Galois Image Research Articles

Abelian varieties over number fields, tame ramification and big Galois image

Abelian varieties over number fields, tame ramification and big Galois image

On uniform large Galois images for modular abelian varieties

We formulate a question regarding uniform versions of "large Galois image properties" for modular abelian varieties of higher dimension, generalizing the well-known case of elliptic curves. We then answer our question affirmatively in the exceptional image case, and provide lower estimates for uniform bounds in the remaining cases.

On the image of Galois $l$-adic representations for abelian varieties of type III

In this paper we investigate the image of the $l$-adic representation attached to the Tate module of an abelian variety defined over a number field. We consider simple abelian varieties of type III in the Albert classification. We compute the image of the $l$-adic and mod $l$ Galois representations and we prove the Mumford-Tate and Lang conjectures for a wide class of simple abelian varieties of type III.

On uniform lower bound of the Galois images associated to elliptic curves

Soit p un nombre premier et K un corps de nombres. Soit ρ E,p :G K →Aut(T p E)≅GL 2 (ℤ p ) la représentation Galoisienne donnée par l’action du groupe de Galois sur le module de Tate p-adique d’une courbe elliptique E définie sur K. Serre a prouvé que l’image de ρ E,p est ouverte si E n’a pas de multiplication complexe. Pour E une courbe elliptique définie sur K et dont l’invariant j n’appartient pas à un ensemble fini exceptionnel (qui est non explicite cependant), nous donnons une minoration uniforme et explicite de la taille de l’image de ρ E,p .

Gonality of modular curves in characteristic~$p$

Let k be an algebraically closed field of characteristic p. Let X(p^e;N) be the curve parameterizing elliptic curves with full level N structure (where p does not divide N) and full level p^e Igusa structure. By modular curve, we mean a quotient of any X(p^e;N) by any subgroup of ((Z/p^e Z)^* x \SL_2(Z/NZ))/{+-1}. We prove that in any sequence of distinct modular curves over k, the k-gonality tends to infinity. This extends earlier work, in which the result was proved for particular sequences of modular curves, such as X_0(N) for p not dividing N. We give an application to the function field analogue of a uniform boundedness statement for the image of Galois on torsion of elliptic curves.

A cuspidality criterion for the functorial product on GL(2) × GL(3) with a cohomological application

For all cusp forms π on GL(3) and π′ on GL(2) over a number field F, H. Kim and F. Shahidi have functorially associated an automorphic form Formula on GL(6) such that L(s, Π) agrees with the Rankin-Selberg L-function of the pair (π, π′). First we establish a criterion as to when Π is cuspidal. Then we apply it to construct non-self-dual, nonmonomial cuspidal cohomology classes for suitable congruence subgroups of SL(3, ℤ). We also analyze the Galois image of certain related l-adic representations.

The Multiple Zeta Value Algebra and the Stable Derivation Algebra

The MZV algebra is the graded algebra over \mathbf Q generated by all multiple zeta values. The stable derivation algebra is a graded Lie algebra version of the Grothendieck–Teichmüller group. We shall show that there is a canonical surjective \mathbf Q -linear map from the graded dual vector space of the stable derivation algebra over \mathbf Q to the new-zeta space, the quotient space of the sub-vector space of the MZV algebra whose grade is greater than 2 by the square of the maximal ideal. As a corollary, we get an upper-bound for the dimension of the graded piece of the MZV algebra at each weight in terms of the corresponding dimension of the graded piece of the stable derivation algebra. If some standard conjectures by Y. Ihara and P. Deligne concerning the structure of the stable derivation algebra hold, this will become a bound conjectured in Zagier’s talk at 1st European Congress of Mathematics. Via the stable derivation algebra, we can compare the new-zeta space with the l-adic Galois image Lie algebra which is associated with the Galois representation on the pro- l fundamental group of \mathbf P\frac 1 {\mathbf Q} − \{0, 1, ∞\} .

The rational function analogue of a question of Schur and exceptionality of permutation representations

In 1923 Schur considered the following problem. Let f(X) be a polynomial with integer coefficients that induces a bijection on the residue fields Z/pZ for infinitely many primes p. His conjecture, that such polynomials are compositions of linear and Dickson polynomials, was proved by M. Fried in 1970. Here we investigate the analogous question for rational functions, also we allow the base field to be any number field. As a result, there are many more rational functions for which the analogous property holds. Some infinite series come from rational isogenies of elliptic curves and deformations. There are several sporadic examples which do not fit in any of the series we obtain. First we translate the arithmetic property to a question about finite permutation groups, and classify those groups which fulfill the necessary conditions. The proofs depend on the classification of the finite simple groups. Then we use arithmetic arguments to either rule out many cases, or to prove that the remaining cases indeed give rise to examples. This part is based on Mazur's classical results about rational points on the modular curves X_0(p) and X_1(p), results about Galois images in GL_2(p) coming from action of the absolute Galois group of Q on p-torsion points of elliptic curves, the theory of complex multiplication, and the techniques used in the inverse regular Galois problem.

Homothéties et images de Galois dans les représentations associées aux motifs

We prove that the image of Galois in the p-adic Galois-representation associated with a pure motive of non-zero weight on a number field contains for p large non-trivial homotheties of order prime to p. The vanishing of cohomology groups of the image of Galois with values in these representations follows.

The Mumford-Tate Conjecture for Drinfeld-Modules

Consider the Galois representation on the Tate module of a Drinfeld module over a finitely generated field in generic characteristic. The main object of this paper is to determine the image of Galois in this representation, up to commensurability. We also determine the Dirichlet density of the set of places of prescribed reduction type, such as places of ordinary reduction.

Rigidity and Semi-invariants in Drinfeld Modules

We show that if φ is a Drinfeld A-module over a field L, then any polynomial g( x) ∈ L[ x] which maps the a-torsion into itself for all a ∈ A must be an endomorphism of φ. In generic characteristic, we prove a stronger result: if there is an infinite A-submodule S of the torsion submodule (over L̄) such that g maps S into S, or if g maps the a-torsion into itself for infinitely many a, then g is an endomorphism followed by a translation. The first of these results generalizes easily to maps between different Drinfeld modules. The second can be generalized as well, assuming a conjecture which would follow from an analogue of Serre′s theorem on the image of Galois. Analogous rigidity results are known to hold for abelian varieties. As one application, we show that the ring of semi-invariants of a Drinfeld module (defined by D. Goss) is nothing more than the ring of endomorphisms.

On the universal power series for Jacobi sums and the vandiver conjecture

We shall study some explicit connections between (1) the Vandiver conjecture on the class number of the real cyclotomic field Q (cos(2π/ l)) and (2) the images of various Galois representations induced from the power series representation (constructed and studied by Ihara, Anderson, Coleman, etc.) of Gal( Q/Q(μ I∞)) which describes universally the Galois action on the Fermat curves of l-power degrees. One such connection was first discovered by Coleman. In the case of the original power series representation, we shall also describe the difference between the “expected image” and the actual Galois image in terms of a certain invariant of Iwasawa type.