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On certain finiteness questions in the arithmetic of Galois representations
WIESE, Gabor


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Abstract :
[en] Let p be a fixed prime number. It has been known for a long time that there are only finitely many Galois extensions K/Q with Galois group a finite irreducible subgroup of GL_2(F_p^bar) that are imaginary and unramified outside p. In contrast, there are infinitely many such with Galois group inside GL_2(Z_p^bar), even if one restricts to ones coming from modular forms (this restriction is believed to be local at p). It is tempting to ask what happens "in between" F_p^bar and Z_p^bar, i.e. whether there is still finiteness modulo fixed prime powers. In the talk, I will motivate and explain a conjecture made with Ian Kiming and Nadim Rustom stating that the set of such Galois extensions `modulo p^m' (a proper definition will be given in the talk) coming from modular forms is finite. I will present partial results and a relation of the finiteness conjecture to a strong question by Kevin Buzzard. The talk is based on joint work with Ian Kiming and Nadim Rustom.
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Author, co-author :
WIESE, Gabor  ;  University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit
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Title :
On certain finiteness questions in the arithmetic of Galois representations
Publication date :
07 June 2017
Event name :
Field Arithmetic Seminar, University of Tel Aviv
Event place :
Tel Aviv, Israel
Event date :
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