Beijing Institute of Mathematical Sciences and Applications

For every triple $F$, $K$, $p$ where $F$ is a classical elliptic eigenform, $K$ is a quadratic imaginary field and $p$ is an odd prime integer which is not split in $K$, I will explain how to attach a $p$-adic $L$-function which interpolates the algebraic parts of the special values of the complex $L$-functions of $F$ twisted by algebraic Hecke characters of $K$ such that the $p$-part of their conductor is $p^n$ (for $n$ large enough and explicitly determined). This construction extends a classical construction of $N$. Katz for $F$ an Eisenstein series and of Bertolini–Darmon–Prasanna for $F$ a cuspform when $p$ is split in $K$. <br>This is joint work with Adrian Iovita.

I will talk about the matrix coefficient conjecture and its relation with a few other conjectures in transcendental number theory and Iwasawa theory. This is joint work with Samit Dasgupta.

Let $E$ be an elliptic curve over $\mathbb{Q}$ with complex multiplication, we discuss some recent work proving unconditional results on $p$-converse theorem and rank $1$ BSD formula for $E$. This involves improving a recent construction by Andreatta-Iovita on certain $p$-adic $L$-functions, and doing some Iwasawa theory for it. This is joint work with Yangyu Fan.