Subconvexity problem is one of the central topics in analytic number theory. In this talk, we report on hybrid subconvexity bounds for GL_2 × GL_2 Rankin--Selberg L-functions twisted by a primitive Dirichlet character χ modulo a prime power, in the t and depth aspects. This is a joint work with Chenchen Shao.

In 1972, Serre proved the celebrated Open Image Theorem, claiming that for a non-CM elliptic curve E over Q, the residue modulo $\ell$ Galois representation associated with E is surjective for sufficiently large prime $\ell$. An effective version of this theorem seeks to bound such least non-surjective prime $\ell$. In the talk, I will review some results concerning the effective version of Serre's Open Image Theorem. Then, I will present a work in progress on the effective open image theorem for pairs of elliptic curves, especially the semistable elliptic curves. This is joint with Tian Wang.

The Katz-Sarnak heuristic predicts that the distribution of low-lying zeros of families of automorphic L-functions are governed by certain classical compact groups determined by the family. In this talk I will present some recent progress concerning low-lying zeros of spinor and standard L-functions of Siegel modular forms. I will first describe these results in the $k$ (weight) aspect, and then explain how to extend the support of Fourier transforms of test functions by averaging over $k$. I will also discuss applications towards the non-vanishing of central L-values.