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Die Serresche Modularitätsvermutung und Computer-Algebra Wiese, Gabor in Computeralgebra-Rundbrief (2010), 47 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. Detailed reference viewed: 23 (0 UL)On mod $p$ representations which are defined over $\Bbb F_p$: II ; Wiese, Gabor in Glasgow Mathematical Journal (2010), 52(2), 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 ... [more ▼] 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. [less ▲] Detailed reference viewed: 30 (0 UL)Computing congruences of modular forms and Galois representations modulo prime powers ; Wiese, Gabor in Arithmetic, geometry, cryptography and coding theory 2009 (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 ... [more ▼] 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. [less ▲] Detailed reference viewed: 42 (2 UL)On modular symbols and the cohomology of Hecke triangle surfaces Wiese, Gabor in 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 ... [more ▼] 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. [less ▲] Detailed reference viewed: 42 (0 UL)On projective linear groups over finite fields as Galois groups over the rational numbers Wiese, Gabor in Modular forms on Schiermonnikoog (2008) 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 ... [more ▼] 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. [less ▲] Detailed reference viewed: 31 (0 UL)On the generation of the coefficient field of a newform by a single Hecke eigenvalue ; ; Wiese, Gabor in Journal de Théorie des Nombres de Bordeaux (2008), 20(2), 373--384 # 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 ... [more ▼] # 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. [less ▲] Detailed reference viewed: 162 (1 UL)Zahlentheorie und Geometrie vereint in der Serre-Vermutung Wiese, Gabor in Essener Unikate (2008), 33 Detailed reference viewed: 56 (1 UL)On the failure of the Gorenstein property for Hecke algebras of prime weight ; Wiese, Gabor in Experimental Mathematics (2008), 17(1), 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 ... [more ▼] 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. The Magma package, instructions for its use, generated tables and the complete data are available as supplemental material. [less ▲] Detailed reference viewed: 49 (0 UL)On the faithfulness of parabolic cohomology as a Hecke module over a finite field Wiese, Gabor in Journal für die Reine und Angewandte Mathematik (2007), 606 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 ... [more ▼] 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. [less ▲] Detailed reference viewed: 37 (1 UL)Multiplicities of Galois representations of weight one Wiese, Gabor in Algebra Number Theory (2007), 1(1), 67--85 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 ... [more ▼] 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. [less ▲] Detailed reference viewed: 46 (2 UL)Dihedral Galois representations and Katz modular forms Wiese, Gabor in Documenta Mathematica (2004), 9 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 ... [more ▼] 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. [less ▲] Detailed reference viewed: 47 (1 UL)A database of invariant rings ; ; et al in Experimental Mathematics (2001), 10(4), 537--542 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 ... [more ▼] 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. [less ▲] Detailed reference viewed: 49 (1 UL) |
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