# Seminars & Events for Princeton University/IAS Number Theory Seminar

##### Heegner Divisors, L-Functions and Harmonic Weak Maass Forms

##### The subconvexity bounds for L-functions

For a general L-function, the bound on its critical line obtained by applying the Phragmen-Lindeloff interpolation method is called the convexity bound. Any bounds with a power saving of the convexitybound are called subconvexity bounds. In this talk we will give the first subconvexity bounds for GL(3) L-functions as well as for GL(3) x GL(2) L-functions. Our methods also recover the subconvexity bounds for GL(2) L-functions in the eigenvalue aspect.

##### Prime Chains and Pratt trees

A sequence of primes $p_1, ..., p_k$ is called a prime chain if $p_j | (p_{j+1}-1)$ for each $j$; e.g. 3, 7, 29, 59. We will discuss problems about counting prime chains with certain properties, and about the existence of prime chains with various properties. The Pratt tree for a prime p is the tree with root node p and below p are the Pratt trees of the odd prime factors of $p_1$.

Example: 79

................................../\

................................3 13

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........................................3

We are concerned with the normal and extremal behavior of the depth of such trees.

##### Integral models of some Shimura varieties

##### Hilbert Spaces of Entire Functions and Automorphic L-Functions

We review the de Branges theory of Hilbert spaces of entire functions. This theory gives a canonical form for a class of operators as multiplication operator together with a generalized Fourier transform taking such an operator to a generalized differential operator. We discuss its relation to other theories of canonical forms for certain non-self adjoint operators, including "model spaces" and Lax-Phillips scattering theory. We present examples, including de Branges spaces associated to automorphic L-functions, and discuss how the Riemann hypothesis may be encoded in this framework.

##### Counting rational points on a cubic surface

A conjecture of Manin predicts precise asymptotic for the density of rational points on del Pezzo surfaces. This has been satisfactorily settled for del Pezzo surfaces of higher degree. But for lower degree not much is known. At present, most research in the field is concentrated around del Pezzo surfaces of degree four (intersections of two quadratics in $\mathbb P4$), and degree three (cubic surfaces). Although many special cases of singular cubic and quartic surfaces have been successfully dealt with, it is generally believed that the problem is much harder for smooth surfaces and for surfaces with `mild' singularities.

##### Arithmetic invariants of discrete Langlands parameters

Let $G$ be a reductive algebraic group over a local field $k$. Hiraga, Ichino and Ikeda have recently proposed a general conjecture for the formal degree of a discrete series representation of $G(k)$, using special values of the adjoint L-function and $\epsilon$ factor of its (conjectural) Langlands parameter. I will reformulate this conjecture using Euler-Poincare measure on $G(k)$ and the motive of $G$, establish a key rationality property of the ratio of special values in the non-Archimedean case, and explore some of its implications for supercuspidal parameters.

This is joint work with Mark Reeder.

##### Iwasawa theory of elliptic curves for supersingular primes

Studying the Selmer groups of elliptic curves for a supersingular prime is difficult. It turned out we should instead use the plus/minus Selmer groups defined by Kobayashi. In this talk, we will see the plus/minus Selmer group theory for supersingular primes is very analogous to the Selmer group theoery for ordinary primes, and as an application, we will prove the parity conjecture of elliptic curves for supersingular primes among other things. We will report some other recent progress as well.

##### Cohen-Lenstra heuristics and the negative Pell equation

For a squarefree integer $d$ we ask, if the negative Pell equation $x2-dy2 = -1$ is solvable over the integers. By easy considerations we see that in this case $d>0$ and that all odd prime divisors of $d$ are congruent to 1 modulo 4. Now we call a $d$ special, if it satisfies those two conditions. We are able to prove that for a positive density of special $d$ we can solve the negative Pell equation. Furthermore there is a positive density of special $d$, where the negative Pell equation cannot be solved. This result gives a big support to a conjecture of Stevenhagen who predicts those densities.

##### On a result of Waldspurger in higher rank

An important result of Waldspurger relates the central value of quadratic base change L-functions for GL(2) to period integrals over tori. Subsequently this result was reproved by Jacquet using the relative trace formula. We will explain some progress on extending Waldspurger's result to higher rank via a generalization of Jacquet's approach.

##### Nonvanishing mod p of Eisenstein series

##### Automorphic lifts of prescribed type

##### Explicit reduction modulo p of certain crystalline representations

We use the $p$-adic local Langlands correspondence for $GL_2(Q_p)$ to explicitly compute the reduction modulo $p$ of crystalline representations of small slope, and give applications to modular forms.

Joint work with Kevin Buzzard.

##### Multizeta values and related structures in function field arithmetic

##### The Divisor Matrix, Dirichlet Series and SL(2,Z)

The divisor matrix is indexed by the natural numbers with $(i,j)$ entry equal to $1$ if $i$ divides $j$ and $0$ otherwise. The convolution of a Dirichlet series with the Riemann zeta function corresponds to multiplication of the sequence of coefficients by the divisor matrix. In this talk, we consider groups which contain the divisor matrix and preserve the space of convergent Dirichlet series. A reduction step is to show that the divisor matrix can be brought to a Jordan normal form by transition matrices which preserve the above space. We then construct an representation of $SL(2,Z)$ on the space of convergent Dirichlet series in which the standard unipotent element is represented by the divisor matrix.