Gabor Wiese

On modular forms and the inverse Galois problem

In this article new cases of the Inverse Galois Problem are established. The main result is that for a fixed integer n, there is a positive density set of primes p such that PSL2(Fpn) occurs as the Galois group of some finite extension of the rational nu mbers. These groups are obtained as projective images of residual modular Galois representations. Moreover, families of modular forms are constructed such that the images of all their resid ual Galois representations are as large as a priori possible. Both results essentially use Khare’s a nd Wintenberger’s notion of gooddihedral primes. Particular care is taken in order to exclud e nontrivial inner twists. 2000 Mathematics Subject Classification: 11F80 (primary); 12F12, 11F11.

On projective linear groups over finite fields as Galois groups over the rational numbers

Ideas and techniques from Khare´s and Wintenberger’s preprint on the proof of Serre’s conjecture for odd conductors are used to establish that for a fixed prime l infinitely many of the groups PSL₂(Flr ), PGL₂(Flr ) (for r running) occur as Galois groups over the rationals such that the corresponding number fields are unramified outside a set consisting of l and only one other prime.

On the failure of the Gorenstein property for Hecke algebras of prime weight

In this article we report on extensive calculations concerning the Gorenstein defect for Hecke algebras of spaces of modular forms of prime weight p at maximal ideals of residue characteristic p such that the attached mod-p Galois representation is unramified at p and the Frobenius at p acts by scalars. The results lead us to ask the question whether the Gorenstein defect and the multiplicity of the attached Galois representation are always equal to 2. We review the literature on the failure of the Gorenstein property and multiplicity one, discuss in some detail a very important practical improvement of the modular-symbols algorithm over finite fields, and include precise statements on the relationship between the Gorenstein defect and the multiplicity of Galois representations.

Compatible systems of symplectic Galois representations and the inverse Galois problem II: Transvections and huge image

This article is the second part of a series of three articles a bout compatible systems of symplectic Galois representations and applications to the inv erse Galois problem. This part is concerned with symplectic Galois representations having a huge residual image, by which we mean that a symplectic group of full dimension over the prime field is contained up to conjugation. We prove a classification result on those sub groups of a general symplectic group over a finite field that contain a nontrivial transvection. Tr anslating this group theoretic result into the language of symplectic representations whose image contains a nontrivial transvection, these fall into three very simply describable classes: the reduci ble ones, the induced ones and those with huge image. Using the idea of an (n,p)-group of Khare, Larsen and Savin we give simple conditions under which a symplectic Galois representation with coefficients in a finite field has a huge image. Finally, we combine this classification result w ith the main result of the first part to obtain a strenghtened application to the inverse Galois pro blem. MSC (2010): 11F80 (Galois representations); 20G14 (Linear algebraic groups over finite fields), 12F12 (Inverse Galois theory).

On the faithfulness of parabolic cohomology as a Hecke module over a finite field

Abstract This article exhibits conditions under which a certain parabolic group cohomology space over a finite field 𝔽 is a faithful module for the Hecke algebra of cuspidal Katz modular forms over an algebraic closure of 𝔽. These results can e.g. be applied to compute cuspidal Katz modular forms of weight one with methods of linear algebra over 𝔽.

A Database of Invariant Rings

We announce the creation of a database of invariant rings. This database contains a large number of invariant rings of finite groups, mostly in the modular case. It gives information on generators and structural properties of the invariant rings. The main purpose is to provide a tool for researchers in invariant theory.

A Short Note on the Bruinier-Kohnen Sign Equidistribution Conjecture and Halasz' Theorem

In this note, we improve earlier results towards the Bruinier–Kohnen sign equidistribution conjecture for half-integral weight modular eigenforms in terms of natural density by using a consequence of Halasz’ theorem. Moreover, by applying a result of Serre, we remove all unproved assumptions.

Computations with modular forms : proceedings of a summer school and conference, Heidelberg, August/September 2011

Part I Summer School: Lectures on Computing Cohomology of Arithmetic Groups by P.E. Gunnells.- Computing with Algebraic Automorphic Forms by D. Loeffler.- Overconvergent Modular Symbols by R. Pollack.- Part II Conference and Research Contributions: Congruence Subgroups, Cusps and Manin Symbols over Number Fields by J. E. Cremona and M. T. Aranes.- Computing Weight One Modular Forms over C and Fp by K. Buzzard.- Lattice Methods for Algebraic Modular Forms on Classical Groups by M. Greenberg and J. Voight.- Efficient Computation of Rankin p-Adic L-Functions by A.G.B. Lauder.- Formes Modulaires Modulo 2 et Composantes Reelles de Jacobiennes Modulaires by L.Merel.- Universal Hecke L-Series Associated with Cuspidal Eigenforms over Imaginary Quadratic Fields by A. Mohamed.- On Higher Congruences Between Cusp Forms and Eisenstein Series by B. Naskrecki.- Arithmetic Aspects of Bianchi Groups by M.H.Sengun.- A Possible Generalization of Maedas Conjecture by P. Tsaknias.- Computing Power Series Expansions of Modular Forms by J. Voight and J. Willis.- Computing Modular Forms for GL2 over Certain Number Fields by D. Yasaki.

ON MODULAR SYMBOLS AND THE COHOMOLOGY OF HECKE TRIANGLE SURFACES

The aim of this article is to give a concise algebraic treatment of the modular symbols formalism, generalized from modular curves to Hecke triangle surfaces. A sketch is included of how the modular symbols formalism gives rise to the standard algorithms for the computation of holomorphic modular forms. Precise and explicit connections are established to the cohomology of Hecke triangle surfaces and group cohomology. A general commutative ring is used as coefficient ring in view of applications to the computation of modular forms over rings different from the complex numbers.

On certain finiteness questions in the arithmetic of modular forms

We investigate certain finiteness questions that arise naturally when studying approximations modulo prime powers of p-adic Galois representations coming from modular forms. We link these finiteness statements with a question by K. Buzzard concerning p-adic coefficient fields of Hecke eigenforms. Specifically, we conjecture that for fixed N , m, and prime p with p not dividing N , there is only a finite number of characters arising from reductions modulo pm of p-adic representations attached to eigenforms on Γ1(N). We consider various variants of our basic finiteness conjecture, prove some results that support it, and give some numerical evidence.