On the images of modular and geometric three-dimensional Galois representations

We prove that the equation A 4 + B 2 = C p has no solutions in coprime positive integers when p ≥ 211. The main step is to show that, for all sufficiently large primes p , every [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /]-curve over an imaginary quadratic field K with a prime of potentially multiplicative reduction greater than 6 has a surjective mod p Galois representation. The bound on p depends on K and the degree of the isogeny between E and its Galois conjugate, but is independent of the choice of E . The proof of this theorem combines geometric arguments due to Mazur, Momose, Darmon, and Merel with an analytic estimate of the average special values of certain L -functions.

Deformations and the rigidity method

We list here Hecke eigenvalues of several automorphic forms for congruence subgroups of Sl(3; Z). To compute such tables, we describe an algorithm that combines techniques developed by Ash, Grayson and Green with the Lenstra–Lenstra–Lovász algorithm. With our implementation of this new algorithm we were able to handle much larger levels than those treated by Ash, Grayson and Green and by Top and van Geemen in previous work. Comparing our tables with results from computations of Galois representations, we find some new numerical evidence for the conjectured relation between modular forms and Galois representations.

We prove the modularity of minimally ramified ordinary residually reducible p-adic Galois representations of an imaginary quadratic field F under certain assumptions. We first exhibit conditions under which the residual representation is unique up to isomorphism. Then we prove the existence of deformations arising from cuspforms on GL2(AF) via the Galois representations constructed by Taylor et al. We establish a sufficient condition (in terms of the non-existence of certain field extensions which in many cases can be reduced to a condition on an L-value) for the universal deformation ring to be a discrete valuation ring and in that case we prove an R=T theorem. We also study reducible deformations and show that no minimal characteristic 0 reducible deformation exists.

Let k be a finite field, and let X be a smooth, projective curve over k with structure sheaf [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]. Let G be a finite group, and write C1([inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /][G]) for the reduced Grothendieck group of the category of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /][ G ]-vector bundles. In this paper we describe explicitly the subgroup of C1([inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /][ G ]) which is generated by the classes arising from G -stable invertible sheaves on tame Galois covers of X which have Galois group G .

The purpose of this course is to give an introduction to the theory of p-adic integration with values in spaces of modular forms (elliptic modular forms, Siegel modular forms, . . .). We show that very general p-adic families of modular forms can be constructed as moments of certain p-adic measures on a profinite group Y = lim ←− Yi with values in a formal q-expansion ring like Zp[[q ]] where B is an additive semi-group, and q = {q |ξ ∈ B} the corresponding formally written multiplicative semi-group (for example B = Bn = {ξ = ξ ∈ Mn(Q)|ξ ≥ 0, ξ half-integral} is the semi-group, important for the theory of Siegel modular forms). We discuss some applications of this theory to the construction of certain new p-adic families of modular forms (families of Klingen-Eisenstein series, families of theta-series with spherical polynomials. . .). Main sources of this theory are: • Serre’s theory of p-adic forms as certain formal q-expansions (J.-P. Serre, Formes modulaires et fonctions zeta p-adiques, LNM 350 (1973) 191-268) [Se73]. • Hida’s theory of p-adic modular forms and p-adic Hecke algebras (H. Hida, Elementary theory of L-functions and Eisenstein series, Cambridge University Press, 1993 [Hi93]). • Construction of p-adic Siegel-Eisenstein series by the author, see [PaSE]. As an application, we describe a solution of a problem of Coleman-Mazur in [PaTV], using the RankinSelberg method and the p-adic integration in a Banach algebra A. An introductory cours given on November 29 in POSTECH (Pohang, Korea) 0 Introduction Let p be a prime number (we often assume p≥ 5). There are two different ways of introducing p-adic modular forms: the first approach uses formal q-expansions with coefficients in a p-adic ring [Se73], and the second approach is the p-adic interpolation of Galois representations attached to classical automorphic forms. The first approach was extensively developped by Katz [Ka78] for the group G = GL2 over a totally real number field, in order to construct p-adic L-functions for CM-fields using p-adic Hilbert-Eisenstein series. In general, in this q-expansion method a typical p-adic family φ of modular (automorphic) forms is an element of the Serre ring: φ ∈ Λ[[q]] where Λ = Zp[[T ]] is the Iwasawa algebra. In the second approach one considers Λ-adic Galois representations of type ρ : Gal(Q/Q) → GLm(Λ) (“Big Galois representations”, see [Hi86], [Til-U]). These two theories are essentially equivalent if we start from holomorphic automorphic forms on the group G = GL2 over a totally real field, but in other cases there is no direct link between φ and ρ. On the other hand there exist interesting examples of p-adic L-functions Lφ,p and Lρ,p attached to φ and to ρ. In general Lφ,p and Lρ,p should belong to the quotient field L = QuotΛ or to its finite extensions. If ρ interpolates a p-adic family of motives then there are conjectural general definitions of Lρ,p (see [Co-PeRi], [Colm98], [PaAdm]). It would be very interesting to formulate a general Langlands-type conjecture relating Λ-adic automorphic forms and Λ-adic Galois representations. As an application, we describe a solution of a problem of Coleman-Mazur, using the Rankin-Selberg method and the theory of p-adic integration with values in a p-adic algebra A. This problem was stated in "The Eigencurve" (1998), R.Coleman and B.Mazur stated the following as follows: Given a prime p and Coleman’s family {fk′} of cusp eigenforms of a fixed positive slope σ = ordp(αp(k )) > 0, to construct a two variable p-adic L-function interpolating on k the Amice-Velu p-adic L-functions Lp(fk′ ). Our p-adic L-functions are p-adic Mellin transforms of certain A-valued measures. Such measures come from Eisenstein distributions with values in certain Banach A-modules M = M(N ;A) of families of overconvergent forms over A.

A BSD formula for high-weight modular forms

Teitelbaum formulated a conjecture relating first derivatives of the Mazur-Swinnerton-Dyer p -adic L -functions attached to modular forms of even weight k ≥ 2 to certain [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]-invariants arising from Shimura curve parametrizations. This article formulates an analogue of Teitelbaum's conjecture in which the cyclotomic [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] extension of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] is replaced by the anticyclotomic [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /]-extension of an imaginary quadratic field. This analogue is then proved by using the Cerednik-Drinfeld theory of p -adic uniformisation of Shimura curves.

We study compatible families of four-dimensional Galois representations constructed in the etale cohomology of a smooth projective variety. We prove a theorem asserting that the images will be generically large if certain conditions are satisfied. We only consider representations with coefficients in an imaginary quadratic field. We apply our result to an example constructed by Jasper Scholten (A non-selfdual 4-dimensional Galois representation, www.math.uiuc.edu/Algebraic-Number-Theory/0183, 1999), obtaining a family of linear groups and one of unitary groups as Galois groups over \({\mathbb{Q}}\) .

In this paper, we study the variation of Selmer groups in families of modular Galois representations that are congruent modulo a fixed prime $p \geq 5$ . Motivated by analogies with Goldfeld’s conjecture on ranks in quadratic twist families of elliptic curves, we investigate the stability of Selmer groups defined over $\mathbb{Q}$ via Greenberg’s local conditions under congruences of residual Galois representations. Let X be a positive real number. Fix a residual representation $\bar{\rho}$ and a corresponding modular form f of weight 2 and optimal level. We count the number of level-raising modular forms g of weight 2 that are congruent to f modulo p , with level $N_g\leq X$ , such that the p -rank of the Selmer groups of g equals that of f . Under some mild assumptions on $\bar{\rho}$ , we prove that this count grows at least as fast as $X (\log X)^{\alpha - 1}$ as $X \to \infty$ , for an explicit constant $\alpha \gt 0$ . The main result is a partial generalisation of theorems of Ono and Skinner on rank-zero quadratic twists to the setting of modular forms and Selmer groups.

Let K be an arbitrary number field, and let ρ : Gal($\math{\bar{K}}$/K) → GL2(E) be a nearly ordinary irreducible geometric Galois representation. In this paper, we study the nearly ordinary deformations of ρ. When K is totally real and ρ is modular, results of Hida imply that the nearly ordinary deformation space associated to ρ contains a Zariski dense set of points corresponding to ‘automorphic’ Galois representations. We conjecture that if K is not totally real, then this is never the case, except in three exceptional cases, corresponding to: (1) ‘base change’, (2) ‘CM’ forms, and (3) ‘even’ representations. The latter case conjecturally can only occur if the image of ρ is finite. Our results come in two flavours. First, we prove a general result for Artin representations, conditional on a strengthening of the Leopoldt Conjecture. Second, when K is an imaginary quadratic field, we prove an unconditional result that implies the existence of ‘many’ positive-dimensional components (of certain deformation spaces) that do not contain infinitely many classical points. Also included are some speculative remarks about ‘p-adic functoriality’, as well as some remarks on how our methods should apply to n-dimensional representations of Gal($\math{\bar{\QQ}}$/ℚ) when n > 2.

Perhaps the most fascinating connection between modular forms and number theory is the way in which they are connected with the existence of non-abelian extensions. Shimura [Sh 3] first established a connection between coefficients of certain modular forms, and the traces of Frobenius elements in extensions K of Q whose Galois group has a representation in GL 2 (F 1 ), and K is the field of l-division points of the curve X0(11), or in the Jacobian of X 1 (N), Theorem 7.14 of [Sh 2].KeywordsConjugate ClassisModular FormElliptic CurveGalois GroupIrreducible CharacterThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We consider families of Siegel eigenforms of genus $2$ and finite slope, defined as local pieces of an eigenvariety and equipped with a suitable integral structure. Under some assumptions on the residual image, we show that the image of the Galois representation associated with a family is big, in the sense that a Lie algebra attached to it contains a congruence subalgebra of non-zero level. We call the Galois level of the family the largest such level. We show that it is trivial when the residual representation has full image. When the residual representation is a symmetric cube, the zero locus defined by the Galois level of the family admits an automorphic description: it is the locus of points that arise from overconvergent eigenforms for $\operatorname{GL}_{2}$, via a $p$-adic Langlands lift attached to the symmetric cube representation. Our proof goes via the comparison of the Galois level with a ‘fortuitous’ congruence ideal. Some of the $p$-adic lifts are interpolated by a morphism of rigid analytic spaces from an eigencurve for $\operatorname{GL}_{2}$ to an eigenvariety for $\operatorname{GSp}_{4}$, while the remainder appear as isolated points on the eigenvariety.

Let X be a smooth projective variety over the complex numbers and let N 1 ( X ) be the real vector space of 1-cycles on X modulo numerical equivalence. As usual denote by NE ( X ) the cone of curves on X , i.e. the convex cone in N 1 ( X ) generated by the effective 1-cycles. One knows by the Cone Theorem [4] that the closed cone of curves [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]( X ) is rational polyhedral whenever c 1 ( X ) is ample. For varieties X such that c 1 ( X ) is not ample, however, it is in general difficult to determine the structure of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /]( X ). The purpose of this paper is to study the cone of curves of abelian varieties. Specifically, the abelian varieties X are determined such that the closed cone [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /]( X ) is rational polyhedral. The result can also be formulated in terms of the nef cone of X or in terms of the semi-group of effective classes in the Néron-Severi group of X .

The goal of this article is to give an explicit classification of the\npossible $p$-adic Galois representations that are attached to elliptic curves\n$E$ with CM defined over $\\mathbb{Q}(j(E))$. More precisely, let $K$ be an\nimaginary quadratic field, and let $\\mathcal{O}_{K,f}$ be an order in $K$ of\nconductor $f\\geq 1$. Let $E$ be an elliptic curve with CM by\n$\\mathcal{O}_{K,f}$, such that $E$ is defined by a model over\n$\\mathbb{Q}(j(E))$. Let $p\\geq 2$ be a prime, let $G_{\\mathbb{Q}(j(E))}$ be the\nabsolute Galois group of $\\mathbb{Q}(j(E))$, and let $\\rho_{E,p^\\infty}\\colon\nG_{\\mathbb{Q}(j(E))}\\to \\operatorname{GL}(2,\\mathbb{Z}_p)$ be the Galois\nrepresentation associated to the Galois action on the Tate module $T_p(E)$. The\ngoal is then to describe, explicitly, the groups of\n$\\operatorname{GL}(2,\\mathbb{Z}_p)$ that can occur as images of\n$\\rho_{E,p^\\infty}$, up to conjugation, for an arbitrary order\n$\\mathcal{O}_{K,f}$.\n