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On Galois Representations of Weight One
A two-dimensional Galois representation into the Hecke algebra of Katz modular forms of weight one over a finite field of characteristic p is constructed and is shown to be unramified at p in most cases.
On modular Galois representations modulo prime powers (with Imin Chen and Ian Kiming)
We study modular Galois representations mod p^m. We show that there are three progressively weaker notions of modularity for a Galois representation mod p^m: we have named these `strongly', `weakly', and `dc-weakly' modular. Here, `dc' stands for `divided congruence' in the sense of Katz and Hida. These notions of modularity are relative to a fixed level M. Using results of Hida we display a `stripping-of-powers of p away from the level' type of result: A mod p^m strongly modular representation of some level Np^r is always dc-weakly modular of level N (here, N is a natural number not divisible by p). We also study eigenforms mod p^m corresponding to the above three notions. Assuming residual irreducibility, we utilize a theorem of Carayol to show that one can attach a Galois representation mod p^m to any `dc-weak' eigenform, and hence to any eigenform mod p^m in any of the three senses. We show that the three notions of modularity coincide when m=1 (as well as in other, particular cases), but not in general.

Refereed Publications

A Computational Study of the Asymptotic Behaviour of Coefficient Fields of Modular Forms
with Marcel Mohyla, 19 pages, to appear in Publications Mathématiques de Besancon, numero dédié aux : Actes de la conférence "Théorie des Nombres et Applications", CIRM, 30/11-4/12 2009. arXiv:math/0910.2251
The article motivates, presents and describes large computer calculations concerning the asymptotic behaviour of arithmetic properties of coefficient fields of modular forms. The observations suggest certain patterns, which deserve further study.
On Modular Forms and the Inverse Galois Problem.
with Luis Dieulefait, accepted for publication in the Transactions of the AMS, 17 pages, arXiv:math/0905.1288
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 $PSL_2(F_{p^n})$ occurs as the Galois group of some finite extension of the rational numbers. 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 residual Galois representations are as large as a priori possible. Both results essentially use Khare's and Wintenberger's notion of good-dihedral primes. Particular care is taken in order to exclude nontrivial inner twists.
Computing Congruences of Modular Forms and Galois Representations Modulo Prime Powers
with Xavier Taixes i Ventosa, in Arithmetic, Geometry, Cryptography and Coding Theory 2009, edited by: David Kohel and Robert Rolland. Contemporary Mathematics 521 (2010), 145-166. arXiv:0909.2724v2
This article starts a computational study of congruences of modular forms and modular Galois representations modulo prime powers. Algorithms are described that compute the maximum integer modulo which two monic coprime integral polynomials have a root in common in a sense that is defined. These techniques are applied to the study of congruences of modular forms and modular Galois representations modulo prime powers. Finally, some computational results with implications on the (non-)liftability of modular forms modulo prime powers and possible generalisations of level raising are presented.
On mod p representations which are defined over F_p: II
with L.J.P. Kilford, Glasgow Math. J. 52 (2010) 391-400 arXiv:0905.4372v1
The behaviour of Hecke polynomials modulo $p$ has been the subject of some study. In this note we show that, if $p$ is a prime, the set of integers $N$ such that the Hecke polynomials $T^{N,\chi}_{l,k} $ for all primes l, all weights $k>1$ and all characters $\chi$ taking values in $\{+1,-1\}$ splits completely modulo $p$ has density 0, unconditionally for $p=2$ and under the Cohen-Lenstra heuristics for odd $p$. The method of proof is based on the construction of suitable dihedral modular forms.
On the generation of the coefficient field of a newform by a single Hecke eigenvalue
with Koopa Tak-Lun Koo and William Stein, in Journal de Théorie des Nombres de Bordeaux 20 (2008), 373-384, arXiv:0711.3405v2
Let $f$ be a non-CM newform of weight $k>1$ without nontrivial inner twists. In this article we study the set of primes $p$ such that the eigenvalue $a_p(f)$ of the Hecke operator $T_p$ acting on $f$ generates the field of coefficients of $f$. We show that this set has density 1, and prove a natural analogue for newforms having inner twists. We also present some new data on reducibility of Hecke polynomials, which suggest questions for further investigation.
On projective linear groups over finite fields as Galois groups over the rational numbers
in 'Modular Forms on Schiermonnikoog' Cambridge University Press (2008), 343-350, arXiv:math/0606732v4
Ideas and techniques from Khare's and Wintenberger's article 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 $PSL_2(F_{l^r})$ (for $r$ running) occur as Galois groups over the rationals such that the corresponding number fields are unramified outside a set consisting of $l$, the infinite place and only one other prime.
Multiplicities of Galois representations of weight one
with an appendix by N. Naumann, Algebra and Number Theory, 1:1 (2007), 67-85 [Preprint version: arXiv:math/0612318v1]
In this article we consider mod $p$ modular Galois representations which are unramified at $p$ such that the Frobenius element at $p$ acts through a scalar matrix. The principal result states that the multiplicity of any such representation is bigger than 1.
On the failure of the Gorenstein property for Hecke algebras of prime weight
with L.J.P. Kilford, Experimental Mathematics 17(1), 2008, 37-52, arXiv:math/0612317v1
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 the ask the question whether the Gorenstein defect and the multplicity 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.
Appendix A: Manual of Magma package HeckeAlgebra,
Appendix B: Tables of Hecke algebras.
On modular symbols and the cohomology of Hecke triangle surfaces
International Journal of Number Theory (2009) 5(1), 89-108, arXiv:math/0511113v2
The aim of this article is to give a concise algebraic treatment of the modular symbols formalism, generalised 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. In all the note 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 the faithfulness of parabolic cohomology as a Hecke module over a finite field
In this article we prove that under certain conditions the Hecke algebra of cuspidal modular forms over $F_p$ coincides with the Hecke algebra of a certain parabolic group cohomology group with coefficients in $F_p$. These results can e.g. be used to compute Katz modular forms of weight one over an algebraic closure of $F_p$ with methods of linear algebra over $F_p$.
On the faithfulness of parabolic cohomology as a Hecke module over a finite field
in Journal of the Inst. of Math. Jussieu (2006) 5(1), 1-34, arXiv:math/0312019v1
We report on an implementation in Magma of the algorithm for calculating Hecke algebras of Katz modular forms exposed in the main article. Moreover, results of some computations are included.
Preprint version of the appendix: [pdf]
Dihedral Galois representations and Katz modular forms
We show that any two-dimensional odd dihedral representation $\rho$ over a finite field of characteristic $p>0$ of the absolute Galois group of the rational numbers can be obtained from a Katz modular form of level $N$, character $\epsilon$ and weight $k$, where $N$ is the conductor, $\epsilon$ is the prime-to-$p$ part of the determinant and $k$ is the so-called minimal weight of $\rho$. In particular, $k=1$ if and only if $\rho$ is unramified at $p$. Direct arguments are used in the exceptional cases, where general results on weight and level lowering are not available.
A Database of Invariant Rings
with G. Kemper, E. Körding, G. Malle, B. Heinrich Matzat, D. Vogel, in Experimental Mathematics 10 (2001), 537-542
Preprint version: [pdf]
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.

Nonrefereed Publications

Die Serresche Modularitätsvermutung und Computer-Algebra
Computeralgebra-Rundbrief, Nr. 47, Oktober 2010, 9-13. [pdf]
In diesem Artikel für Nichtspezialisten wird die kürzlich von Khare, Wintenberger und Kisin bewiesene Serresche Modularitätsvermutung vorgestellt und ihre Bedeutung in der Computer-Algebra erläutert.
Zahlentheorie und Geometrie vereint in der Serre-Vermutung
in Essener Unikate 33, 2008, 72-83, Unpublished unabbreviated version: [pdf]

Computer Programs

MAGMA package, download & info
This package contains functions for handling commutative algebras over fields (e.g. matrix algebras), such as the computation of the decomposition of such an algebra into a direct product of its localisations, and the computation of an isomorphic affine algebra with few relations. Functions testing certain properties (e.g. the Gorenstein property) are included. Some functions only work over finite fields (for technical, not conceptual reasons).
MAGMA package, download & info
This package computes Katz modular forms of weight one over finite fields. It is based on an algorithm by Edixhoven.
Hecke Algebra
MAGMA package, download & info
This package is designed for the computation of Hecke algebras of modular forms over finite fields of characteristic p in the following cases:
  • all eigenforms for a given level $N$, weight $k$ and Dirichlet character of modulus $N$,
  • dihedral modular forms coming from the Hilbert class field of a quadratic number field,
  • icosahedral modular forms (currently only in characteristic $p=2$) given by an A5-polynomial.
by Georg Weber, download & info
Fast computation of the Hecke operator $T_p$ on weight 2 modular symbols on $\Gamma_0(N)$ for "small" $N$ and "large" $p$.
The main function is written in C. A Magma interface is provided. It should be easy to adapt this for Sage.