Let $J_0\left(𝔫\right)$ be the Jacobian variety of the Drinfeld modular curve $X_0\left(𝔫\right)$ over ${𝔽}_q\left(t\right)$, where $𝔫$ is an ideal in ${𝔽}_q\left[t\right]$. Let $0\to B\to J_0\left(𝔫\right)\to A\to 0$ be an exact sequence of abelian varieties. Assume $B$, as a subvariety of $J_0\left(𝔫\right)$ , is stable under the action of the Hecke algebra $𝕋\subset$ End $\left(J_0\left(𝔫\right)\right)$. We give a criterion which is sufficient for the exactness of the induced sequence of component groups $0\to {\Phi }_{B,\infty }\to {\Phi }_{J,\infty }\to {\Phi }_{A,\infty }\to 0$ of the Néron models of these abelian varieties over ${𝔽}_q\left[\left[\frac{1}{t}\right]\right]$. This criterion is always satisfied when either $A$ or $B$ is one-dimensional. Moreover, we prove that the sequence...

Given an odd prime  $p$  and a representation $\varrho$  of the absolute Galois group of a number field $k$ onto ${\mathrm{PGL}}_2\left({𝔽}_p\right)$ with cyclotomic determinant, the moduli space of elliptic curves defined over $k$ with $p$-torsion giving rise to $\varrho$ consists of two twists of the modular curve $X\left(p\right)$. We make here explicit the only genus-zero cases $p=3$ and $p=5$, which are also the only symmetric cases: ${\mathrm{PGL}}_2\left({𝔽}_p\right)\simeq {𝒮}_n$ for $n=4$ or $n=5$, respectively. This is done by studying the corresponding twisted Galois actions on the function field of the curve, for which...

Shimura curves associated to rational nonsplit quaternion algebras are coarse moduli spaces for principally polarized abelian surfaces endowed with quaternionic multiplication. These objects are also known as fake elliptic curves. We present a method for computing equations for genus 2 curves whose Jacobian is a fake elliptic curve with complex multiplication. The method is based on the explicit knowledge of the normalized period matrices and on the use of theta functions with characteristics. As...

Let $E/F$ be a modular elliptic curve defined over a totally real number field $F$ and let $\phi$ be its associated eigenform. This paper presents a new method, inspired by a recent work of Bertolini and Darmon, to control the rank of $E$ over suitable quadratic imaginary extensions $K/F$. In particular, this argument can also be applied to the cases not covered by the work of Kolyvagin and Logachëv, that is, when $[F:\mathbb{Q}]$ is even and $\phi$ not new at any prime.