Mod 4 Galois representations and elliptic curves

Galois representations with cyclotomic determinant all arise from the -torsion of elliptic curves for . For , we show the existence of more than a million such representations which are surjective and do not arise from any elliptic curve.

     

On the semisimplicity conjecture and Galois representations

The semisimplicity conjecture says that for any smooth projective scheme over a finite field , the Frobenius correspondence acts semisimply on , where is an algebraic closure of . Based on the works of Deligne and Laumon, we reduce this conjecture to a problem about the Galois representations of function fields. This reduction was also achieved by Laumon a few years ago (unpublished).

     

Mod Galois representations of solvable image

It is proved that, for a number field and a prime number , there exist only finitely many isomorphism classes of continuous semisimple Galois representations of into of fixed dimension and bounded Artin conductor outside which have solvable images. Some auxiliary results are also proved.

     

Reduction modulo p of cuspidal representations and weights in Serre's conjecture

Let be the ring of integers of a p-adic field and its maximalideal. This paper computes the Jordan–Hölder decompositionof the reduction modulo p of the cuspidal representations ofGL₂(/^e) for e 1. An alternative formulation of Serre's conjecturefor Hilbert modular forms is then provided.     

Torsion p-adic Galois representations and a conjecture of Fontaine

Let p be a prime, K a finite extension of Qp and T a finite free Zp-representation of . We prove that TZp_⊗Qp is semi-stable (resp. crystalline) with Hodge-Tate weights in {0,…,r} if and only if, for all n, T/pnT is torsion semi-stable (resp. crystalline) with Hodge-Tate weights in {0,…,r}.     

Galois representations modulo p and cohomology of Hilbert modular varieties

The aim of this paper is to extend some arithmetic results on elliptic modular forms to the case of Hilbert modular forms. Among these results let us mention:

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control of the image of Galois representations modulo p,

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Hida's congruence criterion outside an explicit set of primes,

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freeness of the integral cohomology of a Hilbert modular variety over certain local components of the Hecke algebra and Gorenstein property of these local algebras.

We study the arithmetic properties of Hilbert modular forms by studying their modulo p Galois representations and our main tool is the action of inertia groups at primes above p. In order to determine this action, we compute the Hodge-Tate (resp. Fontaine-Laffaille) weights of the p-adic (resp. modulo p) étale cohomology of the Hilbert modular variety. The cohomological part of our paper is inspired by the work of Mokrane, Polo and Tilouine on the cohomology of Siegel modular varieties and builds upon geometric constructions of Tilouine and the author.     

Arboreal Galois representations

Let be the absolute Galois group of , and let T be the complete rooted d-ary tree, where d ≥ 2. In this article, we study “arboreal” representations of into the automorphism group of T, particularly in the case d = 2. In doing so, we propose a parallel to the well-developed and powerful theory of linear p-adic representations of . We first give some methods of constructing arboreal representations and discuss a few results of other authors concerning their size in certain special cases. We then discuss the analogy between arboreal and linear representations of . Finally, we present some new examples and conjectures, particularly relating to the question of which subgroups of Aut(T) can occur as the image of an arboreal representation of .      

Computing modular Galois representations

We compute modular Galois representations associated with a newform $f$ , and study the related problem of computing the coefficients of $f$ modulo a small prime $\ell $ . To this end, we design a practical variant of the complex approximations method presented in Edixhoven and Couveignes (Ann. of Math. Stud., vol. 176, Princeton University Press, Princeton, 2011). Its efficiency stems from several new ingredients. For instance, we use fast exponentiation in the modular jacobian instead of analytic continuation, which greatly reduces the need to compute abelian integrals, since most of the computation handles divisors. Also, we introduce an efficient way to compute arithmetically well-behaved functions on jacobians, a method to expand cuspforms in quasi-linear time, and a trick making the computation of the image of a Frobenius element by a modular Galois representation more effective. We illustrate our method on the newforms $\Delta $ and $E_4 \cdot \Delta $ , and manage to compute for the first time the associated faithful representations modulo $\ell $ and the values modulo $\ell $ of Ramanujan’s $\tau $ function at huge primes for $\ell \in \{ 11,13,17,19,29\}$ . In particular, we get rid of the sign ambiguity stemming from the use of a projective representation as in Bosman (On the computation of Galois representations associated to level one modular forms. arxiv.org/abs/0710.1237, 2007). As a consequence, we can compute the values of $\tau (p)~\mathrm{mod}~2^{11} \times 3^6 \times 5^3 \times 7 \times 11 \times 13 \times 17 \times 19 \times 23 \times 29 \times 691 \approx 2.8 \times 10^{19}$ for huge primes $p$ . The representations we computed lie in the jacobian of modular curves of genus up to $22$ .     

On unramified Galois extensions constructed using Galois representations

 For a field k, We denote the maximal abelian extension of k by k _ab and (K _ab ^r−1 _ab by k _ab ^r . In this paper, unramified Galois extensions over k _ab ^r are constructed using Galois representations of arbitrary dimension with larger coefficient rings. Received: 31 August 2001 / Revised version: 22 March 2002 Mathematics Subject Classification (2000): 11R21     

Multiplicities of modp Galois representations

We study the multiplicity with which 2-dimensional modp Galois representations occur in Jacobians of modular curves.     

Galois representations on holomorphic differentials

Using the same methods as in [S2] and [S3] we derive the Brauer trees for the cyclic pblocks for G₂(2^k),G₂(3^k) and the noncyclic pblocks for these groups, where p ≠ 2,3. The results are analogous to those obtained for G₂(q), where q is not divisible by 2 or 3. These results were first announced in [Sl].     

Minimum product set sizes in nonabelian groups of order pq

Let G be a nonabelian group of order pq, where p and q are distinct odd primes. We analyze the minimum product set cardinality μG(r,s)=min|AB|, where A and B range over all subsets of G of cardinalities r and s, respectively. In this paper, we completely determine μG(r,s) in the case where G has order 3p and conjecture that this result can be extended to all nonabelian groups of order pq. We also prove that for every nonabelian group of order pq there exist 1?r,s?pq such that μG(r,s)>μZ/pqZ(r,s).     

Abhyankar's conjecture on Galois groups over curves

Sans résuméOblatum 5-I-1993En hommage à Armand Borel avec notre admiration     

Mod representations of elliptic curves

Explicit equations are given for the elliptic curves (in characteristic ) with mod representation isomorphic to that of a given one.