Seminars Sorted by Series

In 2002, J.B. Conrey and K. Soundararajan showed that there are infinitely many Dirichlet $L$-functions which do not vanish on the critical segment. Let $\mathcal{G}$ be the family of Dirichlet $L$-functions they considered. Following their work, a...

Suppose E is an elliptic curve defined over a number field K, and p is a prime where E has good ordinary reduction. The usual methods of Iwasawa theory give a single Iwasawa module from which one can recover the Selmer groups of E over all finite...

In this talk, I will give applications of Kedlaya's recent results on p-adic differential equations. In particular, I will give a new proof of Colmez-Fontaine's theorem which describes semistable p-adic representations.

For a number field K and a (compact) p-adic Lie groups G, the inverse Galois problem asks whether G can be realized as the Galois group of an extension of K. Already in the case that G is zero-dimensional, this is too difficult. So I propose to...

In this talk, I'd like to describe my recent work with P.Sarnak concerning the equidistribution properties for Hecke igenforms on the modular surface. We evaluate asymptotically the variance for the equidistribution by means of the trace formula and...

Serre conjectured that all continuous, irreducible, odd $\rho:G_{\mathbf{Q}} \to \mathrm{GL}_2(\overline{\mathbf{F}}_p)$ arise from modular forms. If $\rho$ is modular, then proven refinements provide recipes for the possible weights and levels of...

Let $f$ be a primitive cusp form of weight at least 2, and let $\rho_f$ be the $p$-adic Galois representation attached to $f$. If $f$ is $p$-ordinary, then it is known that the restriction of $\rho_f$ to a decomposition group at $p$ is `upper...

It is well-known that to study the space of modular forms of a fixed weight and level over $Q_P$, it is useful to embed this finite-dimensional space in a $p$-adic Banach space of overconvergent modular forms, and use methods of $p$-adic analysis...

After a brief review of the theory of multiple zeta values and zeta functions, we will discuss the multiple Hurwitz zeta function given by $$\zeta(s_1, s_2,..., s_r; x_1, x_2,..., x_r) = \sum_{n_1>n_2>\cdots >n_r\geq 1} {1 \over (n_1+x_1)^{s_1} (n_2...

In this talk, I will explain a new result on the distribution of rational points of bounded height on "wonderful" compactifications of semi-simple groups of adjoint type. A special case of the result is the conjecture of Manin on the distribution of...

Let K be an imaginary quadratic field. Modular forms for K are related to the cohomology of arithmetic 3-manifolds. By using the Galois representations associated to such forms we produce an explicit tower of rational homology three spheres with...

I will discuss relatively new kinds of zeta and L-functions - the Ihara-Selberg-Artin L-functions attached to finite graph coverings. In joint work with Harold Stark, we have found that many of the methods used by number theorists in investigations...

Let E be an elliptic curve over the rationals and p a fixed prime. A (very hard) question of Kolyvagin asks if there exists a quadratic discriminant d such that the Tate-Shafarevich group of the quadratic twist E_d has order prime to p. I will...

Shimura Curves over Q parameterize abelian surfaces with quaternionic multiplication. However the universal families exist only when considerable level structure is added. This makes it difficult to write eqautions for these families, or...

The title refers to a conjecture that Serre made in the early 1970's, that has proved to be very influential. I will sketch the main ideas of the proof of the level 1 case of Serre's conjecture. If time permits, I will indicate how these ideas can...

We will talk about a proof of Manin's conjecture on the asymptotic density of rational points of bounded height for the special case of a compactification of a semisimple algebraic group defined over a number field. The main tool is the mixing...

In his last letter to Hardy, Ramanujan defined 17 peculiar functions which are now referred to as his mock theta functions. Although these mysterious functions have been investigated by many mathematicians over the years, many of their most basic...

In this lecture, I will describe some results from the thesis of Jarad Schofer, (Maryland, 2005), which provide a generalization of the Gross-Zagier factorization of singular moduli or arbitrary Borcherds forms. After a review of the construction of...

I will describe some identities of linear forms and distributions attached to representations in theta correspondence. These identities can be used to prove formula of special values of L-functions.

In this talk I will discuss a particular family of modular units constructed using functional solutions to q-difference equations found in the work of Selberg. Arising in this way, these objects are of interest for various analytic properties and...

Weyl group multiple Dirichlet series are multiple Dirichlet series in several variables whose coefficients involve Gauss sums and also reflect the combinatorics of a given root system. The earliest examples came from Mellin transforms of metaplectic...

In this talk, I will explain how to compute the trace of a power of the Frobenius endomorphism on the intersection cohomology of the Baily-Borel compactification of a Siegel modular variety. The main tools are : - Kottwitz's calculation of the...

In this talk, I shall introduce a sieve method for bounding the average size of shifted convolution summation terms related to the Quantum Unique Ergodicity Conjecture for a fixed Hecke-Maass cusp form. This bound will be uniform in the spectral...

I describe a new simple way to obtain Rankin-Selberg type spectral identities. These include the classical Rankin-Selberg identity, the Motohashi identity for the forth moment of the zeta function and many new identities between various L-functions...

p-adic height pairings are analogues of the real height pairings that show up for example in the Birch-Swinnerton Dyer conjectures on special values of L-functions of Abelian varieties. The p-adic pairings show up when one tries to formulate p-adic...

A well known theorem of Duke states that the collection of all closed geodesics of a given discriminant on SL(2,Z) \ H becomes equidistributed as the discriminant goes to infinity and can be proved using subconvex estimates for L-functions...

Thanks to the Weil conjectures (proved by Deligne) we know that counting points of varieties over finite fields yields topological information about them. In this talk I will describe such a calculation for certain character varieties...

A conjecture of Lang (on elliptic curves) generalized by Silverman on abelian varieties predicts that the Neron-Tate height of a point on an abelian variety should grow at least like the height of the variety itself. We shall suggest higher...

In this talk I will explain some recent results on local-global compatibility for the action of GL_2 of the finite adeles on the p-adically completed cohomology of modular curves, as well as some applications of these results (e.g. to the Fontaine...

In 1957 Erdos asked what is the smallest maximum modulus of a trigonometric polynomial of degree n all whose coefficients have modulus 1. He thought that there should be a c>0 such that this max modulus is larger than (1+c)sqrt(n).In 1966 Littlewood...

We show that Leopoldt's conjecture for totally real number fields implies Greenberg's conjecture on the uniform boundedness of the $p$-primary part of the class groups of the finite extensions along the cyclotomic tower of a totally real number...

In inverse problems in arithmetic combinatorics, one is interested in describing internal properties of those finite subsets $A$ of an algebraic structure that ``barely expand'' under its operations. One of the deepest results in the area is Freiman...