A thousand years old problem is to determine when a square free integer $n$ is a congruent number ,i,e, the areas of right angled triangles with sides of rational lengths. This problem has a some beautiful connection with the BSD conjecture for elliptic curves $E_n : ny^2 = x^3 - x$. In fact by BSD, all $n= 5, 6, 7$ mod $8$ should be congruent numbers, and most of $n=1, 2, 3$ mod $8$ should not be congruent numbers. Recently, Alex Smith has proved that at least 41.9% of $n=1,2,3$ satisfy (refined) BSD in rank $0$, and at least 55.9% of $n=5,6,7$ mod $8$ satisfy (weak) BSD in rank $1$. This implies in particular that at last 41.9% of $n=1,2,3$ mod $8$ are not congruent numbers, and 55.9% of $n=5, 6, 7$ mod $8$ are congruent numbers. I will explain the ingredients used in Smith's proof: including the classical work of Heath-Brown and Monsky on the distribution F_2 rank of Selmer group of E_n, the complex formula for central value and derivative of L-fucntions of Waldspurger and Gross-Zagier and their extension by Yuan-Zhang-Zhang, and their mod 2 version by Tian-Yuan-Zhang.[-]

Let $E$ be an elliptic curve over the rationals, and let $\chi$ be a Dirichlet character of order $\ell$ for some odd prime $\ell$. Heuristics based on the distribution of modular symbols and random matrix theory have led to conjectures predicting that the vanishing of the twisted $L$-functions $L(E, \chi, s)$ at $s = 1$ is a very rare event (David-Fearnley-Kisilevsky and Mazur-Rubin). In particular, it is conjectured that there are only finitely many characters of order $\ell > 5$ such that $L(E, \chi, 1) = 0$ for a fixed curve $E$.
We investigate the case of elliptic curves over function fields. For Dirichlet $L$-functions over function fields, Li and Donepudi-Li have shown how to use the geometry to produce infinitely many characters of order $l \geq 2$ such that the Dirichlet $L$-function $L(\chi, s)$ vanishes at $s = 1/2$, contradicting (the function field analogue of) Chowla's conjecture. We show that their work can be generalized to constant curves $E/\mathbb{F}_q(t)$, and we show that if there is one Dirichlet character $\chi$ of order $\ell$ such that $L(E, \chi, 1) = 0$, then there are infinitely many, leading to some specific examples contradicting (the function field analogue of) the number field conjectures on the vanishing of twisted $L$-functions. Such a dichotomy does not seem to exist for general curves over $\mathbb{F}_q(t)$, and we produce empirical evidence which suggests that the conjectures over number fields also hold over function fields for non-constant $E/\mathbb{F}_q(t)$.[-]

The Ceresa class is the image under a cycle class map of a canonical algebraic cycle associated to a curve in its Jacobian. This class vanishes for all hyperelliptic curves, and is known to be non-vanishing for the generic curve of genus at least 3. It is necessary for the Ceresa class to have infinite order for the Galois action on the fundamental group of a curve to have big image. We will present an algorithm for certifying that a curve over a number field has infinite order Ceresa class.

N.B. This is preliminary joint work with Jordan Ellenberg, Adam Logan and Akshay Venkatesh.[-]

I will discuss recent joint work with Sarah Zerbes in which we use Euler systems and reciprocity laws for GSp(4) to study the analytic rank 0 case of the Birch--Swinnerton-Dyer conjecture for abelian surfaces. Via restriction of scalars, this also includes the BSD conjecture for analytic rank 0 elliptic curves over imaginary quadratic fields. Our main result is a conditional proof of the conjecture subject to an assumption about the local geometry of the GSp4 eigenvariety at non-regular-weight points. I will explain how this conjecture arises and some motivation for why it seems plausible that it should hold. [-]