Algebraic points on Shimura curves of Γ₀(p)-type

Abstract

Abstract. In this article, we classify the characters associated to algebraic points on Shimura curves of Γ0(p)-type, and over a quadratic field we show that there are at most elliptic points on such a Shimura curve for every sufficiently large prime number p. This is an analogue of the study of rational points or points over a quadratic field on the modular curve X 0 ( p ) $X_0(p)$ by Mazur and the second author. We also apply the result to a finiteness conjecture on abelian varieties with constrained prime power torsion by Rasmussen–Tamagawa.

Rational Points on Shimura Curves and the Manin Obstruction

“…Note that is a finite set, while is an infinite set (cf. [3, Remark 4.4]). We have the following criterion of .…”

On 5-torsion of CM elliptic curves

“…For a positive integer N , the Shimura curve X (O N ) := H/O 1 N is similar to the Shimura curve X (1), but with an extra structure of level N . Because of the connection between O N and Γ 0 (N ), the curve X (O N ) is often denoted by X 0 (N ) (and sometimes by M B 0 (N ), as for instance in [2]). We recall the moduli interpretation for X 0 (N ) (see again [8] and also [23]).…”

“…Furthermore, for every positive integer n, we indicate the n-th multiple of P simply by nP . It is well-known that E[m] ≃ (Z/mZ) 2 . Let {P 1 , P 2 } be a Z-basis for E[m]; thus K m = K(x(P 1 ), x(P 2 ), y(P 1 ), y(P 2 )).…”

A Function Field Analogue of the Rasmussen-Tamagawa Conjecture: The Drinfeld Module Case

“…• for QM abelian surfaces over certain imaginary quadratic fields [Ar,Theorem 9.3]. We notice that, under the assumption of the generalized Riemann hypothesis (GRH) for Dedekind zeta functions of number fields, the conjecture is true in general [RT2,Theorem 5.1].…”