DOCSLIB.ORG

ON SELMER RANK PARITY of TWISTS Contents Introduction 1 1. General Notation 2 2. Preliminary Notions and Results 3 2.1. Global M

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
ON SELMER RANK PARITY of TWISTS Contents Introduction 1 1. General Notation 2 2. Preliminary Notions and Results 3 2.1. Global M ON SELMER RANK PARITY OF TWISTS MAJID HADIAN AND MATTHEW WEIDNER Abstract. In this paper we study the variation of the p-Selmer rank parities of p-twists of a self-dual Abelian variety over an arbitrary number field K and show, under certain assumptions, that this parity is periodic with an explicit period. Our result applies in particular to Abelian varieties with full K-rational p-torsion subgroup, arbitrary elliptic curves, and Jacobians of hyperelliptic curves. Assuming the Shafarevich-Tate conjecture, our result allows one to classify the rank parities of all quadratic twists of an elliptic or hyperelliptic curve after a finite calculation. Contents Introduction 1 1. General Notation 2 2. Preliminary Notions and Results 3 2.1. Global Metabolic Structures 3 2.2. Selmer Structures 4 2.3. Twisting 5 2.4. Local Conditions 6 3. Periodicity of Selmer Rank Parity 7 4. Some Examples 11 References 11 Introduction In this paper we study the behaviour of the p-Selmer rank in families of p-twists of self- dual Abelian varieties over arbitrary number fields. Our main result is that, under certain conditions on the Abelian variety, the parity of the p-Selmer rank is periodic with an explicit period (see Theorem 3.6, Corollary 3.7, and the following remarks). This is proved by using known results which write the variation of the p-Selmer rank under p-twist as a sum of local factors and which evaluate those local factors in certain cases (see Theorem 3.4). Calculating arithmetic ranks of elliptic curves, and more generally Abelian varieties, is a very difficult problem in Diophantine geometry. As of today, this problem remains widely open in general, theoretically and computationally. In various forms, it is related to many long-standing open problems, such as the Birch and Swinnerton-Dyer conjecture. Many of the recent attempts at progress in this field have involved studying the Selmer rank and its parity. Given that the Selmer rank provides an upper bound for the arithmetic rank and, assuming the Shafarevich-Tate conjecture, in certain cases (see Remark 3.10) its parity differs 2010 Mathematics Subject Classification. Primary: 11G05, 11G10. Key words and phrases. Elliptic curves, Abelian varieties, Selmer rank parity, p-twist. 1 2 M. HADIAN AND M. WEIDNER by a simple factor from the parity of the arithmetic rank, any information on Selmer ranks and their parities can be used in the study of arithmetic ranks. As an example, consider Goldfeld's conjecture about elliptic curves over Q (see [2, Conjec- ture B]). It predicts that among quadratic twists of a fixed elliptic curve E over Q, half have rank 0 and half have rank 1. Assuming the Shafarevich-Tate conjecture, this would imply that half of the quadratic twists of E have even Selmer rank and half have odd Selmer rank. Now, it would be very interesting if one could prove statements like this concerning Selmer ranks without assuming big conjectures such as Goldfeld or Shafarevich-Tate. In fact, there has been much recent interest in studying the variation of Selmer ranks and their parities. For example, Klagsbrun, Mazur, and Rubin prove the above statement for 2-Selmer rank parities in [6]. As another example, Swinnerton-Dyer in [18] studies the size of 2-Selmer groups of quadratic twists of elliptic curves over Q with full rational 2-torsion subgroup. Several other mathematicians including Kane, Kramer, and Yu have considered more general versions of this problem for p-Selmer groups of p-twists of elliptic curves and Jacobians of hyperelliptic curves (see [5, 7, 9, 10, 13, 19] for example). Our work in this paper is inspired by the above literature and gives uniform results that are finer than some of the existing ones. In particular, under certain conditions on the Abelian variety, the periodicity result of Section 3 reduces the problem of finding what portion of twists have even or odd p-Selmer rank to a finite calculation (see the examples given in Section 4). Under certain conditions when p = 2, assuming the Shafarevich-Tate conjecture, we can actually completely classify the rank parities of twists after a finite calculation (see Remark 3.10). Organization of the paper: In Section 1 we set up some general notations that will be fixed throughout the paper. Section 2 is devoted to reviewing the necessary definitions and results on global metabolic structures, Selmer structures, twisting, and local conditions. This material is gathered from several references including [6, 8, 9, 11, 13, 19], but in order to give a uniform treatment suitable for our applications, we have included our version of it for the reader's convenience. In Section 3 we prove our main results, which imply the periodicity of the p-Selmer ranks together with an explicit period. Finally, in Section 4, we illustrate how our results can be applied to particular examples by calculating the 2-Selmer rank parities for all quadratic twists of two explicit elliptic curves over Q. 1. General Notation Throughout this paper, p denotes a fixed rational prime and K a number field containing a primitive p-th root of 1. Let C(K) := Hom(GK ; µp) denote the group of characters of GK with order dividing p. χ (resp., 1K ) will denote a nontrivial (resp., the trivial) character in C(K). Note that by Kummer theory, we can identify C(K) with K∗=(K∗)p. A will denote a self-dual Abelian variety over K, and when dimK (A) = 1 we use the more standard notation E. Let ep : A[p]×A[p] ! µp be the nondegenerate alternating Weil pairing of A. Let Σ denote a fixed finite set of places of K containing all primes of bad reduction for A, all primes above p, and all archimedean places. Fix an algebraic closure K¯ of K and let GK := Gal(K=K¯ ) be the absolute Galois group of K. For each place v of K, fix an embedding of K¯ into the algebraic closure K¯v of the completion Kv of K at v. This gives an embedding of the absolute Galois group GKv := Gal(K¯v=Kv) into GK . If v is a finite place, Ov denotes the ring of integers of Kv, kv the residue field, Iv the inertia subgroup of GKv , and Gkv = GKv =Iv the absolute Galois group of kv. ON SELMER RANK PARITY OF TWISTS 3 For any GK -module M which is a finite dimensional Fp-vector space and any place v of 1 1 K, let locv : H (GK ;M) ! H (GKv ;M) denote the localization map. If M is unramified at 1 v, that is if Iv acts trivially on M, we define Hur(GKv ;M) to be the unramified subgroup 1 1 H (Gkv ;M) ⊂ H (GKv ;M), where the inclusion is induced by the inflation homomorphism. Finally, M ∗ denotes the Tate twist of the dual of M, i.e. ∗ M := HomGK (M; µp): 2. Preliminary Notions and Results 2.1. Global Metabolic Structures. A critical notion we will use in this paper is that of global metabolic structures as developed in [6]. For the reader's convenience, we briefly outline in this section what we will use. Let V be a finite dimensional vector space over Fp. Recall that a quadratic form on V is a function q : V ! Fp such that: 2 • q(av) = a q(v) for every a 2 Fp and v 2 V ; • the map (v; w)q := q(v + w) − q(v) − q(w) is a bilinear form. A maximal isotropic subspace of V is called a Lagrangian subspace. That is, a subspace ? X of V is Lagrangian if qjX = 0 and X = X with respect to the bilinear form (; )q. The quadratic space (V; q) is called a metabolic space if V has a Lagrangian subspace and (; )q is nondegenerate. All metabolic spaces relevant to our study arise in the following way. Let A be a self-dual Abelian variety defined over K, and let A[p] denote the p-torsion subgroup of A. Then for each place v of K, the cup product and the Weil pairing induce the local Tate pairing: 1 1 2 h; iv : H (GKv ;A[p]) × H (GKv ;A[p]) ! H (GKv ; µp); which is symmetric and nondegenerate (recall that local Tate duality states that for any Ga- 1 1 ∗ 2 lois module M, the pairing H (GKv ;M) × H (GKv ;M ) ! H (GKv ; µp) is nondegenerate). Note that if v is a finite place, then it is a well-known fact of local class field theory that 2 H (GKv ; µp) = Fp. 1 A global metabolic structure on A[p] consists of a quadratic form qv on H (GKv ;A[p]) for every place v of K, such that: 1 (1) the quadratic space (H (GKv ;A[p]); qv) is a metabolic space for every v; 1 (2) for every v2 = Σ, the unramified cohomology group Hur(GKv ;A[p]) is an isotropic subspace with respect to qv; 1 P (3) for any c 2 H (GK ;A[p]), v qv(locv(c)) = 0; (4) the bilinear form induced by qv is the local Tate pairing h; iv for every v. We conclude our discussion of global metabolic structures with a result concerning their existence and uniqueness. For this result, if the fixed rational prime p happens to be 2, we need to assume that the Abelian variety A is the Jacobian of a hyperelliptic curve with affine equation y2 = f(x), where f(x) is an odd-degree separable polynomial.
Load more

Recommended publications
  • Hasse Principles for Higher-Dimensional Fields 3
    Annals of Mathematics 183 (2016), 1{71 http://dx.doi.org/10.4007/annals.2016.183.1.1 Hasse principles for higher-dimensional fields By Uwe Jannsen dedicated to J¨urgen Neukirch Abstract For rather general excellent schemes X, K. Kato defined complexes of Gersten-Bloch-Ogus type involving the Galois cohomology groups of all residue fields of X. For arithmetically interesting schemes, he developed a fascinating web of conjectures on some of these complexes, which generalize the classical Hasse principle for Brauer groups over global fields, and proved these conjectures for low dimensions. We prove Kato's conjecture over number fields in any dimension. This gives a cohomological Hasse principle for function fields F over a number field K, involving the corresponding function fields Fv over the completions Kv of K. For global function fields K we prove the part on injectivity for coefficients invertible in K. Assuming resolution of singularities, we prove a similar conjecture of Kato over finite fields, and a generalization to arbitrary finitely generated fields. Contents 0. Introduction1 1. First reductions and a Hasse principle for global fields7 2. Injectivity of the global-local map for coefficients invertible in K 16 3. A crucial exact sequence, and a Hasse principle for unramified cohomology 26 4. A Hasse principle for Bloch-Ogus-Kato complexes 40 5. Weight complexes and weight cohomology 52 References 67 0. Introduction In this paper we prove some conjectures of K. Kato [Kat86] which were formulated to generalize the classical exact sequence of Brauer groups for a global field K, L (0.1) 0 −! Br(K) −! Br(Kv) −! Q=Z −! 0; v c 2016 Department of Mathematics, Princeton University.
    [Show full text]
  • Selmer Groups in Degree $\Ell$ Twist Families
    $\ell^{\infty}$-Selmer Groups in Degree $\ell$ Twist Families The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Smith, Alexander. 2020. $\ell^{\infty}$-Selmer Groups in Degree $\ell$ Twist Families. Doctoral dissertation, Harvard University, Graduate School of Arts & Sciences. Citable link https://nrs.harvard.edu/URN-3:HUL.INSTREPOS:37365902 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA `1-Selmer groups in degree ` twist families A dissertation presented by Alexander Smith to The Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Mathematics Harvard University Cambridge, Massachusetts April 2020 © 2020 – Alexander Smith All rights reserved. Dissertation Advisor: Professor Elkies and Professor Kisin Alexander Smith `1-Selmer groups in degree ` twist families Abstract Suppose E is an elliptic curve over Q with no nontrivial rational 2-torsion point. Given p a nonzero integer d, take Ed to be the quadratic twist of E coming from the field Q( d). For every nonnegative integer r, we will determine the natural density of d such that Ed has 2-Selmer rank r. We will also give a generalization of this result to abelian varieties defined over number fields. These results fit into the following general framework: take ` to be a rational prime, take F to be number field, take ζ to be a primitive `th root of unity, and take N to be an `- divisible Z`[ζ]-module with an action of the absolute Galois group GF of F .
    [Show full text]
  • Local Galois Cohomology Computation) Let K Be a finite Extension of Qp Or Fp((T)) with Residue field Fq
    Exercise 2 (due on October 26) Choose 4 out of 8 problems to submit. Problem 2.1. (Local Galois cohomology computation) Let K be a finite extension of Qp or Fp((t)) with residue field Fq. Let V be a representation of GK on an F`-vector space. 1 GK ∗ GK (1) Show that when ` 6= p, H (GK ;V ) = 0 unless V 6= 0 or V (1) 6= 0. When ` = p 1 and K = Qp, what is dim H (GQp ;V ) \usually"? (2) When ` 6= p, compute without using Euler characteristic formula, in an explicit way, i n dim H (GK ; F`(n)). Your answer will depend on congruences of q modulo `. Observe that the dimensions coincidence with the prediction of Tate local duality and Euler characteristic formula. i (3) When ` = p and K a finite extension of Qp, compute the dimension of dim H (GK ; Fp(n)). Problem 2.2. (Dimension of local Galois cohomology groups) Let K be a finite extension of Qp, and let V be a representation of GK over a finite dimensional F`-vector space. Suppose that V is irreducible as a representation of GK and dim V ≥ 2. 1 2 (1) When ` 6= p, show that H (GK ;V ) = 0. (Hint: compute H using local Tate duality and then use Euler characteristic.) 1 (2) When ` = p, what is dim H (GK ;V )? Problem 2.3. (An example of Poitou{Tate long exact sequence) Consider F = Q, and let S = fp; 1g for an odd prime p. Determine each term in the Poitou{Tate exact sequence for 2 the trivial representation M = Fp.
    [Show full text]
  • A Tate Duality Theorem for Local Galois Symbols II; the Semi-Abelian Case
    A TATE DUALITY THEOREM FOR LOCAL GALOIS SYMBOLS II; THE SEMI-ABELIAN CASE EVANGELIA GAZAKI* Abstract. This paper is a continuation to [Gaz17]. For every integer n ≥ 1, we consider sn 2 the generalized Galois symbol K(k; G1, G2)/n −→ H (k, G1[n] ⊗ G2[n]), where k is a finite extension of Qp, G1, G2 are semi-abelian varieties over k and K(k; G1, G2) is the Somekawa K-group attached to G1, G2. Under some mild assumptions, we describe the exact anni- 2 hilator of the image of sn under the Tate duality perfect pairing, H (k, G1[n] ⊗ G2[n]) × 0 H (k, Hom(G1[n] ⊗ G2[n],µn)) → Z/n. An important special case is when both G1, G2 are abelian varieties with split semistable reduction. In this case we prove a finiteness result, which gives an application to zero-cycles on abelian varieties and products of curves. 1. Introduction In the early 90’s Somekawa ([Som90]), following a suggestion of K. Kato, defined a new K-group, K(k; G1, ··· ,Gr) attached to a finite family of semi-abelian varieties over a field M k. This group is a generalization of the Milnor K-group, Kr (k), of a field k. In fact, in the most basic case when G1 = ··· = Gr = Gm, Somekawa proved an isomorphism M Kr (k) ≃ K(k; Gm, ··· , Gm). Moreover, similarly to the case of the classical Milnor K-group, for every integer n ≥ 1 invertible in k, there is a well-defined homomorphism to Galois cohomology, r sn : K(k; G1, ··· ,Gr)/n → H (k,G1[n] ⊗···⊗ Gr[n]), known as the generalized Galois symbol.
    [Show full text]
  • Galois Cohomology and Applications
    Galois Cohomology and Applications Shiva Chidambaram, discussed with Prof. Frank Calegari March 17, 2017 1 Notations and Preliminaries We start with some notation. For any field extension K F , GK F denotes the Galois group Gal(K F ). j j j For any field K , K denotes the separable closure of K , and G denotes Gal(K K ). If K is a local field, K j I denotes the inertia subgroup Gal(K K ur ), where K ur is the maximal unramified extension of K . K j If K is a number field and l is a prime in K , Gl denotes the absolute Galois group of the completion Kl , and if l is finite, Il denotes the inertia subgroup. Depending on our choice of embedding K , K , we can regard G as a subgroup of G in different ways. This choice doesn’t affect any ¡! l l K of our statements. Also, if S is a finite set of places of a number field K , KS denotes the maximal extension of K unramified outside S, and G denotes the corresponding Galois group Gal(K K ). K ,S Sj For any finite group G and a G-module M, unless we are dealing with Global Euler Characteristic formula or Wiles-Greenberg formula, H 0(G,M) will denote the modified Tate cohomology group G G 0 M /NG (M) rather than the usual M . In the two exceptional cases, we will use Hˆ (G,M) to denote the modified Tate cohomology groups, distinguishing it from the normal cohomology groups H 0(G,M).
    [Show full text]
  • Non Abelian Class Number Formulas and Adjoint Selmer Groups
    NON ABELIAN CLASS NUMBER FORMULAS AND ADJOINT SELMER GROUPS HARUZO HIDA Contents 1. Introduction 3 1.1. Hilbert class field. 3 ∗ 1.2. Dual class group ClK = Hom(ClK , Q/Z). 4 1.3. Pontryagin dual. 4 1.4. Group cohomology. 4 1.5. Compatible system of Galois representations. 4 1.6. Selmer group. 5 1.7. A variant of Bloch-Kato conjecture. 5 2. Congruence modules 5 2.1. Set up. 5 2.2. Differentials. 6 2.3. Universality. 6 2.4. Functoriality. 6 2.5. An algebra structure on R M and derivation. 7 2.6. Congruence modules. ⊕ 7 3. Galois deformation theory for Gm 8 3.1. Deformation of a character. 8 3.2. Ray class groups of finite level. 8 3.3. Ray class group of infinite level. 9 3.4. Groups algebra is universal. 9 3.5. Universal deformation ring for a Galois character ρ. 9 3.6. Congruence modules for group algebras. 9 3.7. Class group and Selmer group. 10 3.8. Class number formula. 10 4. Number of generators of adjoint Selmger grioups 10 4.1. Tangent spaces of local rings 10 4.2. Tangent space as adjoint cohomology group 11 4.3. p-Frattini condition. 13 5. Adjoint Selmer groups and differentials 13 5.1. p-Ordinarity condition 13 5.2. Ordinary deformation functor. 13 5.3. Fiber products. 14 5.4. Slight generalization. 14 5.5. Tangent space of deformation functors. 14 5.6. Tangent space of rings and deformation functor 14 5.7. Tangent space as cohomology group with local condition.
    [Show full text]
  • Grothendieck's Pairing on Neron Component Groups: Galois Descent
    GROTHENDIECK’S PAIRING ON NERON´ COMPONENT GROUPS: GALOIS DESCENT FROM THE SEMISTABLE CASE TAKASHI SUZUKI Abstract. In our previous study of duality for complete discrete valuation fields with perfect residue field, we treated coefficients in finite flat group schemes. In this paper, we treat abelian varieties. This in particular im- plies Grothendieck’s conjecture on the perfectness of his pairing between the N´eron component groups of an abelian variety and its dual. The point is that our formulation is well-suited with Galois descent. From the known case of semistable abelian varieties, we deduce the perfectness in full generality. We also treat coefficients in tori and, more generally, 1-motives. Contents 1. Introduction 2 1.1. Main results 2 1.2. General picture 4 1.3. Organization 6 2. Site-theoretic preliminaries 8 2.1. Sitesandalgebraicgroups: setupandfirstproperties 9 2.2. Generalities on derived categories of ind-procategories 13 2.3. The derived categories of ind-proalgebraic groups and of sheaves 21 2.4. Serre duality and P-acyclicity 26 3. Local fields with ind-rational base 33 3.1. Basic notions and properties 33 3.2. Topology on rational points of varieties 35 3.3. The relative fppf site of a local field 41 3.4. Cohomologyassheavesontheresiduefield 47 4. Statement of the duality theorem 53 arXiv:1410.3046v4 [math.NT] 22 Mar 2018 4.1. Formulation 53 4.2. Reduction to components groups and the first cohomology 54 5. Relation to Grothendieck’s and Safareviˇc’sˇ conjectures 59 5.1. Grothendieck’s pairing 59 5.2. Bester-Bertapelle’s isomorphism 61 5.2.1.
    [Show full text]
  • On a Generalization of Tate Dualities with Application to Iwasawa Theory Compositio Mathematica, Tome 85, No 2 (1993), P
    COMPOSITIO MATHEMATICA LI GUO On a generalization of Tate dualities with application to Iwasawa theory Compositio Mathematica, tome 85, no 2 (1993), p. 125-161 <http://www.numdam.org/item?id=CM_1993__85_2_125_0> © Foundation Compositio Mathematica, 1993, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Compositio Mathematica 85: 125-161,125 1993. © 1993 Kluwer Academic Publishers. Printed in the Netherlands. On a generalization of Tate dualities with application to Iwasawa theory LI GUO* Department of Mathematics, University of Washington, Seattle, WA98195 Received 8 April 1991; accepted 15 November 1991 Let E be an abelian variety defined over a number field K. Let p be a prime number. Let (K, E)p~ be the p-Tate-Shafarevich group of E and SclassEp~(K) the p°°-Selmer group of E. Thirty years ago, Tate proved a local duality theorem for E and used it to establish a global duality for E, later called Cassels-Tate pairing [3, 14]. It states that there is a pairing between (K, E)p~ and the p-Tate- Shafarevich group (K, E *)p. for the dual abelian variety E* of E and that this pairing is nondegenerate modulo the maximal divisible subgroups.
    [Show full text]
  • On the P-Converse of the Kolyvagin-Gross-Zagier Theorem
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by AIR Universita degli studi di Milano On the p-converse of the Kolyvagin-Gross-Zagier theorem Rodolfo Venerucci Universität Duisburg-Essen Fakultät für Mathematik, Mathematikcarrée Thea-Leymann-Straße 9, 45127 Essen. E-mail address: [email protected] Abstract. Let A=Q be an elliptic curve having split multiplicative reduction at an odd prime p. Under some mild technical assumptions, we prove the statement: rankZA(Q) = 1 and # X(A=Q)p1 < 1 =) ords=1L(A=Q; s) = 1; thus providing a ‘p-converse’ to a celebrated theorem of Kolyvagin-Gross-Zagier. Contents Introduction 1 1. Hida Theory 5 2. The theorem of Skinner-Urban7 3. Restricting to the central critical line 10 4. Bertolini-Darmon’s exceptional zero formula 13 5. Bounding the characteristic ideal via Nekovář’s duality 14 6. Proof of the main result 22 References 25 Introduction Let A be an elliptic curve defined over Q, let L(A=Q; s) be its Hasse-Weil L-function, and let X(A=Q) be its Tate-Shafarevich group. The (weak form of the) conjecture of Birch and Swinnerton-Dyer predicts that X(A=Q) is finite, and that the order of vanishing ords=1L(A=Q; s) of L(A=Q; s) at s = 1 equals the rank of the Mordell- Weil group A(Q). The main result to date in support of this conjecture comes combining the fundamental work of Kolyvagin [Kol90] and Gross-Zagier [GZ86] (KGZ theorem for short): ran := ords=1L(A=Q; s) ≤ 1 =) rankZA(Q) = ran and # X(A=Q) < 1: Let p be a rational prime, let ralg 2 f0; 1g, and let X(A=Q)p1 be the p-primary part of X(A=Q).
    [Show full text]
  • Euler Systems and Iwasawa Theory Sarah Livia Zerbes
    Euler systems and Iwasawa theory Sarah Livia Zerbes Introduction The theory of Euler systems is one of the most powerful tools available for studying the arithmetic of global Galois representations. However, constructing Euler systems is a difficult problem, and the list of known constructions was until recently accordingly rather short. In these lecture notes, we outline a general strategy for constructing new Euler systems in the cohomology of Shimura varieties: these Euler systems arise via pushforward of certain units on subvarieties. We study in detail two special cases of this construction: the Euler system of Beilinson{Flach elements, where the underlying Shimura variety is the fibre product of two modular curves; and the Euler system of Lemma{Flach elements, arising in the cohomology of Siegel modular threefolds. The lecture notes are structured as follows. • In Chapter 1, we will review the definition of Euler systems for Galois representations, and their arithmetic application to the Bloch{Kato conjecture. • In Chapter 2, we introduce some general tools for constructing global cohomology classes for Galois representations arising in geometry, assuming the existence of a supply of subvarieties of appropriate codimension and units on them. We also introduce Siegel units, which are the key for all the Euler system constructions to follow. • Chapter 3 is largely motivational (and can be skipped at a first reading): it explains how one can use Rankin{Selberg-type integral formulas for L-functions as a guide to where to look for Euler systems. • Chapter 4 is devoted to the construction of the Beilinson{Flach Euler system for pairs of modular forms of weight 2; and in Chapter 5, we discuss how to adapt this construction to pairs of modular forms of higher weight, using cohomology with coefficients.
    [Show full text]
  • Geometric Deformations of Orthogonal and Symplectic Galois Representations
    Geometric Deformations of Orthogonal and Symplectic Galois Representations Jeremy Booher submitted July 2016, corrections from paper version incorporated October 2016 Contents 1 Introduction 3 1.1 Motivation and Results . 4 1.1.1 Background about Galois Representations . 4 1.1.2 Geometric Lifts . 5 1.1.3 Relation to Previous Work . 6 1.2 Overview of the Proof . 6 1.2.1 Generalizing Ramakrishna's Method . 6 1.2.2 Fontaine-Laffaille Deformation Condition . 8 1.2.3 Minimally Ramified Deformation Condition . 9 1.3 Acknowledgements . 11 2 Producing Geometric Deformations 12 2.1 Preliminaries about Algebraic Groups . 12 2.1.1 Good and Very Good Primes . 12 2.1.2 Finite Groups of Lie Type . 13 2.2 Review of Galois Cohomology and Deformations . 15 2.2.1 Results about Galois Cohomology . 15 2.2.2 Deformations of Galois Representations . 17 2.3 Big Representations . 20 2.3.1 Big Representations . 20 2.3.2 Checking the Assumptions . 22 2.4 Ramakrishna's Deformation Condition . 23 2.4.1 Constructing the Deformation Condition . 24 2.4.2 Shrinking the Dual Selmer Group . 27 2.5 Generalizing Ramakrishna's Method . 31 2.5.1 Odd Galois Representations . 31 2.5.2 Local Lifting Implies Global Lifting . 32 2.5.3 Choosing Deformation Conditions . 33 2.5.4 Lifting Representations valued in Tori . 35 3 Fontaine-Laffaille Deformations with Pairings 37 3.1 Fontaine-Laffaille Theory with Pairings . 37 3.1.1 Covariant Fontaine-Laffaille Theory . 38 3.1.2 Tensor Products and Freeness . 40 3.1.3 Duality . 43 3.2 Fontaine-Laffaille Deformation Condition .
    [Show full text]
  • UNIVERSITY of CALIFORNIA Los Angeles Structure of Various
    UNIVERSITY OF CALIFORNIA Los Angeles Structure of various Lambda-adic arithmetic cohomology groups A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Jaehoon Lee 2019 c Copyright by Jaehoon Lee 2019 ABSTRACT OF THE DISSERTATION Structure of various Lambda-adic arithmetic cohomology groups by Jaehoon Lee Doctor of Philosophy in Mathematics University of California, Los Angeles, 2019 Professor Haruzo Hida, Chair In this thesis, we study the structure of various arithmetic cohomology groups as Iwasawa modules, made out of two different p-adic variations. In the first part, for an abelian va- riety over a number field and a Zp-extension, we study the relation between structure of the Mordell-Weil, Selmer and Tate-Shafarevich groups over the Zp-extension as Iwasawa modules. In the second part, we consider the tower of modular curves, which is an analogue of the Zp-extension in the first part. We study the structure of the ordinary parts of the arithmetic cohomology groups of modular Jacobians made out of this tower. We prove that the ordinary parts of Λ-adic Selmer groups coming from one chosen tower and its dual tower have almost the same Λ-module structures. This relation of the two Iwasawa modules explains well the functional equation of the corresponding p-adic L-function. We also prove the cotorsionness of Λ-adic Tate-Shafarevich group under mild assumptions. ii The dissertation of Jaehoon Lee is approved. Romyar Thomas Sharifi Chandrashekhar Khare Don M. Blasius Haruzo
    [Show full text]