# Joint IAS/PU Number Theory

## Singularities in reductions of Shimura varieties

The singularities in the reduction modulo $p$ of the modular

curve $Y_0(p)$ are visualized by the famous picture of two curves

meeting transversally at the supersingular points. It is a fundamental

question to understand the singularities which arise in the reductions

modulo $p$ of integral models of Shimura varieties. For PEL type

Shimura varieties with parahoric level structure at $p$, this question

has been studied since the 1990's. Due to the recent construction of

## How does the rank of an elliptic curve grow in towers of number fields?

## Singular moduli for real quadratic fields

The theory of complex multiplication describes finite abelian extensions of imaginary quadratic number fields using singular moduli, which are special values of modular functions at CM points. I will describe joint work with Henri Darmon in the setting of real quadratic fields, where we construct p-adic analogues of singular moduli through classes of rigid meromorphic cocycles. I will discuss p-adic counterparts for our proposed RM invariants of classical relations between singular moduli and analytic families of Eisenstein series.

## Local aspects of Venkatesh's thesis.

The thesis of Akshay Venkatesh obtains a ``Beyond Endoscopy'' proof of stable functorial transfer from tori to ${\rm SL}(2)$, by means of the Kuznetsov formula. In this talk, I will show that there is a local statement that underlies this work; namely, there is a local transfer operator taking orbital measures for the Kuznetsov formula to test measures on the torus. The global comparison of trace formulas is then obtained as a Poisson summation formula for this transfer operator.

## An Application of a Conjecture of Mazur-Tate to Supersingular Elliptic Curves

In 1987, Barry Mazur and John Tate formulated refined conjectures of the "Birch and Swinnerton-Dyer type", and one of these conjectures was essentially proved in the prime conductor case by

Ehud de Shalit in 1995. One of the main objects in de Shalit's work is the so-called *refined* $\mathscr{L}$*invariant*, which happens to be a Hecke operator. We apply some results of the theory of Mazur's

Eisenstein ideal to study in which power of the Eisenstein ideal $\mathscr{L}$ belongs. One corollary of our