Galois Representations Over Fields of Moduli and Rational Points on Shimura Curves

Abstract

Abstract. The purpose of this note is introducing a method for proving the existence of no 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 on the work of Ellenberg and Skinner on the modularity of Q-curves, is that to a point P = [A] ∈ X(K) represented by an abelian variety A/K one may still attach a Galois representation of Gal(K/K) with values in the quotient group GL(T ℓ (A))/ Aut(A), provided Aut(A) lies in the centre of GL(T ℓ (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.

Rational Points on Shimura Curves and the Manin Obstruction

“…Lemma 7.2 and a similar study combined with Proposition 7.1, [8, Table 1] help us prove Proposition 2.6 as follows.…”

“…If has odd degree, then has a real place, and so .In [2], we proved only in the setting of Theorem 2.3.Theorem 2.3 for imaginary quadratic fields was proved in [7, Theorem 6.3], [8, Theorem 1.1], [11, Theorem 3.1].…”

“…Theorem 2.3 for imaginary quadratic fields was proved in [7, Theorem 6.3], [8, Theorem 1.1], [11, Theorem 3.1].…”

“…In the situation of [7, Theorem 6.3], Skorobogatov proved that counterexamples to the Hasse principle for is accounted for by the Manin obstruction in the sense of [10, Section 5.2] (see [11, Theorem 3.1]). Rotger and de Vera-Piquero expanded these results to the case where by using projective Galois representations (see [8, Theorem 1.1]). Note that a point of is represented by a QM-abelian surface by over if and only if (see [7, Theorem 1.1]).…”

On 7-division fields of CM elliptic curves

“…For j ∈ {0, 1}, we will denote by X j (7)(K) the set of the K-rational points of X j (7) and by X j (7)(K) CM the set of the K-rational CM points of X j (7). For some Shimura varieties and certain fields F it is known that the set of F -rational points is empty (see for instance [6], [22] and [26]). Here we can show an infinite number of fields K such that X 0 (7)(K) = ∅ and X 1 (7)(K) = ∅.…”