X) and a modulus m supported on S, saying (g) is prime to S, denoted g ≡ 1 mod m, will mean vP (1 − g) ≥ nP for all P ∈ S. In addition, saying g ≡ 1 mod m at P will mean vP (1 − g) ≥ nP . Let Div(X − S) denote the set of divisors of X that are prime to S. The goal in this chapter is to study maps f : X r S → G where G is a commutative algebraic group. Any such map extends to a map f : Div(X − S) → G by X X D = nP P 7−→ f(D) := nP f(P ). P ∈X P 6∈S For instance, this map is given on principal divisors by X (g) 7→ f((g)) = vP (g)f(P ). P 6∈S Definition. A modulus for a map f : X r S → G is a modulus m supported on S such that f((g)) = 0 for all functions g ≡ 1 mod m. Definition. Let f : XrS → G be a map to a commutative algebraic group. A local symbol × is the assignment of an element (f, g)P ∈ G for each g ∈ k(X) and P ∈ X satisfying: 0 0 (a) (f, gg )P = (f, g)P + (f, g )P .

(b) If P ∈ S, then (f, g)P = 0 for any g ≡ 1 (mod m) at P .

P × (d) P ∈X (f, g)P = 0 for all g ∈ k(X) . Proposition 3.1.1. Let f : X r S → G be a map and m any eﬀective divisor supported on S. Then m is a modulus for f if and only if there exists a local symbol associated to f and m. Further, if such a local symbol exists, it is unique. Proof. First suppose a local symbol exists for f, m. If g ≡ 1 mod m, we have X X f((g)) = vP (g)f(P ) = (f, g)P by (c) P 6∈S P 6∈S X = − (f, g)P by (d) P ∈S = 0 by (b).

34 3.1 Moduli and Local Symbols 3 Maps From Curves

Hence m is a modulus for f by definition. Conversely, suppose m is a modulus for f, which P may be viewed as an effective divisor m = nP P . We will define a local symbol (f, g)P × for all g ∈ k(X) . For any P 6∈ S, condition (c) forces us to define (f, g)P = vP (g)f(P ). × For P ∈ S, there is a function gP ∈ k(X) such that gP ≡ 1 mod m at any Q ∈ S r {P } and g ≡ 1 mod m at P , by the weak approximation theorem (see ANT). Define the local gP symbol at P ∈ S by X (f, g)P = − vQ(gP )f(Q). Q6∈S 0 × 0 0 Now if g, g ∈ k(X) with associated functions gP , gP as above, the product gP gP satisfies the weak approximation theorem for gg0. Thus

0 X 0 (f, gg )P = − vP (gP gP )f(Q) Q6∈S X X 0 = − vP (gP )f(Q) − vP (gP )f(Q) Q6∈S Q6∈S 0 = (f, g)P + (f, g )P .

So (a) is satisfied. Next, when g ≡ 1 mod m at P ∈ S, the function gP = 1 satisfies the P weak approximation theorem for g so by definition (f, g)P = − Q6∈S vP (1)f(Q) = 0. Thus (b) holds. Finally, note that X X X (f, g)P = − vQ(gP )f(Q) P ∈X P ∈X Q6∈S X X = − vQ(gP )f(Q) since G is associative Q6∈S P ∈X ! X Y = − vQ gP f(Q). Q6∈S P ∈X Q g P Set h = P ∈X gP and k = h . Then k ≡ 1 mod m so Q6∈S vQ(k)f(Q) = 0 by (b). This gives us X X 0 = (f, g)Q = − vQ(g)f(Q) by (c) Q6∈S Q6∈S X X = − vQ(g)f(Q) + vQ(k)f(Q) Q6∈S Q6∈S X g X = − v f(Q) = − v (h)f(Q) Q k Q Q6∈S Q6∈S X = (f, g)P P ∈X from above. Hence (d) is satisfied. Remark. If m, m0 are two moduli supported on S and m ≥ m0 (in the sense of effective divisors), then g ≡ 1 mod m0 implies g ≡ 1 mod m, which means a modulus for f is not unique. However, local symbols are unique by Proposition 3.1.1, so they must not depend on m.

35 3.1 Moduli and Local Symbols 3 Maps From Curves

Suppose θ : G → G0 is a homomorphism of commutative algebraic groups. × Lemma 3.1.2. Local symbols are natural, i.e. θ(f, g)P = (θ ◦ f, g)P for all g ∈ k(X) and P ∈ X. Proposition 3.1.3. If m is a modulus of f : X r S → G, then m is also a modulus of θ ◦ f : X r S → G0. Proof. Follows from Lemma 3.1.2.

For a morphism of curves π : X → X0, set π(S) = S0. Then π : X r S → X0 r S0 is a cover with pullback divisors ∗ 0 X π (P ) = eP P P →P 0 0 0 0 ∗ 0 P defined for every P ∈ X r S . Thus f(π (P )) = P →P 0 eP f(P ) is well-defined. Definition. The trace of a map f : X r S → G along π : X → X0 is the map 0 0 Trπ f : X r S −→ G P 0 7−→ f(π∗(P )). Proposition 3.1.4. If m is a modulus for f : X r S → G, then there is a modulus m0 for Trπ f and 0 X 0 (Trπ f, g )P 0 = (f, g ◦ π)P P →P 0 for any P 0 ∈ X0 and g0 ∈ k(X0)×. 0 Proof. It suffices to show that (Trπ f, g )P 0 satisfies the conditions of a local symbol, since 0 then Proposition 3.1.1 provides a modulus m for Trπ f. (a) We have

0 X 0 (Trπ f, gg )P 0 = (f, (gg ) ◦ π)P P →P 0 X X 0 = (f, g ◦ π)P + (f, g ◦ π)P P →P 0 P →P 0 0 = (Trπ f, g)P 0 + (Trπ f, g )P 0 . 0 0 0 0 0 (b) Suppose g ≡ 1 mod m at P . Then for any P → P , vP 0 (1 − g ) ≥ nP 0 and 0 0 0 0 0 vP (1−g ◦π) = eP vP 0 (1−g ) ≥ eP nP ≥ nP , so g ◦π ≡ 1 mod m at P . Hence (f, g ◦π)P = 0 0 0 0 for each P ∈ S and if P 6∈ S, vP (g ◦ π) = 0 so (f, g ◦ π)P = vP (g ◦ π)f(P ) = 0. This implies

0 X 0 (Trπ f, g )P 0 = (f, g ◦ π)P = 0. P →P 0 (c) Let P 0 6∈ S0. Then

0 X 0 X 0 −1 0 (Trπ f, g )P 0 = (f, g ◦ π)P = vP (g ◦ π)f(P ) since P 6∈ S = π (S ) P →P 0 P →P 0 X 0 0 X = eP vP 0 (g )f(P ) = vP 0 (g ) eP f(P ) P →P 0 P →P 0 0 0 = vP 0 (g ) Trπ f(P )

36 3.1 Moduli and Local Symbols 3 Maps From Curves

by deﬁnition of the trace. (d) Finally, for any g0 ∈ k(X0)×,

X 0 X X 0 X 0 (Trπ f, g )P 0 = (f, g ◦ π)P = (f, g ◦ π)P = 0 P 0∈X0 P →P 0 P 0∈X0 P ∈X

0 0 by (d) for the local symbol (f, −)P . Hence (Trπ f, −)P 0 is a local symbol on X r S . Let π : X → X0 be a ramiﬁed covering of curves and suppose f 0 : X0 r S0 → G0 is a morphism to a commutative algebraic group G0.

Definition. For g ∈ k(X), the norm of g is defined to be its norm in the field extension k(X)/π(k(X0)): Nπg := Nk(X)/π(k(X0))(g).

Proposition 3.1.5. Suppose f 0 : X0 r S0 → G0 has modulus m0. Then f 0 ◦ π has a modulus m with local symbol 0 X 0 (f ,Nπg)P 0 = (f ◦ π, g)P P →P 0 for all P 0 ∈ X0 and g ∈ k(X)×.

Proof. By Proposition 3.1.1, it’s enough to show the formula above defines a local symbol. Q Note that the norm map is continuous and satisfies a local-global principle N = P ∈X NP , 0 where NP is the norm of the extension of complete local fields at P → P . Thus if a 0 modulus m supported on S exists and g ≡ 1 mod m then Nπg ≡ 1 mod m . Moreover, P P for any g ≡ 1 mod m, if (g) = vP (g)P then π(g) = vP (g)π(P ). We must show P vP 0 (Nπg) = P →P 0 vP (g) for all g ≡ 1 mod m. If w1, . . . , wd are the distinct extensions of 0 0 0 vP from k(X )vP to k(X) ⊗k(X ) k(X )vP , then we have

d ! d Y X vP 0 (Nπg) = vP 0 Ni(g) = vP 0 (Ni(g)) i=1 i=1 d X = wi(g) by algebraic number theory i=1

= π(vP (g)).

0 0 0 0 0 So f ◦ π((g)) = f (Nπg) = 0 since m is a modulus for f and Nπg ≡ 1 mod m . This shows that if a modulus m supported on S exists, it is a modulus for f 0 ◦ π. Now assume P 0 6∈ S0 and P → P 0, so that P 6∈ S. Then

0 0 0 (f ,Nπg)P 0 = vP 0 (Nπg)f (P ) X 0 = vP (g)(f ◦ π)(P ) P →P 0 X 0 = (f ◦ π, g)P P →P 0

37 3.2 The Main Theorem 3 Maps From Curves

0 0 × g as desired. On the other hand, if P ∈ S , choose h ∈ k(X) such that h ≡ 1 mod m at all P → P 0 and h ≡ 1 mod m at all P ∈ S,P 6→ P 0, by weak approximation. Then by the same argument as above, Nπg ≡ 1 mod m0 at every P 0 ∈ S0 and N h ≡ 1 mod m0 at every Nπh π P 0 6∈ S0. Hence

0 0 (f ,Nπg)P 0 = (f ,Nπh)P 0 X = − (f, Nπh)Q0 by condition (d) for the local symbol Q06∈S0 X 0 = − (f ◦ π, h)P by the above Q06∈S0 X 0 = (f ◦ π, h)P by condition (d) again P →P 0 X = (f ◦ π, g)P . P →P 0 Therefore the local symbol formula holds for all P 0 ∈ X0.

3.2 The Main Theorem

Let G be a commutative, connected algebraic group and suppose f : X → G is a rational map, i.e. regular away from a ﬁnite set S ⊆ X. We will prove: Theorem 3.2.1. The map f has a modulus m supported on S.

Example 3.2.2. Let G = Ga = k, the additive group of the field. We can show Theo- rem 3.2.1 holds for any curve X over k and any map f : X → Ga. For this, it suffices to prove the existence of a local symbol (by Proposition 3.1.1), which we claim is given by f dg (f, g)P = resP g where resP denotes the residue at P , as in Section 2.4. For each P ∈ S, P f dg set nP = 1−vP (f) and define m = P ∈S nP P . Let’s verify that (f, g)P = resP g defines a local symbol for this map. (a) follows from the definition of resP . (b) If g ≡ 1 mod m at P , then vP (1 − g) ≥ nP and

vP (dg) = vP (d(1 − g)) = vP (1 − g) − 1 ≥ nP − 1 = −vP (f).

f dg f dg Thus, since vP (g) = 0, we have vP g ≥ 0 so resP g = 0 by Proposition 2.4.4. dg f dg (c) Let P 6∈ S. Then f is regular at P and g has a simple pole at P , so g has a simple pole at P as well. By (4) of Proposition 2.4.4, f dg dg res = f(P )res = f(P )v (g). P g P g P

(d) follows immediately from the residue formula (Theorem 2.4.6). Hence (f, g)P = f dg resP g is a local symbol and m is a modulus for f.

38 3.2 The Main Theorem 3 Maps From Curves

One consequence of this construction on G = Ga is that the local symbol (f, g)P = f dg resP g commutes with the Frobenius:

Proposition 3.2.3. If char k = p > 0, then for any map f : X r S → k, g ∈ k(X)× and P ∈ X, f p dg f dg p res = res . P g P g

p Proof. Let θ : Ga → Ga, x 7→ x be the Frobenius map. Then f factors through θ: f X r S Ga

Ga Apply Lemma 3.1.2.

× Example 3.2.4. Let G = Gm = k be the multiplicative group of the ﬁeld and suppose f : X 99K Gm is a rational map and S is the ﬁnite set of points of X at which f is not P regular, i.e. the set of zeroes and poles of f. We claim m = P ∈S P is a modulus for f, that is, nP = 1 for all P ∈ S works. To show this, we show the assignment f n (f, g) = (−1)mn (P ) where n = v (g), m = v (f) P gm P P

f n is a local symbol for f. Note that vP gm = mn − mn = 0 so this symbol is well-deﬁned. Also, (f, g)P satisﬁes the following properties:

(i)( f, g)P (g, f)P = 1

(ii)( −f, f)P = 1

(iii) (1 − f, f)P = 1. P To prove this is a local symbol corresponding to the modulus m = P ∈S P , keep in mind that the abelian group structure on Gm is multiplicative with identity 1. 0 0 (a) becomes (f, gg )P = (f, g)P (f, g )P in multiplicative notation, which is clear from the deﬁnition of the symbol. −m (b) Suppose vP (1−g) ≥ nP = 1. Then vP (g) = 0 and g(P ) = 1, so (f, g)P = g(P ) = 1, the multiplicative identity in Gm. n (c) Take P 6∈ S. Then by the deﬁnition of S, vP (f) = 0 and we have (f, g)P = f(P ) = f(P )vP (g), the correct equality in multiplicative notation. × 1 (d) View g ∈ k(X) as a rational function g : X 99K P . If g ≡ a 6= 0 is constant, then −vP (f) −vP (f) for all P ∈ X,(f, g)P = g(P ) = a so we have

P Y Y −vP (f) −vP (f) − deg((f)) (f, g)P = a = a P ∈X = a P ∈X P ∈X

39 3.2 The Main Theorem 3 Maps From Curves

which is a0 = 1 by Proposition 2.1.2. So the formula holds for constant functions. More generally, any g ∈ k(X)× may be viewed as a branched cover g : X → P1. Set E = k(P1) = k(t) and F = k(X) so that g induces a ﬁeld extension F/E with norm map × × N = NF/E : F → E .

Lemma 3.2.5. Let t denote the identity map on E. Then for all P ∈ P1, Y (f, g)Q = (Nf, t)P . Q→P

× × Proof. Extend the deﬁnition of (f, g)P to the completions EbP = kd(t)P and FbQ = k[(X)Q. It is immediate that (f, g)P is still multiplicative in both arguments. By algebraic number theory, N = N = Q N where N = N . Thus we have F/E Q→P Q Q FbQ/EbP Y (Nf, t)P = (NQf, t)P Q→P

−1 so it suﬃces to show (f, g)Q = (NQf, t)P for all Q → P . Fix such a Q ∈ g (P ). By multiplicativity of the local norm NQ, we may assume f and g are uniformizers of FbQ and EbP , respectively, i.e. that vQ(f) = vP (g) = 1, FbQ = k((f)) and EbP = k((g)). Then (N f)n N f (N f, g) = (−1)nm Q (P ) = − Q (P ) Q P gm g

since vP (NQf) = vQ(f) = 1. On the other side, vQ(g) = e = [FbQ : EbP ], the ramiﬁcation index of Q/P , so viewing EbP ⊂ FbQ, we have f e f e (f, g) = (−1)em (Q) = (−1)e (Q). Q gm g

NQf e−1 Considering the minimal polynomial of f over EbP , we see that f e (P ) = (−1) so we get (f, g)Q = (NQf, t)P as desired.

0 1 × Y 0 Lemma 3.2.6. For every function f ∈ k(P ) , (f , t)P = 1. P ∈P1

0 Q nλ × Proof. Write f = µ λ∈k(t − λ) for µ ∈ k . Then

!nλ 0 Y (f , t)P = (µ, t)P (t − λ, t)P , λ∈k

(µ, t) = (µ(P ))vP (g) = µn and Q (µ, t) = 1. For the remaining pieces, if λ = 0 we have P P ∈P1 P ( n2 n2 −1,P = 0, ∞ (t, t)P = (−1) 1(P ) = (−1) = 1,P 6= 0, ∞.

40 3.2 The Main Theorem 3 Maps From Curves

Thus Q (t, t) = 1. On the other hand, for λ 6= 0, we have P ∈P1 P −λ, P = 0  n  1 nm (t − λ) λ ,P = λ (t − λ, t)P = (−1) m (P ) = t −1,P = ∞  1,P 6= 0, ∞, λ. Thus Q (t − λ, t) = 1 so we get Q (f 0, t) = 1. P ∈P1 P P ∈P1 P

These lemmas give us condition (d) and therefore show that Theorem 3.2.1 holds for Gm, since Y Y Y Y (f, g)Q = (f, g)Q = (Nf, g)P = 1. Q∈X Q→P P ∈P1 P ∈P1 × Proposition 3.2.7. Suppose f, g ∈ k(X) and Sf ,Sg are, respectively, their sets of zeroes and poles. If Sf ∩ Sg = ∅, then f((g)) = g((f)). nm f n Proof. Let (f, g)P = (−1) gm (P ) be the local symbol from Example 3.2.4. Then

Y vP (g) Y f((g)) = f(P ) = (f, g)P

P 6∈Sf P 6∈Sf 1 Y vP (f) Y Y and g((f)) = g(P ) = (g, f)P = P 6∈ Sg . (f, g)P P 6∈Sg P 6∈Sg

Also, for all P outside Sf ∪ Sg,(f, g)P = 1 so putting these together, we have Y Y Y 1 1 = (f, g) = (f, g) (f, g) = g((f)). P P P f((g)) P ∈X P ∈Sf P ∈Sg Hence f((g)) = g((f)). Proposition 3.2.8. If G is any commutative algebraic group and f : X rS → G is a regular 0 0 0 map, then for all ramiﬁed covers π : X → X , Trπ f is a regular map X r S → G. Proof. Let n = deg π and consider the n-fold symmetric product X(n) (see Section 3.3 for more details). We will see that X(n) is irreducible, normal and nonsingular if X is. Consider the pullback map π∗ : X0 −→ X(n) 0 X P 7−→ eP P. P →P 0 Set Y = X r S,Y 0 = X0 r S0 and deﬁne F : Y n −→ G n X (P1,...,Pn) 7−→ f(Pi). i=1 Clearly the sum on the right is invariant under the symmetric group action, so F descends to a map F 0 : Y (n) → G. Consider the diagram

41 3.2 The Main Theorem 3 Maps From Curves

Y Y n F π π∗ F 0 Y 0 Y (n) G

Trπ f

Since F 0 is regular by deﬁnition, it suﬃces to show π∗ is regular which we will do in Sec- tion 3.3.

× P Let g ∈ k(X) , m = P ∈S nP P and suppose vP (g) ≥ nP for all P ∈ S (this condition will 1 be denoted g ≡ 0 mod m). Then by Proposition 3.2.8, Trg f is a regular map P r{0} → G. P Proposition 3.2.9. Let m = P ∈S nP P be a modulus and f : X r S → G a regular map. × Then m is a modulus for f if and only if for all g ∈ k(X) such that g ≡ 0 mod m, Trg f is constant.

Proof. For a 6= ∞, let (g)a be the divisor of zeroes of g − a and let (g)∞ be the divisor of poles of g. Then for all a ∈ P1 r {0}, ! X Trg f(a) = f eP P = f((g)a). P →a We now proceed to the proof. 1 g ( =⇒ ) Suppose m is a modulus for f. For a ∈ P r {0, ∞}, write g − a = −a 1 − a and notice that since −a ∈ k×, we have g g v 1 − 1 − = v = v (g) ≥ n P a P a P P g so in particular 1 − a ≡ 1 mod m. Applying f to the divisor (g − a), we get g 0 = f((g − a)) = f 1 − = f((g) − (g) ) = f((g) ) − f((g) ), a a ∞ a ∞ 1 so f((g)∞) = f((g)a) = Trg f(a). As this holds for any a ∈ P r {0}, Trg f is constant. ( ⇒ = ) Suppose Trg f is constant for every g ≡ 0 mod m. Setting h = 1 − g, we have h ≡ 1 mod m and (h) = (g)1 − (g)∞, so

f((h)) = f((g)1 − (g)∞) = f((g)1) − f((g)∞) = Trg f(1) − Trg f(∞) = 0. Thus m is a modulus for f. Let us now move towards a proof of Theorem 3.2.1. We will prove it in the case that char k = 0; although it also holds when char k > 0, the proof is quite diﬀerent and can be found in Serre. Let f : X r S → G be a regular map, set r = dim G and take ω1, . . . , ωr to be a basis of degree 1 invariant diﬀerential forms on G, that is, a basis of the Lie algebra ∗ associated to G. Set αi = f (ωi) for each 1 ≤ i ≤ r. For P ∈ S, choose nP ≥ 0 such that P vP (αi) ≥ −nP for all 1 ≤ i ≤ r and set m = P ∈S nP P .

42 3.2 The Main Theorem 3 Maps From Curves

Theorem 3.2.10. m is a modulus for f.

Proof. Fix g ≡ 0 mod m. By Proposition 3.2.9 it is enough to show that the function 0 0 ∗ f = Trg f is constant. To do this, we will show (f ) ωi = 0 for all 1 ≤ i ≤ r. To see why this 0 ∗ 0 implies the trace function is constant, notice that 0 = (f ) ωi(P ) = ωi(f (P )) so the map of tangent spaces induced by f 0 is the zero map and thus f 0 is constant. 0 ∗ Our ﬁrst step is to show (f ) ωi = Trg ωi for all 1 ≤ i ≤ r. Let π : Y → X be a ﬁnite Galois cover and consider the Galois, separable map g ◦π : Y → P1. Since it is separable, the ∗ ∗ ∗ 0 ∗ ∗ ∗ induced map ω 7→ (g ◦ π) ω is injective. Thus if we can show that π g (f ) ωi = π g Trg ωi, 0 ∗ it will follow that (f ) ωi = Trg ωi. To do this, we observe that all lifts along π are Galois conjugate, or more generally: g Proposition 3.2.11. Suppose Y −→π X −→ X0 are Galois covers of curves with G = Gal(k(Y )/k(X0)) and H = Gal(k(Y )/k(X)) ≤ G. Then for all P ∈ X,

−1 0 X g (g(P )) = [k(X): k(X )]i π ◦ σj(Q) j

0 where [k(X): k(X )]i denotes the inseparable degree, π(Q) = P and {σj} is a transversal of G/H. Proof. Set h = g ◦ π and note that

−1 X X h (g(P )) = nσσ(Q) = nσ(Q) σ∈G σ∈G for a ﬁxed n ∈ N, since π is Galois. On the other hand, h(h−1(P 0)) = [k(Y ): k(X0)]P 0 for 0 0 −1 P −1 all P ∈ X , so we must have h (g(P )) = σ∈G σ(Q). Now to get g (g(P )), apply π: X π(h−1(g(P ))) = π ◦ σ(Q) σ∈G X X = π ◦ α ◦ σj(Q) α∈H j X X = π ◦ σj(Q) since α ∈ H α∈H j X = [k(Y ): k(X)] π ◦ σj(Q). j

On the other hand, π(h−1(g(P ))) = [k(Y ): k(X)]g−1(g(P )) so dividing by [k(Y ): k(X)], we get the result.

× P Corollary 3.2.12. For any f ∈ k(X) and g ∈ k(X) , (Trg f) ◦ g ◦ π = j f ◦ π ◦ σj. π g 1 0 P Applying this to our maps Y −→ X −→ P , we have f ◦ g ◦ π = j f ◦ π ◦ σj, so

∗ ∗ X ∗ ∗ π g Trg αi = σj π αi. j

43 3.3 A Word About Symmetric Products 3 Maps From Curves

On the other hand,

∗ ∗ 0 ∗ X ∗ ∗ ∗ X ∗ ∗ π g (f ) ωi = σj π f (ωi) = σj π (αi) j j

∗ ∗ 0 ∗ ∗ ∗ by definition of the αi. Thus π g (f ) ωi = π g Trg ωi as claimed. 0 ∗ 0 ∗ Finally, we use this to show each (f ) ωi = 0. We start by showing (f ) ωi has at most a simple pole at 0. Let t : P1 → P1 denote the identity map once again and suppose 0 ∗ m 0 ∗ v0((f ) ωi) = −m − 1 for m ∈ Z. Then v0(t (f ) ωi) = m − m − 1 = −1 so by definition m 0 ∗ 1 1 res0(t (f ) ωi) 6= 0. The function g : X → P induces a field extension k(P ) ,→ k(X), t 7→ g, in which m m m m 0 ∗ Tr(g αi) = Tr(t αi) = t Tr(αi) = t (f ) αi. By the residue formula (Theorem 2.4.6),

X m m m 0 ∗ resQ(g αi) = res0(Tr(g αi)) = res0(t (f ) ωi) 6= 0. Q→0

m For any Q → 0, if Q 6∈ S, g and αi are regular at Q, so g αi is regular at Q. If Q ∈ S, we have m vQ(g αi) = mvQ(g) + vQ(αi) ≥ mnQ − nQ = (m − 1)nQ since g ≡ 0 mod m. Since the residue formula holds, we must have m − 1 ≤ −1, which 0 ∗ means m ≤ 0, that is, (f ) ωi has at most a simple pole at 0. 0 1 0 ∗ By Proposition 3.2.8, f = Trg f is regular on P r {0}. It follows that (f ) ωi is regular 1 0 ∗ on P r {0}. Moreover, since v0((f ) ωi) ≥ −1 and the residue theorem tells us that

0 ∗ X 0 ∗ res0((f ) ωi) = resP ((f ) ωi) = 0, P ∈P1

0 ∗ 0 ∗ (f ) ω must be regular at 0 as well. Hence (f ) ωi = 0, which ﬁnishes the proof of Theo- rem 3.2.10.

3.3 A Word About Symmetric Products

Let X be an algebraic variety such that every finite subset of X is contained in an affine (n) n open subset (for example, this is true when X is projective). Let X = X /Sn denote the n-fold symmetric product of X. This can be given the structure of a variety. More generally: Proposition 3.3.1. If G is a finite group acting on a variety V such that every orbit of G is contained in any affine open subset of V , then the quotient V/G is a variety.

Proof. Let S1,...,Sk denote the orbits. Since each orbit is invariant under the G-action, we 0 0 0 can cover V by a collection of affine open sets {Ui } with Si ⊆ Ui . Set Ui = Ui /G. Then Ui 0 ∼ G is affine: if Ui = Spec Ai for a ring Ai, then Ui = Spec Ai , the affine scheme on the ring of G-invariants of Ai. It is easy to see that V/G is covered by these affine open sets Ui. T Finally, we must show the diagonal map ∆V/G is closed. It suffices to show ∆V/G i,j Ui × Uj is closed in Uk × U` for all k, `. Consider the commutative diagram

44 3.3 A Word About Symmetric Products 3 Maps From Curves

π V V/G

∆V ∆V/G

V × V V/G × V/G π × π

T 0 0 0 0 Since V is a variety, ∆V i,j Ui × Uj is closed in Uk × U` for all k, `. Since π : V → V/G T 0 0 is a ﬁnite map, (π × π) ◦ ∆V i,j Ui × Uj is closed in Uk × U`, but by commutativity, this T is precisely ∆V/G i,j Ui × Uj .

Remark. If dim V > 1, V/G may be a singular variety. For example, when X is a curve and dim X > 1, the symmetric product X(n) is always singular.

Proposition 3.3.2. Suppose X is a variety and n ≥ 2. Then

(a) If X is irreducible, so is X(n).

(b) If X is normal, so is X(n).

(n) Deﬁne the map δn : X → X which sends P 7→ nP := [P,P,...,P ]. This factors as n (n) δn = π ◦ ∆ : X → X → X . An important result for us in Chapter 5 will be:

(n) Proposition 3.3.3. When char k = 0, the map δn : X → δn(X) ⊆ X is a birational homeomorphism.

45 4 Curves with Singularities

4 Curves with Singularities

4.1 Normalization

To discuss singular curves, we introduce a process of “normalization” to get rid of singularities in some way. The goal will be to prove a Riemann-Roch theorem and describe a canonical divisor by way of meromorphic differentials on a singular curve. As a motivating example, consider the irreducible algebraic curve X0 defined by the equation y2 − x3 = 0 on an affine patch and compactified by adding a point O at infinity. Let O0 be the structure sheaf of X0, which has local rings

( 2 3 0 k[x , x ],P = O OP = k[x]P ,P 6= O.

Then k[x2, x3] is not a regular local ring (the dimension of the tangent space at P = O is 2), so X0 is singular at O. In general, let X0 be an irreducible algebraic curve with singular locus S0 and ﬁeld of 0 0 0 rational functions K = k(X ). For each Q ∈ X , let OQ denote the integral closure of OQ in K.

Lemma 4.1.1. For any irreducible algebraic curve X0,

0 (a) OQ is a ﬁnitely generated k-algebra for all Q ∈ X .

0 0 0 (b) [OQ : OQ] < ∞ with OQ = OQ if and only if Q is a nonsingular point of X .

0 (c) The assignment Q 7→ OP defines a sheaf of k-algebras on X . Definition. A normalization of X0 is an algebraic curve X and a finite rational morphism ϕ : X → X0 such that

(a) X is nonsingular and normal.

(b) ϕ is a bijection X r S → X0 r S0, where S = ϕ−1(S0). (c) ϕ is a birational equivalence.

If U 0 ⊆ X0 is an open affine subset, let A0 = k[U 0] be the corresponding coordinate ring and denote by A the integral closure of A0 in K, which by commutative algebra is also finitely generated over k. This allows us to explicitly normalize X0 on affine patches: replace each U 0 with an affine curve U corresponding to A and glue these U together using the same gluing information as an affine cover X0 = S U 0 to obtain X. For each U 0, there is a morphism U → U 0 induced by the inclusion A0 ,→ A which glue together to give a morphism ϕ : X → X0. Note that for an open V ⊆ U with p(V ) ∩ S = ∅, the projection p|V : V → p(V ) is a birational isomorphism. Lemma 4.1.2. The X just constructed is an algebraic curve. Moreover, if X0 is a projective curve then so is X.

46 4.1 Normalization 4 Curves with Singularities

0 T For any Q ∈ S , note that OQ = P ∈ϕ−1(Q) OP where OP is the local ring of X at P . Theorem 4.1.3. For every algebraic curve X0, there exists a normalization ϕ : X → X0.

Deﬁnition. The conductor of the normalization X → X0 is the sheaf c deﬁned as the annihilator of O/O0 as O0-modules, or equivalently, as the assignment of local rings

0 0 0 Q 7→ c := {f ∈ O | fO ⊆ O } = Ann 0 (O /O ). Q Q Q Q OQ Q Q

Given the singular locus S0 ⊆ X0, we deﬁne a relation R on S = ϕ−1(S0) by aRb if ϕ(a) = ϕ(b) in S0. Then normalization may be viewed as the assignment

{algebraic curves over k} −→ {nonsingular curves /k} × {ﬁnite sets} × {relations} X0 7−→ (X,S,R).

0 For any Q ∈ S , deﬁne the radical of OQ:

−1 rQ = {f ∈ OQ | f(P ) = 0 for all P ∈ ϕ (Q)}.

0 Then we have k + cQ ⊆ OQ ⊆ k + rQ ⊆ OQ and each quotient is ﬁnite dimensional over k. n Hence for some n ≥ 0, k + rQ ⊆ k + cQ. Example 4.1.4. For X0 deﬁned by y2 − x3 = 0 and the point Q = O, the chain above (with n = 2) is 2 0 k + rQ ⊆ k + cQ ⊆ OQ ⊆ k + rQ ⊆ OQ k + hxi2 ⊆ k[x2, x3] = k[x2, x3] ⊆ k + hxi ⊆ k[x].

For every singular curve X0, we have constructed a nonsingular normalization X → X0. Conversely, every nonsingular curve can be made into a singular curve by taking the quotient with respect to an equivalence relation at a finite set of points, giving an assignment in the opposite direction. Let (X,S,R) be a triple consisting of a nonsingular curve X, a finite set S ⊆ X and a relation R on S. Define X0 = (X r S) ∪ S/R. To make X0 into a singular 0 0 0 0 curve, we define a sheaf of rings O on X as follows. For Q ∈ X , set OQ = OQ if Q 6∈ S 0 n 0 0 and OQ to be any ring such that k + rQ ⊆ OQ ⊆ k + rQ ⊆ OQ. Next, we declare U ⊆ X to 0 0 T be open if its complement in X is finite. Set O (U) = Q∈U OQ. Lemma 4.1.5. (X0, O0) is an algebraic curve which is singular at every Q ∈ S/R.

Proof. The sheaf axioms for O0 are easily checked. To prove X0 is a curve, it suffices to 0 T 0 prove the case when X is affine, say X = Spec A. In this case let A = Q∈X0 OQ and set 0 0 0 Y = Spec A . It is clear that A ⊆ A. Moreover, by the chain condition on OQ, there exists an f ∈ A such that fA ⊆ A0 ⊆ A so in particular A0 is finitely generated as a k-algebra. Since A is an A0-algebra and is finitely generated, it follows by commutative algebra that A/A0 is an integral extension. Hence dim A0 = dim A = 1 so Y is a curve. In fact, since A is integrally closed (X is nonsingular), A is precisely the integral closure of A0 in its fraction field, which shows X = Spec A is the normalization of Y . Finally, one can show that X0 ∼= Y so X0 is also an affine curve, finishing the proof.

47 4.2 Singular Riemann-Roch 4 Curves with Singularities

Theorem 4.1.6. The assignment

{algebraic curves over k} −→ {nonsingular curves /k} × {ﬁnite sets} × {relations} X0 7−→ (X,S,R)

is a bijection.

Therefore we may treat singular curves formally as nonsingular curves equipped with ‘singular data’ in the form of S ⊆ X with a relation R on S.

Example 4.1.7. Let X be a complete, irreducible, nonsingular curve over k. For a modulus P m = nP P on X with support S = {P ∈ X | nP 6= 0} (we will assume |S| ≥ 2), deﬁne the singular curve X0 associated to the data (X,S,R) where R identiﬁes the entire set S to 0 0 a point Q ∈ X . Let cQ = {f ∈ K | f ≡ 0 mod m} and OQ = k + cQ, so that c is the 0 conductor of O in O. We denote the resulting singular curve by Xm, which has a unique singular point Q. 0 2 3 0 In our example X : y − x = 0, the singular curve X is isomorphic to Xm where X is the normalization of X0 and m = 2O.

4.2 Singular Riemann-Roch

Let X0 be a complete, irreducible algebraic curve over k with normalization X and let g be 0 P the genus of X. For each Q ∈ X, let δQ = dim(OQ/OQ) and set δ = Q∈S δQ. Also put π = g + δ. For a divisor D on X which is prime to S, let L(D) be the sheaf deﬁned by D as in Section 2.2. On X0, setting ( L(D) ,Q 6∈ S0 L0(D) = Q Q 0 0 OQ,Q ∈ S

deﬁnes a sheaf L0(D) since X r S → X0 r S0 is a birational isomorphism. In analogy with the nonsingular Riemann-Roch theory, deﬁne

L0(D) = H0(X0, L0(D)) I0(D) = H1(X0, L0(D)) `0(D) = h0(X0, L0(D)) = dim L0(D) i0(D) = h1(X0, L0(D)) = dim I0(D).

0 When X = Xm for a modulus m, we will also write these as Lm(D),Im(D), `m(D) and im(D), respectively.

Proposition 4.2.1. For any divisor D on X, `0(D) and i0(D) are ﬁnite.

0 Proof. Both L(D) and OQ are coherent sheaves, so this follows from the proof of Proposi- tion 2.2.6.

Proposition 4.2.2. For all q ≥ 0, Hq(X, O) = Hq(X0, O).

48 4.3 Diﬀerentials on Singular Curves 4 Curves with Singularities

Proof. See FAC. Theorem 4.2.3 (Riemann-Roch for Singular Curves). Let X0 be a singular curve with normalization X. Then for any divisor D on X prime to S,

`0(D) − i0(D) = deg(D) + 1 − π.

Proof. For a divisor D prime to S and any point P 6∈ S0, let Q0 = L0(D + P )/L0(D) be the cokernel sheaf in the exact sequence

0 → L0(D) → L0(D + P ) → Q0 → 0.

Then as in the proof of Theorem 2.2.7, Q0 is 0 away from P and has dimension 1 at P . Thus the same argument as in that proof shows that χ0(D + P ) = χ0(D) + 1, where χ0(D) = `0(D) − i0(D). Thus it suﬃces to prove the theorem for D = 0. In this case, we have L0(D) = O0 and since O0 is the pushforward of O along the normalization map X → X0, there is a short exact sequence

0 → O0 → O → O/O0 → 0.

Thus by additivity, χ0(O0) + χ0(O/O0) = χ0(O). Further, as in the proof of Theorem 2.2.7, O/O0 is a skyscraper sheaf concentrated on S0, so

0 0 0 X 0 X dim H (X , O/O ) = dim(OQ/OQ) = δQ = δ. Q∈S Q∈S

This shows that χ0(O0) + δ = χ0(O), but by the standard Riemann-Roch theorem, χ(O) = 1 − g so to ﬁnish, it suﬃces to observe χ0(O) = χ(O) by Proposition 4.2.2. Corollary 4.2.4. π = i0(0) = h1(X0, O0).

4.3 Diﬀerentials on Singular Curves

Once again, let X0 be a complete, irreducible curve over k with normalization X. For Q 6∈ S0, 0 0 let ΩQ = ΩQ = Dk(OX,Q), the ring of regular diﬀerential forms on X at Q. For Q ∈ S , set

0 0 0 ΩQ = {ω ∈ Dk(OX,Q) | hf, ωiQ = 0 for all f ∈ OQ}, P where hf, ωiQ = P ∈ϕ−1(Q) resP (fω) is the residue pairing at Q. Lemma 4.3.1. For any Q ∈ X0,

0 (a) ΩQ ⊆ ΩQ.

0 0 (b) hOQ, ΩQiQ = 0.

0 0 Thus the vector spaces OQ/OQ and ΩQ/ΩQ are dual to each other with respect to h·, ·iQ.

49 4.3 Diﬀerentials on Singular Curves 4 Curves with Singularities

0 P Example 4.3.2. When X = Xm for a modulus m = nP P , with unique singular point Q, 0 the condition hf, ωi = 0 for all f ∈ OQ is equivalent to h1, ωi = 0 and vP (ω) ≥ −nP for all P ∈ S.

Let ϕ : X → X0 be the normalization map and let g : X r S → X0 r S0 be the corresponding birational equivalence. For an algebraic group G and a map f : X r S → G, deﬁne the trace of f along g (as in Section 3.1) to be X Trg f(Q) = f(P ). P →Q

Extend this to a meromorphic diﬀerential ω on X by putting Trg ω(Q) = f where ω = f dtQ.

0 0 Proposition 4.3.3. Let ω be a meromorphic differential on X. Then ω ∈ ΩQ for all Q ∈ X 0 0 if and only if Trg ω = 0 for all g ∈ OQ and Q ∈ S . Now for a divisor D on X prime to S, set ( Ω(D) ,Q 6∈ S Ω0(D) = Q Q 0 ΩQ,Q ∈ S. This defines a sheaf of differentials Ω0(D) on X0 for which Ω0(D) = H0(X0, Ω0(D)). As with Serre duality, we have: Theorem 4.3.4. For any divisor D on X prime to S, Ω0(D) is isomorphic to the dual of I0(D) = H1(X0, L0(D)). Proof. Let R denote the répartitionsof X and define the subsets

0 R = {r ∈ R | rP1 = rP2 whenever ϕ(P1) = ϕ(P2)} ⊆ R 0 0 0 0 R (D) = {r ∈ R | vP (rP ) + vP (D) ≥ 0 for P 6∈ S and rP ∈ OP for P ∈ S} ⊆ R .

0 As in Section 2.4, Dk(K) is isomorphic to the dual of R/K, so for any ω ∈ Dk(K), ω ∈ Ω (D) 0 ∼ 0 ∗ if and only if ω|R0(D) = 0. Hence Ω (D) = (R/(k(X) + R (D))) . Consider the short exact sequence of sheaves 0 → L0(D) → k(X) → Q → 0 where k(X) is the constant sheaf and Q is the sheaﬁﬁcation of k(X)/L0(D). Note that since X0 is irreducible, H0(X0, k(X)) = k(X) and H1(X0, k(X)) = 0. Taking the long exact sequence in sheaf cohomology, we get k(X) → H0(X0, Q) → I0(D) → 0. As in the nonsingular case, Q is a skyscraper sheaf which allows us to write

0 0 ∼ M 0 ∼ 0 0 H (X , Q) = (k(X)/L (D))Q = R /R (D). Q∈X0 Further, k(X) → R0/R0(D) is injective since k(X) is a ﬁeld, so the second exact sequence gives us I0(D) ∼= R0/(k(X) + R0(D)). One ﬁnishes by showing R/(k(X) + R0(D)) and R0/(k(X) + R0(D)) have the same dual, from which it follows that Ω0(D) ∼= I0(D)∗.

50 4.3 Diﬀerentials on Singular Curves 4 Curves with Singularities

0 0 Corollary 4.3.5. i (D) = dimk Ω (D).

0 0 0 Corollary 4.3.6. If Ω is the vector space of regular diﬀerential forms on X , then dimk Ω = π = g + δ.

Assume Ω0 is a locally free sheaf on X0. Set K0 = K + c where c is the conductor of the map X → X0. Then K0 is a canonical divisor on X0 and we obtain the full Riemann-Roch theorem for singular curves:

Theorem 4.3.7. Suppose Ω0 is locally free. For any divisor D on X prime to S,

`0(D) − `0(K0 − D) = deg(D) + 1 − π.

51 5 Constructing Jacobians

5 Constructing Jacobians

Recall that every algebraic curve X over the complex numbers has a Jacobian J(X) = Cg/Λ where Λ is a full lattice in Cg. In this chapter, we construct Jacobians for more general algebraic curves, including so-called generalized Jacobians for singular curves.

5.1 The Jacobian of a Curve

Let X be a complete, nonsingular algebraic curve of genus g over a ﬁeld k. We will prove the following result:

Theorem. There exists an abelian variety J(X) of dimension g over k and a rational map ϕ : X → J(X) such that the group law of J(X) corresponds to a ‘rational group law’ on the gth symmetric power of X.

We adopt Weil’s conventions in Foundations of Algebraic Geometry (FAC):

Assume X is a projective algebraic curve which is irreducible and nonsingular over k.

Assume for the moment k is algebraically closed, although the results will hold in general.

Fix a universal domain Ω, which is an algebraically closed ﬁeld extension Ω/k of inﬁnite transcendence degree within which all other extensions we consider will be contained.

A point of X will be treated as a point with coordinates in Ω.

The field of definition k(P ) for a point P ∈ X is the finite subextension k(P )/k of Ω/k generated by the coordinates of P .

For a subfield Ω ⊇ L ⊇ k and a point P ∈ X, we will say P is L-rational if k(P ) ⊆ L. P Let D = nP P be a divisor on X and choose an affine open set U ⊆ X containing the support of D. Then U = Spec A for some Dedekind ring A/k. Set AΩ = A ⊗k Ω so that the Ω-points U(Ω) correspond to the ring AΩ. Then D corresponds to a fractional ideal a of AΩ P Q nP via nP P ↔ P . Lemma 5.1.1. There exists a smallest field extension K/k such that a is generated by polynomials with coefficients in K.

Definition. Such a field K is called the field of rationality of D, written K = k(D). For any field L/k, we say D is rational over L if k(D) ⊆ L.

Lemma 5.1.2. If D1,D2 ∈ Div(X) are rational over L/k, then D1 − D2 is also rational over L.

Proof. This follows from the fact that k(D1 − D2) ⊆ k(D1) ∩ k(D2).

52 5.1 The Jacobian of a Curve 5 Constructing Jacobians

Lemma 5.1.3. Let P1,...,Pn be independent generic points of X, i.e. k(P1,...,Pn)/k has P transcendence degree n. Then the divisor D = Pi has ﬁeld of rationality

k(D) = k(P1,...,Pn)s,

the subﬁeld of k(P1,...,Pn) ﬁxed by all permutations of {P1,...,Pn}.

Proof. Set K = k(P1,...,Pn)s. If σ is an automorphism of Ω/K, then σ(D) = D so D is rational over some purely inseparable extension of K. Since k(P1,...,Pn)/K is separable (it is Galois), D must be rational over K, i.e. k(D) ⊆ K. On the other hand, any automorphism σ ∈ Aut(Ω/k(D)) ﬁxes D and hence ﬁxes K, so K is purely inseparable over k(D). However, k(D)(P1,...,Pn)/k(D) is separable, so K/k(D) is also separable. Hence K = k(D). Take D ∈ Div(X). Following the conventions of Chapter 2, let L(D) = {f ∈ k(X) | (f)+D ≥ 0} be the Riemann-Roch space of D and set `(D) = dimk L(D). By the Riemann- Roch theorem (2.5.4), `(D) = deg(D) + 1 − g + `(C − D) where C is any divisor in the canonical class of X.

Lemma 5.1.4. Suppose `(C − D) 6= 0 and D is rational over some L/k. Then for any generic point P ∈ X which is L-rational, we have `(C − (D + P )) = `(C − D) − 1.

Proof. If this were not the case, Riemann-Roch would imply `(C − (D + P )) = `(C − D) but then every f ∈ L(C − D) vanishes at P , which is impossible since P is generic.

0 Lemma 5.1.5. Fix a degree 0 divisor D0 ∈ Div (X) and let P1,...,Pg be independent P generic points of X deﬁned over some L/k for which D0 is rational. Set D = Pi. Then P there is a unique eﬀective divisor of the form E = Qi, where Q1,...,Qg are independent generic points, such that E ∼ D0 + D. Moreover,

L(P1,...,Pg)s = L(Q1,...,Qg)s.

Proof. By the Riemann inequality, deg(D0) = 0 implies `(D0) = 0 or 1. Using Riemann- Roch and Lemma 5.1.4, if `(D0) = 0, then `(C − D0) = g − 1 and `(C − (D0 + D)) = 0. On the other hand, if `(D0) = 1, then `(C − D0) = g and `(C − (D0 + D)) = 0 once again. This means there is a unique effective divisor E in the complete linear system for D0 +D. Moreover, since D0 and D are rational over L(P1,...,Pg)s, Lemma 5.1.2 shows that D0 + D is as well. Thus E is rational over L. It follows that D and E have the same field of rationality over L, and since the transcendence degree of this extension is g, we must have P E = Qi for independent generic points Q1,...,Qn. The final statement about fixed fields is immediate from Lemma 5.1.3. Assume X has an effective divisor D with deg(D) = g which is rational over k. To construct the Jacobian of X, we define what Weil calls a normal composition law (see FAC) on the field k(D) = k(P1,...,Pg)s. Since k(D)/k has transcendence degree g, we may choose a model V/k for k(D), i.e. an open subset of Xg with function field k(D) and dim V = g. There is a rational map ϕ : X(g) → V on the symmetric product (see Section 3.3) such that

53 5.1 The Jacobian of a Curve 5 Constructing Jacobians

−1 0 P for any generic point q ∈ V , ϕ (q) can be uniquely identiﬁed with a divisor D = Pi for P P1,...,Pg independent generic points of X over k such that k ( Pi) = k(q). Thus we can P 0 write q = ϕ ( Pi) = ϕ(D ). P Now if P1,...,Pg,Q1,...,Qg are independent generic points of X/k, set D1 = Pi and P D2 = Qi. Then by Lemma 5.1.5, `(D1 + D2 − D) = 1 and D1 + D2 − D ∼ E for some P positive divisor E = Mi which is unique for D1,D2. Set q1 = ϕ(D1), q2 = ϕ(D2) and q3 = ϕ(E). We deﬁne the additive composition law on V by V × V −→ V

(q1, . . . , q2) 7−→ q1 + q2 := q3.

Proposition 5.1.6. Let q0 = ϕ(D0) for a divisor D0 ∈ Div(X) and let q1, q2, q3 as above. Then (1) V × V → V is a rational map which is rationally surjective.

(2) q1 + q2 = q2 + q1.

(3) q0 + (q1 + q2) = (q0 + q1) + q2.

Proof. (1) and (2) are straightforward. For (3), consider the divisor class [D0 +D1 +D2 −2D]. Then associativity follows easy. Theorem 5.1.7 (Weil, FAC). There is an algebraic group G/k and a birational isomorphism V → G such that (1) The group law on G is obtained from V by this map. (2) G is unique up to birational isomorphism. Thus for an effective divisor D ∈ Div(X) of degree deg(D) = g, we may choose a model over k for k(D) which is an algebraic group. Definition. The algebraic group thus defined is called the Jacobian of X, denoted J = J(X). Theorem 5.1.8. For a complete, nonsingular curve X/k of genus g ≥ 1, the Jacobian J = J(X) is an abelian variety of dimension g equipped with a rational map f : X → J such that for any field L/k over which f is defined, and for independent generic points P1,...,Pg ∈ X P over L, the point Q = f(Pi) is a generic point of J such that L(Q) = L(P1,...,Pg)s. Moreover, every generic point of J is of this form and the Pi are uniquely determined up to permutation. (g) P Remark. The morphism f : X → J factors through ϕ : X → J, (P1,...,Pg) 7→ f(Pi). Also, J satisfies a universal property f X J(X)

A for abelian varieties A. We will show this in Section 5.3 in the case of generalized Jacobians.

54 5.2 Generalized Jacobians 5 Constructing Jacobians

5.2 Generalized Jacobians

In this section, we generalize the construction of the ordinary Jacobian of a curve X to an abelian variety Jm, for any modulus m on X, which is universal among algebraic groups G admitting a morphism f : X → G for which m is a modulus. Fix a curve X, a modulus m with support S and assume k is algebraically closed and the points of S are k-rational.

Deﬁnition. We say two divisors D,D0 ∈ Div(X) prime to S are m-equivalent, written 0 0 D ∼m D , if D = D + (g) for some function g ≡ 1 mod m. For m = 0, we take ∼0 to be the ordinary linear equivalence relation on Div(X).

Lemma 5.2.1. For any modulus m, ∼m is an equivalence relation on Div(X) which forms a ﬁner partition than ∼.

0 Let Cm be the group of m-equivalence classes of divisors prime to S and let Cm be the subgroup of degree 0 divisor classes in Cm. Following the construction in Section 4.2, for a 0 divisor D ∈ Div(X) prime to S, let Lm(D) be the k-vector space Lm(D) = H (Xm, Lm(D)) for the sheaf ( 0 OQ,Q ∈ S Lm(D)Q = L(D)Q,Q 6∈ S.

0 0 0 Proposition 5.2.2. Fix D ∈ Div(X) prime to S. The set {D ∈ Div(X): D ≥ 0 and D ∼m D} is in one-to-one correspondence with P(Lm(D)) r P(L(D − m)). Proof. Such a divisor D0 = D + (g) corresponds to a function g ∈ k(X) satisfying g ≡ 1 0 mod m and (g) ≥ −D outside S. This means g ∈ OQ for the unique singular point Q ∈ Xm. On the other hand, suppose g ∈ Lm(D) and assume g 6= 0 on S. Then D + (g) ∼m D. Finally, g ∈ Lm(D) is nonzero on S if and only if g ∈ L(D − m) so we have the desired correspondence.

Lemma 5.2.3. Let D ∈ Div(X) be prime to S and suppose L/k is a ﬁeld containing k(D). Then

(1) Lm(D) has a basis of functions which are rational over L.

∗ (2) Im(D) = Ωm(D) has a basis of diﬀerential forms which are rational over L.

Proof. (1) Let f ∈ k(X). Then f ∈ Lm(D) is equivalent to f ≡ 1 mod m and (f) ≥ −D by deﬁnition, but each of these conditions is L-linear since D is rational over L. Hence a basis of L-rational functions can be chosen. (2) For m = 0, Serre duality (2.5.2) gives us I(D) ∼= L(C − D) for a divisor C in the canonical class. Since C can be chosen to be L-rational, the result follows. The case for general m is similar, using the results of Chapter 4.

Corollary 5.2.4. Fix D ∈ Div(X) prime to S. Suppose there is a unique positive divisor 0 0 D ∼m D. Then D is rational over k(D).

55 5.2 Generalized Jacobians 5 Constructing Jacobians

0 Proof. Since D is unique, dim Lm(D) = 1 and dim L(D − m) = 0 so by Lemma 5.2.3, 0 we can choose g ∈ Lm(D) which is rational over k(D). Write D = D + (g) and apply Lemma 5.1.2. Fix D ∈ Div(X) prime to S and set

`m(D) = dimk Lm(D) = h0(Xm, Lm(D))

im(D) = dimk Im(D) = h1(Xm, Lm(D)) ( g, m = 0 π = g + deg(m) − 1, m 6= 0.

Then by singular Riemann-Roch (Theorem 4.2.3), `m(D) − im(D) = deg(D) + 1 − π. Lemma 5.2.5. Let K be a ﬁeld over which D is rational and let P ∈ X be a generic point over K. Then im(D + P ) = im(D) − 1.

Proof. Basically the same as Lemma 5.1.4, as that proof only needed L(C − D) ∼= I(D).

0 Lemma 5.2.6. Let D0 ∈ Div (X) be rational over K and suppose P1,...,Pπ are independent generic points over a field K/k. Then there exists a unique effective divisor of the form E = Pπ i=1 Qi, where Q1,...,Qπ are independent generic points over K, such that E ∼m D0 + D, Pπ where D = i=1 Pi. Moreover, K(D) = K(E). Proof. This is similar to the proof of Lemma 5.1.5, which is the m = 0 case. For general m, note that since deg(D) = 0, `m(D) ≤ 1 so by singular Riemann-Roch, im(D) ≤ π. Applying Lemma 5.2.5 multiple times, we see that im(D0 + D) = 0 implies `m(D0 + D) = 1. Thus there is a unique (up to scaling) function g ∈ Lm(D0 + D). When m 6= 0, we must also check L(D0 + D − m) = 0 so that g 6∈ L(D0 + D − m). Suppose `(D0 + D − m) ≥ 1. Since deg(D0 +D−m) = g−1, ordinary Riemann-Roch implies i(D0 +D−m) = `(D0 +D−m) ≥ 1. On the other hand, i(D0 +D −m) = i(D0 −m)−π by Lemma 5.2.5, so we get i(D0 −m) > π. However, deg(D0 − m) < 0 implies `(D0 − m) = 0 so by Riemann-Roch, i(D0 − m) = π, a contradiction. Thus E can be chosen to be the unique effective divisor in the complete linear system for D0 + D and the above shows that E ∼m D0 + D. Finally, Lemma 5.2.3 allows us to choose g rational over K(D0+D) = K(P1,...,Pπ)s. By Corollary 5.2.4, E is rational over K(P1,...,Pπ)s. On the other hand, the above paragraph implies `m(D) = 1, so D is also the unique effective divisor that is m-equivalent to E − D0. Thus by the same argument, K(D) ⊆ K(E − D0) = K(E). This completes the proof. Let X(π) be the symmetric product of π copies of X. We now construct a birational composition on an open subset of X(π) as we did in the nonsingular case. Fix a field K/k and a point P0 ∈ X which is rational over K. Lemma 5.2.7. Let P,Q ∈ X(π) be independent generic points over K. Then there exists a unique divisor R ∈ Div(X) such that R ∼m P + Q − πP0 as divisors and K(P,Q) = K(R,P ) = K(R,Q).

56 5.3 The Canonical Map 5 Constructing Jacobians

Proof. Every generic point of X(π) is identiﬁed with a divisor of X (consisting of independent Pπ generic points) and conversely, so we can write P = i=1 Pi, where P1,...,Pπ are independent generic points of X over K(Q) = K(Q − πP0). By Lemma 5.2.6, R = E satisﬁes the desired properties.

Proposition 5.2.8. The map X(π) × X(π) → X(π) given by (P,Q) 7→ P + Q := R is a unique birational composition law making X(π) into a birational group.

Proof. One need only check that π is deﬁned over k – for this, look at the Gal(Ω/k)-action – and that π is associative – as in Proposition 5.1.6, this can be done by considering the 0 00 divisor P + P + P − 2πP0. Thus by Weil’s theorem (5.1.7), there exists a unique (up to birational equivalence) algebraic group G over k birationally equivalent to X(π) with group law induced from the one constructed above on X(π).

Deﬁnition. The algebraic group Jm = Jm(X) associated to the described birational group law on X(π) is called the generalized Jacobian of X for the modulus m.

(π) The birational isomorphism ϕm : X → Jm guaranteed by Weil’s theorem satisﬁes ϕ(P · Q) = ϕ(P ) + ϕ(Q) where · is the composition law on X(π).

5.3 The Canonical Map

Let X be an algebraic curve, m a modulus on X with support S and fix a point P0 ∈ X r S. (π) Let Jm = Jm(X) be the generalized Jacobian of X for m and let ϕ = ϕm : X → Jm be the canonical map given by Weil’s theorem. We would like to extend this to a map 0 θ : Div (X) → Jm, in analogy with the Abel-Jacobi theorem (see Section 1.4). Pπ For any divisor D ∈ Div(X) rational over a field K/k, let M = i=1 Mi be a generic point of X(π) over K. By Lemma 5.2.6, there exists a unique point N ∈ X(π) such that as a divisor P P on X, N ∼m D −deg(D)P0 + Mi and K(N) = K ( Mi). Define θM (D) = ϕ(N)−ϕ(M). Lemma 5.3.1. Let K/k be a field, suppose D,D0 ∈ Div(X) are rational over K and set D00 = D + D0. Then if M,M 0,M 00 are independent generic points of X(π) defined over K, then 00 0 θM 00 (D ) = θM (D) + θM 0 (D ). Proof. Lemma 5.2.6 allows us to write down three unique points N,N 0,N 00 ∈ X(π) for D,D0,D00, respectively, satisfying

0 0 0 0 00 00 00 00 N ∼m D − deg(D)P0 + M,N ∼m D − deg(D )P0 + M ,N ∼m D − deg(D )P0 + M .

Then ϕ(N 00) − ϕ(M 00) = ϕ(N) − ϕ(M) + ϕ(N 0) − ϕ(M 0), and so ϕ(N) + ϕ(N 0) + ϕ(M 00) = ϕ(N 00) + ϕ(M) + ϕ(M 0). Since k(N) = k(M), k(N 0) = k(M 0) and k(N 00) = k(M 00), we see 0 00 0 00 00 0 that N,N ,N are independent over k. Therefore N + N + M ∼m N + M + M , which 00 0 implies θM 00 (D ) = θM (D) + θM 0 (D ).

57 5.3 The Canonical Map 5 Constructing Jacobians

Lemma 5.3.2. For D ∈ Div(X), the deﬁnition of θM (D) is independent of the choice of M ∈ X(π).

0 0 Proof. When D = 0, N ∼m M implies N = M, so θM (D ) = 0. Thus by Lemma 5.3.1, 00 (π) 0 θM (D) = θM 00 (D) for any independent generic points M,M ∈ X . In general, if M,M ∈ (π) 00 X are generic points over K, we can ﬁnd a generic point M such that θM 0 (D) = θM (D) = θM 00 (D).

In light of this, we will write θ(D) = θM (D) = ϕ(N) − ϕ(M) for any generic point M ∈ X(π).

Pπ (π) Lemma 5.3.3. If P = i=1 Pi is a generic point of X , then θ(P ) = ϕ(P ). Proof. If M is a generic point of X(π) which is independent of P , then by Lemma 5.2.7 there (π) exists a unique point N ∈ X satisfying N ∼m P + M − πP0. Thus N = P · M using the birational group law on X(π), and we have θ(P ) = ϕ(N) − ϕ(M) = ϕ(P ) by deﬁnition of ϕ. Lemma 5.3.4. If D ∈ Div(X) is rational over K, then θ(D) is rational over K.

(π) Proof. By Lemma 5.2.6, θ(D) = ϕ(M)−ϕ(N) for N ∈ X satisfying N ∼m D−deg(D)P0 + M and K(M) = K(N). Thus θ(D) is rational over K(N). By Lemma 5.3.2, we may vary M, (π) T so this holds for any generic point N ∈ X , so θ(D) is rational over N∈X(π) K(N) = K. Proposition 5.3.5. Let DivS(X) be the divisors on X which are prime to S. Then θ : S Div (X) → Jm is a surjective group homomorphism with kernel

S ker θ = {D ∈ Div (X) | D ∼m nP0 for some n ∈ Z}. Proof. Lemmas 5.3.1 and 5.3.3 prove that θ is a homomorphism. For surjectivity, note that Lemma 5.3.3 shows that the image of θ is open and hence generates Jm. Since θ is a homomorphism, we have im θ = Jm. Lastly, θ(D) = 0 if and only if ϕ(N) = ϕ(M) for all (π) (equivalently, for any) generic points M ∈ X , which is equivalent to N ∼m D−deg(D)P0 + M. Hence θ(D) = 0 if and only if D ∼m deg(D)P0. We next describe the universal property of ϕ and θ. Lemma 5.3.6. For any separable cover g : X → X0 of nonsingular, irreducible, projective curves of degree deg(g) = n + 1 and for any P ∈ X, the divisor corresponding to g−1(g(P )) (n) is of the form P + HP for an eﬀective divisor HP ∈ X . Therefore P 7→ HP is a regular map X → X(n). g Proof. Let Y be the Galois closure of X → X0, which is a Galois covering Y −→π X −→ X0. Set g = Gal(Y → X0) and h = Gal(Y → X) ≤ g. Then the coset space g/h has coset n+1 representatives {σi}i=1 where [σi+1] = h and we can write

n+1 −1 X g (g(P )) = π ◦ σi(Q) i=1

58 5.3 The Canonical Map 5 Constructing Jacobians

−1 Pn (n) for a fixed Q ∈ π (P ). Set HP = i=1 π ◦ σi(Q), which is an effective divisor on X . Next, define a map

h : Y −→ Xn

Q 7−→ (π ◦ σ1(Q), . . . , π ◦ σn(Q)).

Composing this with Xn → X(n), we get a map h0 : Y → X(n) which is regular. But n (n) (n) explicitly, X → X takes (P1,...,Pn) to the point of X that identifies with the divisor Pn i=1 Pi, so passing to the quotient Y/h = X, we get precisely the map P 7→ HP . S The map θ : Div (X) → Jm defines a rational map θ : X r S → Jm. We will prove that θ is in fact regular. Lemma 5.3.7. For any generic point M of X(π) over K, there exists a rational map θ0 : X → Jm defined over K(M) and regular over X r S which is rational over K, and such that θ0 = θ on X r S over K.

Proof. For π = 0, this is trivial. Otherwise, choose P1 ∈ X r S such that P1 6= P0. (π) By Lemma 5.2.7, there is a unique point N ∈ X satisfying N ∼m −P1 + P0 + M and K(P1) = K(N). Take g ∈ k(X) such that (g) = N + P1 − P0 − M and g ≡ 1 mod m. By Lemma 5.2.7, (g) is unique, so `m(−P0 + P1 + M) = 0 and thus v∞(g) = P0 + M. This 1 shows g : X → P is a covering of degree π + 1. Moreover, P0 is a simple pole and g is not −1 a pth power, so g is separable (of degree π + 1). By Lemma 5.3.6, g (g(P )) = P + HP (n) (π) for an eﬀective divisor HP ∈ X and the map s : X → X ,P 7→ HP is regular. Set 0 θ (P ) = ϕ(N) − ϕ(HP ), which is rational over K(M) by Lemma 5.2.7. Then we must show θ0 agrees with θ on X r S. Suppose P 6∈ S is rational over K; set a = g(P ). When P 6= P0, a 6= ∞ since v∞(g) = P0 + M by the above. Thus (g − a) ≥ P − P0 − M, so by Lemma 5.2.7, g − a is unique. Also 0 g−a 0 note that g − a 6= 0 on S, so the function g = 1−a satisﬁes g ≡ 1 mod m and

0 −1 (g ) = g (a) − v∞(g) = P + HP − (P0 + M).

Thus HP ∼m −P + P0 + M, but by uniqueness, HP is also generic over K. Thus ϕ(HP ) is 0 deﬁned. Finally, θ(P ) = ϕ(M) − ϕ(HP ) follows by deﬁnition, so θ = θ on X r S. (π) (π) Lemma 5.3.8. The extension of θ to X coincides with the birational map ϕ : X → Jm.

(π) Pπ Pπ Proof. Deﬁne Sθ :(X − S) → Jm by i=1 Mi 7→ i=1 θ(Mi). Then Sθ is regular, so it extends to a rational map (π) Sθ : X 99K Jm. Pπ (π) We must show Sθ(M) = ϕ(M) for any generic point M = i=1 Mi of X . By deﬁnition and Pπ Lemma 5.3.3, and viewing M as a divisor on X prime to S, we have Sθ(M) = i=1 θ(Mi) = ϕ(M) = θ(M). We have proven: Theorem 5.3.9. Let X be a nonsingular curve over k, m a modulus on X with support S and ϕm : X → Jm the canonical map. Then

59 5.3 The Canonical Map 5 Constructing Jacobians

(1) ϕm is a rational map over k which is regular over X r S. S 0 ∼ (2) The extension of ϕm to Div (X) descends to an isomorphism Cm −→ Jm. (π) (π) (3) The extension of ϕm to X deﬁnes a birational map X → Jm.

Theorem 5.3.10. Let X be a nonsingular curve over k, P0 ∈ X and G a commutative algebraic group. For any map f : X → G with modulus m and such that f(P0) = 1G, there exists a unique morphism of algebraic groups F : Jm → G such that f = F ϕm + 1G:

ϕm F

X G f

0 Proof. Assume 1G = 0. Then for D ∈ Div (X) which is prime to S, Theorem 5.3.10(2) says 0 ∼ Cm = Jm, so the natural map 0 Cm −→ G D 7−→ f(D) ∼ 0 determines a homomorphism of algebraic groups F : Jm = Cm → G. It is clear that f = F ◦ ϕm by construction. Consider the following extension of f: π π (π) X X Sf : X −→ G Pi 7−→ f(Pi). i=1 i=1 (π) (π) (π) Then Sf is rational on X and regular on (X r S) , so Sf = F ◦ ϕ where ϕ : X → Jm is the extension of ϕm to the symmetric product. By Theorem 5.3.10(3), ϕ is biregular on a −1 −1 nonempty open set U ⊆ Jm. Thus F coincides with Sf ◦ ϕ on U, and Sf ◦ ϕ is regular on U, so F is as well. Translating U to cover all of Jm, we get that F is regular on Jm.

Corollary 5.3.11. The construction of (Jm, ϕm) is independent of the basepoint P0 ∈ X, up to translating ϕm. 0 0 Proof. Suppose P0 ∈ X is another basepoint with associated canonical map ϕm : X → Jm. (By Weil’s theorem, Jm is unique already.) Then by Theorem 5.3.10, there are unique maps 0 F,F : Jm → Jm making the following diagram commute:

Jm ϕm X F F 0 ϕ0 m Jm

0 0 0 Then F ◦ F = idJm = F ◦ F , so ϕm = ϕm + ϕm(P0). Corollary 5.3.12. Every rational map f : X → G into a commutative algebraic group G factors through ϕm : X → Jm for some modulus m on X. Proof. By Theorem 3.2.10, such an f has a modulus m. Now apply Theorem 5.3.10.

60 5.4 Invariant Diﬀerential Forms 5 Constructing Jacobians

5.4 Invariant Diﬀerential Forms

Let G be any algebraic group.

Definition. An invariant differential form on G is a differential form ω ∈ Ωk(G) which is invariant under translation by elements of G. Lemma 5.4.1. If dim G = n, then the invariant differential forms on G form a vector space of dimension n.

In our case, we take G = Jm to be the generalized Jacobian of a curve with modulus m, which is an algebraic group of dimension π = deg m + g − 1. Let ϕm : X → Jm be the canonical map. Then for any invariant differential form ω ∈ Ω(Jm), there is an induced ∗ invariant differential form ϕm(ω) ∈ Ω(X). ∗ Proposition 5.4.2. The map ω 7→ ϕm(ω) is a bijection between the invariant differential forms on Jm and the differential forms α ∈ Ω(X) such that (α) ≥ −m. ∗ Proof. First suppose ϕm(ω) = 0. Consider the maps π π g : X −→ Jm and hi : X −→ Jm π X (x1, . . . , xπ) 7−→ ϕ(xi)(x1, . . . , xπ) 7−→ ϕ(xi) for each i. i=1 Pπ ∗ Pπ ∗ Then g = i=1 hi, so g (ω) = i=1 hi (ω) = 0 by hypothesis. Note that g factors as π (π) X → X → Jm, where the first map is surjective and separable and the second map is the birational isomorphism from Weil’s theorem (5.1.7). It follows that g is generically surjective ∗ ∗ and separable, so g is injective and thus ω = 0 to begin with. Therefore ω 7→ ϕm(ω) is injective. Next, note that Ω(−m) = {α ∈ Ω(X) | (α) ≥ −m}, using the notation of Section 2.4. By Riemann-Roch (2.5.4), `(−m) − dim Ω(−m) = deg(−m) + 1 − g =⇒ 0 − dim Ω(−m) = − deg(m) + 1 − g =⇒ dim Ω(−m) = deg(m) + g − 1 = π.

Thus it’s enough to show that for each ω ∈ Ω(Jm), ϕm(ω) lands in Ω(−m) and count di- mensions. Notice that any element of Ω(−m) has no poles outside S, so by the residue P ∗ theorem (2.4.6), P ∈S resP (α) = 0. Thus to show α = ϕ (ω) ∈ Ω(−m), it suﬃces by Propo- sition 3.2.9 to show that Trg(α) = 0 for all rational maps g : X → k such that g ≡ 0 mod m 1 (which are not pth powers). Let h = Trg(ϕm): P → Jm, where ϕm is the map coming from the diagram

ϕm X Jm

g Trg(ϕ)

P 1

61 5.5 The Structure of Generalized Jacobians 5 Constructing Jacobians

∗ ∗ By the proof of Theorem 3.2.10, Trg(α) = h (ω) where α = ϕ (ω). On the other hand, ∗ since m is a modulus for ϕm, Proposition 3.2.9 implies h is constant. Thus h (ω) = 0 as required.

Corollary 5.4.3. If deg(m) ≥ 2, then the map ϕm is not regular at any point P ∈ S. Proof. Given any P ∈ S, we have deg(m − P ) > 0, so by singular Riemann-Roch (Theo- rem 4.2.3), dim Ω(−m + P ) = π − 1. On the other hand, from the proof of Proposition 5.4.2 P we know that dim Ω(−m) = π, so there is some α ∈ Ω(−m) r Ω(−m + P ). If m = nP P , ∗ then this α has a pole of order nP at P . On the other hand, α = ϕm(ω) for some ω ∈ Ωk(Jm), so ϕm cannot be regular at P .

5.5 The Structure of Generalized Jacobians

In this section, we explore the structure of generalized Jacobians and in addition give some important examples.

0 0 Proposition 5.5.1. Let m and m be moduli on X such that m ≥ m and ﬁx a point P0 ∈ X. Then there exists a unique surjective map F : Jm → Jm0 with connected kernel Hm/m0 such that ϕm0 = F ◦ ϕm.

0 0 Proof. Since m is a modulus for ϕm0 , m ≥ m implies m is also a modulus for ϕm0 . Thus by Theorem 5.3.10, there exists a unique map F : Jm → Jm0 such that F ϕm + ϕm0 (P0) = ϕm0 . To show F is onto, we construct a birational section of F . Consider the map Xπ0 → Xπ given (π0) (π) by (M1,...,Mπ0 ) 7→ (M1,...,Mπ0 ,P0,...,P0). This descends to a map S : X → X , 0 (π) which in turn determines S : Jm0 → Jm via the birational isomorphisms X → Jm and (π0) 0 X → Jm0 . By construction, F ◦ S = 1, so F is surjective. As a result, anytime m ≥ m0, the generalized Jacobian for m0 can be decomposed as ∼ 0 Jm0 = Jm × Hm/m0 . For the special case m = 0, J0 = J is the ordinary Jacobian and we write Lm = Hm/0. A point d ∈ Jm corresponds to a linear equivalence class of divisors D prime to S. When D = (g) for some g ∈ k(X), vP (g) = 0 for all P ∈ S, so the image of d in J via the map F is 0. On the other hand, the class of D determines the origin 0 ∈ Jm when D = (h) for some h ≡ 1 mod m, so g = λh for some λ ∈ k×. For a ﬁxed point P ∈ X, deﬁne the groups

× (n) UP = {f ∈ k(X) | vP (f) = 0} and UP = {f ∈ UP | vP (1 − f) ≥ n}.

(n) For each P ∈ S and g ∈ k(X), we get a function gP ∈ UP /UP for some n ≥ 0, and Q (nP ) conversely, every collection of cosets (gP )P ∈S ∈ P ∈S UP /UP is the image of such a g ∈ × Q (nP ) × k(X) . Set Rm = P ∈S UP /UP , let ∆ be the image of k in Rm via λ 7→ (λ, . . . , λ) and deﬁne Hm = Rm/∆. We have proven the following.

Proposition 5.5.2. The map θ : Hm → Lm, (g) 7→ (gP ) is an isomorphism of groups.

62 5.5 The Structure of Generalized Jacobians 5 Constructing Jacobians

We next describe an algebraic group structure on Lm. To begin, let k((t)) be the ﬁeld of formal (Laurent) power series over k, which is a local ﬁeld when equipped with the usual power series valuation v. Let

U = {f ∈ k((t)) | v(f) = 0} and U (n) = {f ∈ U | v(1 − f) ≥ n}.

(n) n n+1 Then any f ∈ U can be written f = 1 + ant + an+1t + ... so that the elements of the (n) n−1 quotient U/U are truncated power series: f = a0 + a1t + ... + an−1t , a0 6= 0. Therefore U/U (n) has the structure of a variety, and pointwise multiplication, inversion and change of uniformizer t are all regular functions on U/U (n). Thus:

Proposition 5.5.3. U/U (n) is an algebraic group under pointwise multiplication and inversion.

(1) (n) (n) Now let V(n) = U /U ⊆ U/U be the subgroup of truncated power series of the form n−1 1 + a1t + ... + an−1t . This too has the structure of an algebraic group.

(n) ∼ Lemma 5.5.4. There is an isomorphism of algebraic groups U/U = Gm × V(n).

(n) 0 0 (1) (n) Proof. Every function f ∈ U/U is of the form f = a0f for some f ∈ U /U and × (n) ∼ a0 ∈ k , so we have the abstract isomorphism U/U = Gm × V(n). Since multiplication by −1 a0 and a0 are regular, the isomorphism preserves the algebraic group structure.

Lemma 5.5.5. For each 1 ≤ i ≤ n−1, let gi ∈ k((t)) be a formal power series with v(gi) = i. Then every element g ∈ V(n) can be written uniquely as

g = (1 + a1g1)(1 + a2g2) ··· (1 + an−1gn−1)

for a1, . . . , an−1 ∈ k.

n−1 2 b1 Proof. Write g = 1 + b1t + ... + bn−1t and g1 = c1t + c2t + .... Then setting a1 = c and g 1 h1 = , we see that v(1 − h1) ≥ 2. The subsequent ai are deﬁned similarly. 1+a1g1 ∼ n−1 Corollary 5.5.6. For each n ≥ 1, V(n) = Ak as varieties. For the rest of the section, we will assume char k = 0. The structure theory in characteristic p > 0 can be found in Serre.

Deﬁnition. For a formal power series g ∈ k((t)), deﬁne the exponential of g to be the formal expansion g2 g3 exp(g) = 1 + g + + + ... ∈ k((t)). 2 6

Lemma 5.5.7. For all g1, g2 ∈ k((t)), exp(g1 + g2) = exp(g1) exp(g2).

Lemma 5.5.8. For gi ∈ k((t)) with v(gi) = i as above, we may write every g ∈ V(n) uniquely as g = exp(a1g1) exp(a2g2) ··· exp(an−1gn−1)

for ai ∈ k.

63 5.5 The Structure of Generalized Jacobians 5 Constructing Jacobians

Proof. Same proof as for Lemma 5.5.5.

(n) ∼ n−1 By Lemma 5.5.4 and Corollary 5.5.6, we have U/U = Gm ×Ga as an algebraic group. (nP ) The preceding analysis applies to each of the ‘local groups’ UP /UP in the decomposition of Rm, so we have ∼ Y (nP ) ∼ |S| Y nP −1 Rm = UP /UP = Gm × Ga . P ∈S P ∈S ∼ Then since ∆ = Gm, we may write

∼ |S|−1 Y nP −1 Hm = Rm/∆ = Gm × Ga . P ∈S

1 Let g, h ∈ k(X) and for every z ∈ Pk, let Dz be the principal divisor deﬁned by Dz = (g+zh) for z 6= ∞ and D∞ = (h). Let

1 T = {z ∈ Pk | Dz ∩ S 6= ∅}.

1 Lemma 5.5.9. The assignment z 7→ ϕ(Dz) deﬁnes a regular map Pk r T → Lm.

Proof. Notice that ϕ(Dz) ∈ Lm since Dz is the divisor of a rational function. To prove 1 regularity, we may assume Dz = (g − z). Let ψ = Trg(ϕ), which is regular on Pk r T by Proposition 3.2.8. Then ϕ(Dz) = ψ(z) − ψ(∞), so z 7→ ϕ(Dz) is regular oﬀ of T .

Lemma 5.5.10. θ : Hm → Lm is a regular map. |S|−1 ∼ Q nP −1 Proof. Write Hm = Gm × P ∈S Ga . Then it is enough to check regularity on each com- ponent of the product. For each P ∈ S, let Gm,P be the multiplicative group corresponding to P in the ﬁrst product. Let v be a function with v ≡ 1 mod m at P and v ≡ 0 mod m on S r {P }. Then for λ ∈ Gm,P , uλ = (λ − 1)v + 1 satisﬁes uλ ≡ λ mod m at P and uλ ≡ 1 mod m on S r{P }. Moreover, θ(λ) = ϕ((uλ)) for (uλ) the principal divisor associated to uλ, nP −1 so by Lemma 5.5.9, θ is regular on Gm,P . A similar proof works for each factor Ga .

Theorem 5.5.11. The map θ : Hm → Lm is an isomorphism of algebraic groups. In |S|−1 ∼ Q nP −1 particular, Lm = Gm × P ∈S Ga . Proof. By Proposition 5.5.2, θ is a bijection and by Lemma 5.5.10, it’s regular, so we at least know that Hm and Lm have the same dimension. Thus [k(Hm): k(Lm)] < ∞. This means θ is biregular on an open subset, so it is a bijective, birational isomorphism, hence an isomorphism of algebraic groups. Remark. The proof in the case char k = p > 0 requires a bit more care, as a bijective, birational isomorphism is not necessarily a global isomorphism (it may correspond to an inseparable extension). However, an argument using the trace map and residue formula show the result; see Serre for the details.

Corollary 5.5.12. For any curve X with modulus m, the generalized Jacobian Jm is a group extension of the ordinary Jacobian J of X, i.e. there is a short exact sequence of groups

1 → Lm → Jm → J → 1.

64 5.5 The Structure of Generalized Jacobians 5 Constructing Jacobians

1 1 Example 5.5.13. For X = Pk and the modulus m = 0 + ∞, J(Pk) = 0 so the generalized 1 ∼ Jacobian is Jm = Pk r {0, ∞} = Gm. More generally, if m = (P ) + (Q) for distinct points 1 ∼ P,Q ∈ Pk, then Jm = Gm. Let us explicitly describe the multiplication on Jm. First, we have 0 1 Divm(X) = {D = (g) | g : P → k, ordP (g) = ordQ(g) = 0}.

Suppose D1 = (f1) and D2 = (f2). Then

1 × D1 ∼m D2 ⇐⇒ there is some f ∈ k(P ) with (f) = D1 − D2 and f ≡ 1 mod m f ⇐⇒ there is some f with (f) = 1 and f(P ) = f(Q) = 1 f2 f (P ) f (Q) ⇐⇒ 1 = 1 f2(P ) f2(Q) f (P ) f (P ) ⇐⇒ 1 = 2 . f1(Q) f2(Q)

0 1 Thus the map ϕm : Cm → Gm = P r {P,Q} is given by [(g)] 7→ [g(P ), g(Q)]. It is clear that 1 this ϕm is both well-deﬁned and injective. To see that it is surjective, take [α, β] ∈ P r{0, ∞}. Then the function α (t − Q) t − P g(t) = β − P − Q Q − P 0 deﬁnes an m-equivalence class [(g)] ∈ Cm and ϕ([(g)]) = [α, β] by construction. Example 5.5.14. Let X = E be an elliptic curve with distinguished point O ∈ E. Then Pic0(E) = E by the Abel-Jacobi map (as in Chapter 1):

E −→∼ Pic0(E) P 7−→ [(P ) − (O)].

P 0 × For D = nP P ∈ Div (E), there is some function f ∈ k(E) such that D = D − (O) + (f). 0 Consider the modulus m = (M) + (N) for distinct points M,N ∈ E r {O}. If D ∈ Divm(E), then either D = (M) or D = (N); or

D 6= (M), (N).

In the ﬁrst case, D = (M), so that ordM (f) = −1. In the second case, ordM (f) = ordN (f) = P 0, so D ∼ S − (O) ∼ (S + T ) − (T ) for S = nP P and any T ∈ E. Choosing T 6∈ {O,M,N,M − N,N − M}, we may assume (M + T ) − (T ) and (N + T ) − (T ) have support which is disjoint from the support of m. Set ( O, if S 6= M,N R = T, if S = M or N.

× Then there is a function f ∈ k(E) such that D = (S + R) − (R) + (f) and ordM (f) = ordN (f) = 0.

65 5.5 The Structure of Generalized Jacobians 5 Constructing Jacobians

0 Repeating this for two divisors D1,D2 ∈ Divm(E), we can write Di = (Si+Ri)−(Ri)+(fi) for i = 1, 2. Then

× D1 ∼m D2 ⇐⇒ there is an f ∈ k(E) with (f) = D1 − D2, f ≡ 1 mod m f1 ⇐⇒ (f) = (S1 + R1) − (S2 + R2) + (R2) − (R1) + f2 f for some f ∈ k(E)× with f = 1 and f(M) = f(N) = 1 f2

⇐⇒ S1 + R1 − (S2 + R2) + R2 − R1 = O in E

f1(M) f2(M) ⇐⇒ S1 = S2 in E and = . f1(N) f2(N) Thus the canonical map is given by

0 ϕm : Cm(E) −→ Jm = Gm × E f(M) [D] 7−→ ,S if D = (S + R) − (R) + (f). f(N)

We need the following result to describe the group structure of Jm in this case. Theorem 5.5.15. Let (G, +) and (G, ⊕) be two groups and

p 1 → A −→i G −→ G → 0 a group extension by an abelian group A. Then

(1) There is a section s : G → G such that G = A × G as sets.

(2) G acts on A via the map

A × G −→ A (a, σ) 7−→ aσ := x ⊕ a x

where x ∈ G is any element such that p(x) = σ.

(3) There is a 2-cocycle c : G × G → A such that

c(σ, τ) · c(σ + τ, ρ) = c(τ, ρ)σ · c(σ, τ + ρ) and (a, σ) ⊕ (b, τ) = (a · b · c(σ, τ), σ + τ)

for all σ, τ, ρ ∈ G and a, b ∈ A. Explicitly, c(σ, τ) = s(σ) ⊕ s(τ) s(σ + τ), where s is the section from (1).

In our situation, we have a group extension

1 → Gm → Jm → E → 0

66 5.5 The Structure of Generalized Jacobians 5 Constructing Jacobians

and Jm = Gm × E as sets, with (b1,P ) ⊕ (b2,Q) = (b1b2 · cm(P,Q),P + Q) for all b1, b2 ∈ Gm,P,Q ∈ E and cm : E × E → Gm a 2-cocycle. To determine this cocycle, let D1 = (P1) − (O) + (f1) and D2 = (P2) − (O) + (f2) be two divisors on E prime to the support 0 of m. Then under the canonical map ϕm : Cm(E) → Jm, we have ϕm([Di]) = (bi,Pi) for i = 1, 2 and (b1,P1) ⊕ (b2,P2) = (b3,P3) for some b3 ∈ Gm,P3 ∈ E. Notice that if D3 = (P1) + (P2) − 2(O) + (f1f2), then it will be enough to determine f3 such that D3 = (P3) − (O) + (f3). By the Abel-Jacobi isomorphism, we have

(P1) + (P2) − 2(O) = (P1 + P2) − (O) + (`P1,P2 )

× for some `P1,P2 ∈ k(E) . Set f3 = f1f2`P1,P2 ,

f3(M) `P1,P2 (M) `P1,P2 (M) b3 = = b1b2 and cm(P1,P2) = . f3(N) `P1,P2 (N) `P1,P2 (N)

We claim (`P1,P2 ) = (P1) + (P2) − (P1 + P2) − (O) will work. Let LP1,P2 be the linear form 2 deﬁning the line in E ⊆ P containing P1,P2 and −(P1 + P2).

−(P1 + P2)

P2 P1

LP1,P2 P1 + P2

LP1+P2,O

Then dividing out by z gives a function with the following zeroes and poles:

L (z) P1,P2 = (P ) + (P ) + (−(P + P )) − 3(O). z 1 2 1 2

On the other hand, the linear form LP1+P2,O deﬁning the line containing P1 + P2, −(P1 + P2) and O yields L (z) P1+P2,O = (P + P ) + (−(P + P )) − 2(O). z 1 2 1 2

LP1,P2 `P1,P2 (M) So taking `P1,P2 = , we get cm(P1,P2) = as the desired 2-cocycle. LP1+P2,O `P1,P2 (N)

67 6 Class Field Theory for Curves

6 Class Field Theory for Curves

The goal of class field theory is to classify all abelian extensions L/K in terms of the arithmetic of K itself. To this end, let I(K) be the set of fractional ideals of K and set GL/K = Gal(L/K) for any abelian Galois extension. Definition. The Artin map of an unramified abelian extension L/K is the map on fractional ideals given by

θL/K : I(K) −→ GL/K

Y ordv(a) a 7−→ Frobv v where the product runs over all places v of K, Frobv is the Frobenius element at v and ordv(a) is the valuation of a at v.

Theorem 6.0.1. For any principal ideal (α) ∈ I(K) and any unramified abelian extension L/K, θL/K ((α)) = 1. Therefore θL/K descends to a map on the class group: CK = I(K)/P (K) → GL/K . Class field theory for number fields is summarized in the following theorem.

Theorem 6.0.2. Let L/K be a ﬁnite abelian extension with Galois group GL/K . Then

(1) (Artin Reciprocity) The Artin map θL/K : CK → GL/K is surjective with kernel NL/K (CL), where NL/K is the norm map.

(2) For any normal subgroup N ≤ CK , there exists a finite abelian extension L/K for ∼ which N = ker θL/K . In this case, CK /N = GL/K . (3) The Artin map is natural, i.e. for all finite unramified abelian extensions M ⊇ L ⊇ K, there is a commutative diagram

θM/L CL GM/L

NL/K

CK GM/K θM/K

There is an analogous theory for covers of curves, constructed as follows. Let X be a smooth, geometrically connected curve over a perfect ﬁeld k and specify a set of closed points S ⊂ X.

Theorem 6.0.3. Let H any connected commutative algebraic group with principal homogeneous space H1 admitting a regular map f : X r S → H1. Then

68 6.1 Coverings and Isogenies 6 Class Field Theory for Curves

(1) There exists a modulus m on X for f supported on S and a commutative diagram deﬁned over k:

ϕ m 1 X r S Jm

f 1 f m H1

1 where Jm is a principal homogeneous space for the generalized Jacobian Jm and ϕm : 1 X r S → Jm is the restriction of the canonical map.

1 1 1 (2) There exists a map fm : Jm → H with respect to which fm : Jm → H is an equivariant map of principal homogeneous spaces.

1 (3) For every modulus m of f, the map fm is unique and thus there is a unique modulus f dividing all such m.

(4) Any geometrically connected abelian cover of H1 pulls back along f to give a geometrically connected cover of X which is unramiﬁed outside S.

Deﬁnition. The modulus f is called the conductor of the cover f : X r S → H1.

6.1 Coverings and Isogenies

0 Let k = Fq be a finite field and π : X → X a finite, unramified cover of curves defined over k. This corresponds to a field extension K0/k(X) of transcendence degree 1 fields. If Gal(K0/k(X)) is abelian, we say the cover π is abelian.

Theorem 6.1.1. Let π : X0 → X be an abelian cover of curves corresponding to a field extension K0/k(X), with ramification locus S. Then there exists a modulus m supported on 1 1 S and a finite connected abelian cover G → Jm such that the diagram

X0 r S0 G1

1 X Jm ϕm

0 −1 1 commutes, where S = π (S) and Jm is the principal homogeneous space for Jm given by Theorem 6.0.3. Moreover, there exists a unique modulus f dividing all moduli m for π, and 1 1 0 for this f, the cover G → Jm is geometrically connected if and only if X is geometrically connected.

69 6.2 Class Field Theory 6 Class Field Theory for Curves

0 0 0 Suppose π : X → X is an abelian cover such that X (k) 6= ∅. Then for any y0 ∈ X (k), −1 0 with image x0 = π(y0) ∈ X(k), all elements of π (x0) are k-points of X since π is abelian. Let ϕ : X −→ J = J(X)

x 7−→ x − x0 be the ordinary Abel map. Then by Theorem 6.1.1, there is a ﬁnite, connected abelian cover G → J such that the diagram X0 G

X ϕ J

commutes. Then J = G/N for some normal subgroup N ⊆ G. In other words, G → J is an isogeny. In the case when G = J, the cover J → J is an Artin-Schreier cover, that is, it is q q of the form x 7→ x − x. The map Fq : x 7→ x is called the qth Frobenius on J (or on any algebraic group G over k). Then the map G = J → J can be written as Fq − 1. Take a point ξ ∈ X with ﬁeld of deﬁnition Fqr , r ≥ 1. Then ξ determines a Galois orbit ¯ (ξ = ξ1, . . . , ξr) with ξi ∈ X(k),Fq(ξi) = ξi+1 for 1 ≤ i ≤ r − 1 and Fq(ξr) = ξ1. Take 0 0 y ∈ X = {y ∈ J | Fq(y) − y ∈ X} lying above ξ. Note that J acts on X by z · y = y + z, 0 0 and through this action, we may lift Fq to a Frobenius Frobξ : X → X . Explicitly, r r Frobξ(y) = Fq (y), but notice that z = Fq (y) − y ∈ J:

r n n r r X i i−1 X i−1 X i−1 X Fq (y) − y = (Fq (y) − Fq (y)) = Fq (Fq(y) − y) = Fq (ξ) = ξi. i=1 i=1 i=1 i=1 Pn Thus Frobξ may be identified with the element z = i=1 ξi. Definition. The Artin map of X is defined by θ : Div0(X) −→ J X X D = nξξ 7−→ nξξ.

Note that if D is a principal divisor, then θ(D) = 0 in J. This is a completely trivial consequence of the geometry of J, whereas the proof of the analogous Theorem 6.0.1 is one of the most diﬃcult parts of global class ﬁeld theory. As a consequence, θ descends to a map on the Picard group: Pic0(X) = Div0(X)/ PDiv0(X) → J.

6.2 Class Field Theory

We next give a little more detail on covers of curves before stating the analogue of class field theory for curves over a finite field. Let k be any field and V a normal irreducible

70 6.2 Class Field Theory 6 Class Field Theory for Curves

variety over k with function field K = k(V ). Given a finite separable extension L/K, let W be the normalization of V in L. This defines a covering π : W → V corresponding to L/K. If L/K is Galois with G = Gal(L/K), then we say pi : W → V is Galois, write Gal(W/V ) = Gal(L/K), and we have V ∼= W/G. For a point P ∈ V , we say P is unramified in π if it has exactly n = [L : K] preimages. Now suppose f : X → V is a rational map. Then there is an induced map of varieties f ∗W → W given by pullback:

f ∗W X

f π W V

In fact, we have the following amazing result, which can be found in Serre.

Theorem 6.2.1. Every abelian cover W → V is the pullback of an isogeny f : G0 → G of smooth, abelian varieties with kernel ker f ∼= Gal(W/V ).

Given a ﬁnite group N, let A(N) be the group algebra of N (over some universal domain Ω, e.g.) and let G(N) be the subgroup of invertible elements:

A(N) = {(as)s∈N | as ∈ Ω}

G(N) = {(as)s∈N ∈ A(N) | det(ast) 6= 0}.

Then N embeds as a subgroup of G(N) via s 7→ (at) where at = δst for all t ∈ N. Proposition 6.2.2. If π : W → V is a Galois cover with Galois group N, then there exist a rational map f : V → G(N)/N such that W ∼= f ∗G(N). Moreover, if W and π are deﬁned over k, then both f and the isomorphism W ∼= f ∗G(N) are also deﬁned over k.

Proof. Deﬁne g : W → A(N) by x 7→ (ϕs(x)), where ϕ ∈ k¯(W ) is any ﬁxed rational function. Then since det(ϕst) 6= 0, g has image in G(N). Consider the diagram g W G(N)

π f V G(N)/N

Then by the normal basis theorem from Galois theory, there exists an element ψ ∈ k(W ) s such that {ϕ }s∈N is a basis of the ﬁeld extension k(W )/k(V ). It follows that f is equal to g after taking quotients, so the diagram commutes.

Proposition 6.2.3. If P ∈ V is unramiﬁed in π : W → V , then f : V → G(N)/N from above may be chosen to be regular on π−1(P ).

71 6.2 Class Field Theory 6 Class Field Theory for Curves

s Proof. To prove this, we must ﬁnd a normal basis {ψ }s∈N where ψ is regular on the set −1 π (P ). Let OP be the local ring at P in K = k(V ) with maximal ideal mP and as in 0 Chapter 4, let OP be the integral closure of OP in L = k(W ), so that

0 \ OP = OQ. Q∈π−1(P )

0 0 0 0 Set k (P ) = OP /mP OP and k(P ) = OP /mP . Since P is unramiﬁed, k (P ) is an ´etalealgebra of degree n = [L : K] over k(P ). Thus by the normal basis theorem, we can choose λ ∈ k0(P ) s 0 0 such that {λ }s∈N is a normal basis of k (P )/k(P ). Now λ may be lifted to OP for all P , giving a rational function ψ on W which is regular on π−1(P ). This proves Theorem 6.2.1. ∼ n Example 6.2.4. Let N = Z/nZ where (n, p) = 1. Then A(N) = k[t]/(t − 1) and if k ∼ ×n contains a primitive nth root of unity, then G(N) = Gm . Let θn : Gm → Gm be the isogeny x 7→ xn (called the nth Kummer cover). Then G(N) → G(N)/N is the pullback of this isogeny:

G(N) Gm

θn

G(N)/N Gm

∼ p ∼ Example 6.2.5. Let N = Z/pZ. Then A(N) = k[t]/(t − 1), G(N) = Ga and G(N) → p G(N)/N is the pullback of the Artin-Schreier cover ℘ : Ga → Ga, x 7→ x − x:

G(N) Ga ℘

G(N)/N Ga

The general case is given by Artin-Schreier-Witt theory. Now let X be a complete, irreducible, nonsingular curve over k with ﬁeld of rational functions K = k(X). Corollary 6.2.6. If π : Y → X is an abelian cover of curves with Galois group N = Gal(Y/X), then there exists a separable isogeny G → H of abelian varieties with kernel N such that Y ∼= f ∗G for some f : X → H: Y G

π f X H

72 6.2 Class Field Theory 6 Class Field Theory for Curves

Proof. Follows from Theorem 6.2.1. Let S be the ramiﬁcation locus of π : Y → X. By Theorems 3.2.1 and 5.3.10, f : X → H has a modulus m supported on S such that f factors through the generalized Jacobian Jm = Jm(X): ϕm f¯ f : X −→ Jm −→ H. 0 ¯∗ ∼ ∗ 0 Thus we can pull back the isogeny G → H to an isogeny J = f G → Jm such that Y = ϕmJ . This proves:

0 Corollary 6.2.7. Every abelian cover Y → X is the pullback of a separable isogeny J → Jm for some modulus m on X. Moreover, there exists a modulus f such that f ≤ m for all such moduli m.

Deﬁnition. The modulus f is called the conductor of the cover Y → X.

Corollary 6.2.8. There is a one-to-one correspondence

{unramiﬁed abelian covers Y → X} ←→ {isogenies J 0 → J(X)}.