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Publications and communications of Gabor Wiese [gabor.wiese@uni.lux]
On modular Galois representations modulo prime powers ; ; Wiese, Gabor in International Journal of Number Theory (2013), 9(1), 91--113 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 ... [more ▼] 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. [less ▲] Detailed reference viewed: 188 (4 UL)Modulformen und das inverse Galois-Problem Wiese, Gabor Scientific Conference (2012, September 19) Detailed reference viewed: 34 (0 UL)Symplectic Galois representations and applications to the inverse Galois problem Wiese, Gabor Presentation (2012, May 30) Detailed reference viewed: 27 (0 UL)On modular Galois representations modulo prime powers Wiese, Gabor Presentation (2012, May 30) Detailed reference viewed: 76 (1 UL)Modulare Galois-Darstellungen und Computeralgebra Wiese, Gabor Scientific Conference (2012, May 12) Detailed reference viewed: 34 (0 UL)Four lectures on modular forms and Galois representations Wiese, Gabor Presentation (2012, February) Detailed reference viewed: 46 (0 UL)Symplectic Galois representations and applications to the inverse Galois problem Wiese, Gabor Presentation (2012, January 19) Detailed reference viewed: 32 (0 UL)Algèbre 3 (théorie des corps et théorie de Galois) Wiese, Gabor Learning material (2012) Lecture notes written in French from the Algebra 3 lecture in the 3rd term of the Bachelor programme BASI branch Mathematics (old version) at the University of Luxembourg. The lecture covers field theory ... [more ▼] Lecture notes written in French from the Algebra 3 lecture in the 3rd term of the Bachelor programme BASI branch Mathematics (old version) at the University of Luxembourg. The lecture covers field theory and Galois theory and includes a treatment of the solvability of equations by radicals and a treatment of classical construction problems with ruler and compass. [less ▲] Detailed reference viewed: 151 (8 UL)Lectures on Modular Galois Representations Modulo Prime Powers Wiese, Gabor Presentation (2011, December) Detailed reference viewed: 65 (3 UL)On modular forms and the inverse Galois problem ; Wiese, Gabor in Transactions of the American Mathematical Society (2011), 363(9), 4569--4584 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 ... [more ▼] 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. [less ▲] Detailed reference viewed: 117 (2 UL)A computational study of the asymptotic behaviour of coefficient fields of modular forms ; Wiese, Gabor in Actes de la Conférence ``Théorie des Nombres et Applications'' (2011) 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 ... [more ▼] 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. [less ▲] Detailed reference viewed: 53 (0 UL)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: 42 (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: 161 (2 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: 46 (0 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: 150 (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: 143 (0 UL)Zahlentheorie und Geometrie vereint in der Serre-Vermutung Wiese, Gabor in Essener Unikate (2008), 33 Detailed reference viewed: 79 (2 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: 116 (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: 238 (1 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: 109 (1 UL) |
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