1. A Database of Invariant Rings (with Gregor Kemper, Elmar Körding, Gunter Malle, B. Heinrich Matzat, Denis Vogel), (Experimental Mathematics 10 (2001), 537-542). [Preprint version].
    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.
  2. Dihedral Galois representations and Katz modular forms (DOCUMENTA MATHEMATICA, Vol. 9 (2004), 123-133)
    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.
  3. Computing Hecke algebras of weight 1 in MAGMA, Appendix B of the article by Bas Edixhoven entitled Comparison of integral structures on spaces of modular forms of weight two, and computation of spaces of forms mod 2 of weight one (arXiv:math.NT/0312019), Journal of the Inst. of Math. Jussieu (2006) 5(1), 1-34. [Preprint version of the appendix].
    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.
  4. On the faithfulness of parabolic cohomology as a Hecke module over a finite field. Journal für die reine und angewandte Mathematik 606 (2007), 79-103. Preprint version on the arXiv.
    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.
  5. On modular symbols and the cohomology of Hecke triangle surfaces. International Journal of Number Theory (2009) 5(1), 89-108.
    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.
  6. 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.
    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.
  7. Multiplicities of Galois representations of weight one (with an appendix by Niko Naumann). Algebra and Number Theory, 1:1 (2007), 67-85. Preprint version on arXiv.
    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.
  8. On projective linear groups over finite fields as Galois groups over the rational numbers. In: 'Modular Forms on Schiermonnikoog' edited by Bas Edixhoven, Gerard van der Geer and Ben Moonen. Cambridge University Press, 2008, 343-350.
    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.
  9. On the generation of the coefficient field of a newform by a single Hecke eigenvalue (with Koopa Tak-Lun Koo and William Stein). Journal de Théorie des Nombres de Bordeaux 20 (2008), 373-384. Preprint version: arXiv:0711.3405.
    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.
  10. Zahlentheorie und Geometrie vereint in der Serre-Vermutung. Essener Unikate 33, 2008, 72-83. [Unpublizierte Langversion]
  11. On mod p representations which are defined over F_p: II (with L.J.P. Kilford). Glasgow Math. J. 52 (2010) 391--400.
    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.
  12. 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).
    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.
  13. On Modular Forms and the Inverse Galois Problem (with Luis Dieulefait). Trans. Amer. Math. Soc. 363 (2011), 4569-4584. Preprint version arXiv: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.
  14. A Computational Study of the Asymptotic Behaviour of Coefficient Fields of Modular Forms (with Marcel Mohyla). Publications Mathématiques de Besancon, numero dédié aux : Actes de la conférence "Théorie des Nombres et Applications", CIRM, 30/11-4/12 2009, pp. 75-98, 2011. Preprint version: arXiv: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.
  15. Die Serresche Modularitätsvermutung und Computer-Algebra. Computeralgebra-Rundbrief, Nr. 47, Oktober 2010, 9--13.
    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.
  16. On modular Galois representations modulo prime powers (with Imin Chen and Ian Kiming). Int. J. Number Theory, Vol. 9, No. 1 (2013), 91--113. Preprint version: arXiv:1105.1918v1 [math.NT].
    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.
  17. An Application of Maeda's Conjecture to the Inverse Galois Problem. Mathematical Research Letters, Volume 20 (2013), Number 5, 985-993. Preprint version: arXiv:1210.7157v3 [math.NT].

    It is shown that Maeda's conjecture on eigenforms of level 1 implies that for every positive even d and every p in a density-one set of primes, the simple group PSL_2(F_{p^d}) occurs as the Galois group of a number field ramifying only at p.

  18. Equidistribution of Signs for Modular Eigenforms of Half Integral Weight (with Ilker Inam). Archiv der Mathematik: Volume 101, Issue 4 (2013), Page 331-339. Preprint version: arXiv:1210.2319 [math.NT].

    Let f be a cusp form of weight k+1/2 and at most quadratic nebentype character whose Fourier coefficients a(n) are all real. We study an equidistribution conjecture of Bruinier and Kohnen for the signs of a(n). We prove this conjecture for certain subfamilies of coefficients that are accessible via the Shimura lift by using the Sato-Tate equidistribution theorem for integral weight modular forms. Firstly, an unconditional proof is given for the family {a(tp^2)}_p where t is a squarefree number and p runs through the primes. In this case, the result is in terms of the natural density. To prove it for the family {a(tn^2)}_n where t is a squarefree number and n runs through all natural numbers, we assume the existence of a suitable error term for the convergence of the Sato-Tate distribution, which is weaker than one conjectured by Akiyama and Tanigawa. In this case, the results are in terms of the Dedekind-Dirichlet density.

  19. On conjectures of Sato-Tate and Bruinier-Kohnen (with Sara Arias-de-Reyna and Ilker Inam). The Ramanujan Journal, 2015, 36(3), 455-481, DOI 10.1007/s11139-013-9547-2. Preprint version arXiv:1305.5443

    This article covers three topics. (1) It establishes links between the density of certain subsets of the set of primes and related subsets of the set of natural numbers. (2) It extends previous results on a conjecture of Bruinier and Kohnen in three ways: the CM-case is included; under the assumption of the same error term as in previous work one obtains the result in terms of natural density instead of Dedekind-Dirichlet density; the latter type of density can already be achieved by an error term like in the prime number theorem. (3) It also provides a complete proof of Sato-Tate equidistribution for CM modular forms with an error term similar to that in the prime number theorem.

  20. A Short Note on the Bruinier-Kohnen Sign Equidistribution Conjecture and Halász' Theorem (with Ilker Inam) International Journal of Number Theory Vol. 12, No. 02, pp. 357-360 (2016), DOI: 10.1142/S1793042116500214, Preprint version arXiv:1408.2210

    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 Hal\'asz' Theorem. Moreover, applying a result of Serre we remove all unproved assumptions.

  21. Hilbertian fields and Galois representations (with Lior Bary-Soroker and Arno Fehm) J. reine angew. Math. 712 (2016), 123-139, DOI 10.1515 / crelle-2013-0116 Preprint version arXiv:1203.4217
    We prove a new Hilbertianity criterion for fields in towers whose steps are Galois with Galois group either abelian or a product of finite simple groups. We then apply this criterion to fields arising from Galois representations. In particular we settle a conjecture of Jarden on abelian varieties.
  22. Applying modular Galois representations to the Inverse Galois Problem, Oberwolfach Reports, Volume 11, Issue 1, 2014, 305-309.

    For many finite groups, the Inverse Galois Problem can be approached through modular/automorphic Galois representations. This is a report explaining the basic strategy, ideas and methods behind some recent results. It focusses mostly on the 2-dimensional case and underlines in particular the importance of understanding coefficient fields.

  23. On Galois Representations of Weight One. Documenta Mathematica, 19 (2014), 689-707.
    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.
  24. Compatible systems of symplectic Galois representations and the inverse Galois problem I. Images of projective representations (with Sara Arias-de-Reyna and Luis Dieulefait). Trans. Amer. Math. Soc. 369 (2017), 887-908, Preprint version arXiv:1203.6546

    This article is the first part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem.

    In this first part, we determine the smallest field over which the projectivisation of a given symplectic group representation satisfying some natural conditions can be defined. The answer only depends on inner twists. We apply this to the residual representations of a compatible system of symplectic Galois representations satisfying some mild hypothesis and obtain precise information on their projective images for almost all members of the system, under the assumption of huge residual images, by which we mean that a symplectic group of full dimension over the prime field is contained up to conjugation. Finally, we obtain an application to the inverse Galois problem.

  25. Compatible systems of symplectic Galois representations and the inverse Galois problem II. Transvections and huge image (with Sara Arias-de-Reyna and Luis Dieulefait). Pacific Journal of Mathematics, Vol. 281, No. 1, 2016, 1-16,, Preprint version arXiv:1203.6552

    This article is the second part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse 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. A key ingredient is a classification of symplectic representations whose image contains a nontrivial transvection: these fall into three very simply describable classes, the reducible 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 with the main result of the first part to obtain a strenghtened application to the inverse Galois problem.

  26. Compatible systems of symplectic Galois representations and the inverse Galois problem III. Automorphic construction of compatible systems with suitable local properties (with Sara Arias-de-Reyna, Luis Dieulefait and Sug Woo Shin). Mathematische Annalen (2015), 361(3), 909-925. Preprint version arXiv:1308.2192

    This article is the third and last part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. This part proves the following new result for the inverse Galois problem for symplectic groups. For any even positive integer n and any positive integer d, PSp_n(F_{l^d}) or PGSp_n(F_{l^d}) occurs as a Galois group over the rational numbers for a positive density set of primes l. The result is obtained by showing the existence of a regular, algebraic, self-dual, cuspidal automorphic representation of GL_n(A_Q) with local types chosen so as to obtain a compatible system of Galois representations to which the results from Part II of this series applies.

  27. Classification of subgroups of symplectic groups over finite fields containing a transvection (with Sara Arias-de-Reyna and Luis Dieulefait), Demonstratio Mathematica, 49(2), 2016, 129-148. Preprint version arXiv:1405.1258

    In this note we give a self-contained proof of the following classification (up to conjugation) of subgroups of the general symplectic group of dimension n over a finite field of characteristic l, for l at least 5, which can be derived from work of Kantor: G is either reducible, symplectically imprimitive or it contains Sp(n, l). This result is for instance useful for proving "big image" results for symplectic Galois representations.

  28. On certain finiteness questions in the arithmetic of modular forms (with Ian Kiming and Nadim Rustom), J. London Math. Soc. (2016) 94 (2): 479-502. doi: 10.1112/jlms/jdw045. Preprint version arXiv:1408.3249

    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 reductions modulo p^m of normalized eigenforms on \Gamma_1(N). We consider various variants of our basic finiteness conjecture, prove a weak version of it, and give some numerical evidence.

  29. Topics on modular Galois representations modulo prime powers (with Panagiotis Tsaknias) in: Böckle, Decker, Malle (Editors). Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory, Springer-Verlag, 2018, 741-763, Preprint version arXiv:1612.05017

    This article surveys modularity, level raising and level lowering questions for two-dimensional representations modulo prime powers of the absolute Galois group of the rational numbers. It contributes some new results and describes algorithms and a database of modular forms orbits and higher congruences.

  30. Unramifiedness of Galois representations attached to weight one Hilbert modular eigenforms mod p (with Mladen Dimitrov), J. Inst. Math. Jussieu (2018), 1-26, doi:10.1017/S1474748018000026, Preprint version arXiv:1508.07722

    The main result of this article states that the Galois representation attached to a Hilbert modular eigenform over F_p^bar of parallel weight one and level prime to p is unramified above p. This includes the important case of eigenforms that do not lift to Hilbert modular forms in characteristic 0 of parallel weight one. The proof is based on the observation that parallel weight one forms in characteristic p embed into the ordinary part of parallel weight p in two different ways per place above p, namely via `partial' Frobenius operators. These are defined in the article along with and based on Hecke operators T_P for P dividing p. The theorem is deduced from known local properties of the Galois representation attached to ordinary eigenforms in characteristic 0.

Last modification: 5 June 2018.