# Quantum modular and mock modular forms

## Speaker:

## Speaker Link:

## Institution:

## Time:

## Location:

In 2010, Zagier defined the notion of a "quantum modular form,'' and offered several diverse examples, including Kontsevich's 'strange' function. Here, we construct infinite families of quantum modular forms, and prove one of Ramanujan's remaining claims about mock theta functions in his last letter to Hardy as a special case of our work. We will show how quantum modular forms underlie new relationships between combinatorial mock modular and modular forms due to Dyson and Andrews-Garvan. This is joint work with Ken Ono (Emory U.) and Rob Rhoades (CCR-Princeton).

# Average of the First Invariant Factor of the Reductions of Abelian Varieties of CM Type

## Speaker:

## Institution:

## Time:

## Location:

For a field of definition $k$ of an abelian variety $\Av$ and prime ideal $\ip$ of $k$ which is of a good reduction for $\Av$, the structure of $\Av(\F_{\ip})$ as abelian group is:

\Av(\F_{\ip})\simeq \Z/d_1(\ip)\Z\oplus\cdots\oplus\Z/d_g(\ip)\Z\oplus\Z/e_1(\ip)\Z\oplus\cdots\oplus\Z/e_g(\ip)\Z,

where $d_i(\ip)|d_{i+1}(\ip)$, $d_g(\ip)|e_1(\ip)$, and $e_i(\ip)|e_{i+1}(\ip)$ for $1\leq i<g$.

We are interested in finding an asymptotic formula for the number of prime ideals $\ip$ with $N\ip<x$, $\Av$ has a good reduction at $\ip$, $d_1(\ip)=1$. We succeed in this under the assumption of the Generalized Riemann Hypothesis (GRH). Unconditionally, we achieve a short range asymptotic for abelian varieties of CM type, and the full cyclicity theorem for elliptic curves over a number field containing CM field.

# On the non-triviality of the p-adic Abel-Jacobi image of generalised Heegner cycles

## Speaker:

## Institution:

## Time:

## Location:

Generalised Heegner cycles are associated to a pair of an elliptic Hecke eigenform and a Hecke character over an imaginary quadratic extension K. Let p be an odd prime split in K. We describe the non-triviality of the p-adic Abel-Jacobi image of generalised Heegner cycles modulo p over anticyclotomic extensions of K.

# Explicit reciprocity law for generalised Heegner cycles

## Speaker:

## Institution:

## Time:

## Location:

We will describe an explicit reciprocity law for generalised Heegner cycles, relating the images of certain twists of these classes under the Bloch-Kato dual exponential map to certain Rankin-Selberg L-values, and explain the applications of this formula to the proof of certain rank 0 cases of the Bloch-Kato conjecture. This is a joint work with M.-L. Hsieh.

# Discrete logarithms in small characteristic

## Speaker:

## Speaker Link:

## Institution:

## Time:

## Location:

In this talk, I will explain the mathematical ideas and questions arising from the recent breakthrough BGJT algorithm for discrete logarithms over finite fields of small characteristic. This is joint work with Q. Cheng and J. Zhuang (ANTS 2014).

# Igusa’s conjecture for exponential sums

## Speaker:

## Speaker Link:

## Institution:

## Time:

## Host:

## Location:

In joint work with Raf Cluckers, we propose a conjecture for exponential sums which generalizes both a conjecture by Igusa and a local variant by Denef and Sperber, in particular, it is without the homogeneity condition on the polynomial in the phase, and with new predicted uniform behavior. The exponential sums have summation sets consisting of integers modulo p^m lying p-adically close to y, and the proposed bounds are uniform in p, y, and m. We give evidence for the conjecture, by showing uniform bounds in p, y, and in some values for m. On the way, we prove new bounds for log-canonical thresholds which are closely related to the bounds predicted by the conjecture.

# Slopes of modular forms

## Speaker:

## Institution:

## Time:

## Location:

I will explain several conjectures and results regarding the slope distribution of Up operator action on the space of modular forms. Most notably, we prove that the slopes of modular forms with a highly p-divisible characters roughly form unions of arithmetic progressions. This is a joint work with Daqing Wan and Jun Zhang.

# Local newforms for SO(2n+1)

## Speaker:

## Institution:

## Time:

## Location:

When a holomorphic modular form is a newform, its L-function has nice analytic properties and associates a cuspidal automorphic representation, which is a restricted product of local representations. To recover the newform from the representation, Casselman considered the fixed line of the congruence subgroups of GL(2) at the conductor level on the local representations. A vector on this line shall encode the conductor, the L-function and the \epsilon-factor of the representation. This is called the theory of newforms for GL(2). Similar theory has been established for some groups of small ranks as well as GL(n). In this talk I will introduce one for SO(2n+1).

# Overconvergent Eichler-Shimura morphisms for modular curves

## Speaker:

## Institution:

## Time:

## Location:

We construct the Eichler-Shimura morphisms for families of overconvergent modular forms via Scholze's theory of pro-etale site, as well as the Hodge-Tate period maps on modular curves of infinite level. We follow some of the main ideas in the work of Andreatta-Iovita-Stevens. In particular, we reprove the main result in their paper. Since we work entirely on the generic fiber of the modular curve, log structures will not be needed if we only consider the Eichler-Shimura morphism for cusp forms. Moreover, the well-established theory of the Hodge-Tate period map for Shimura varieties of Hodge type may allow us to generalize the construction to more general Shimura varieties. This is a joint work with Hansheng Diao.