Visibility for analytic rank one
Let $E$ be an optimal elliptic curve of conductor~$N$, such that the $L$-function~$L_E(s)$ of~$E$ vanishes to order one at~$s=1$. Let $K$ be a quadratic imaginary field in which all the primes dividing~$N$ split. The Gross-Zagier theorem gives a formula that expresses the Birch and Swinnerton-Dyer conjectural order the Shafarevich-Tate group of~$E$ over~$K$ as a rational number. We extract an integer factor from this formula and relate it to certain congruences of the newform associated to~$E$ with eigenforms of odd analytic rank bigger than one. We use the theory of visibility to show that, under certain hypotheses (which includes the first part of the Birch and Swinnerton-Dyer conjecture on rank), if an odd prime~$q$ divides this factor, then $q^2$~divides the actual order of the Shafarevich-Tate group. This provides theoretical evidence for the Birch and Swinnerton-Dyer conjecture in the analytic rank one case. Rational torsion in elliptic curves and the cuspidal subgroup Abstract: Let~$A$ be an elliptic curve over~$\Q$ of square free conductor~$N$. We prove that if $A$ has a rational torsion point of prime order~$r$ such that $r$ does not divide~$6N$, then $r$ divides the order of the cuspidal subgroup of~$J_0(N)$.