Galois Representations Over Fields of Moduli and Rational Points on Shimura Curves
| Autor: | Victor Rotger, Carlos de Vera-Piquero |
|---|---|
| Rok vydání: | 2014 |
| Předmět: |
Shimura variety
Abelian variety Pure mathematics Galois cohomology General Mathematics 010102 general mathematics Galois group Galois module 01 natural sciences Algebra Moduli of algebraic curves 0103 physical sciences 010307 mathematical physics Hilbert's twelfth problem 0101 mathematics Abelian group Mathematics |
| Zdroj: | Canadian Journal of Mathematics. 66:1167-1200 |
| ISSN: | 1496-4279 0008-414X |
| DOI: | 10.4153/cjm-2013-020-3 |
| Popis: | The purpose of this note is to introduce a method for proving the non-existence of rational points on a coarse moduli space X of abelian varieties over a given number field K in cases where the moduli problem is not fine and points in X(K) may not be represented by an abelian variety (with additional structure) admitting a model over the field K. This is typically the case when the abelian varieties that are being classified have even dimension. The main idea, inspired by the work of Ellenberg and Skinner on the modularity of ℚ-curves, is that one may still attach a Galois representation of Gal(/K) with values in the quotient group GL(Tℓ(A))/ Aut(A) to a point P = [A] ∈ X(K) represented by an abelian variety A/, provided Aut(A) lies in the centre of GL(Tℓ(A)). We exemplify our method in the cases where X is a Shimura curve over an imaginary quadratic field or an Atkin–Lehner quotient over ℚ. |
| Databáze: | OpenAIRE |
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