Daniel Bump

An introduction to the Langlands program

Preface.- E. Kowalski - Elementary Theory of L-Functions I.- E. Kowalski - Elementary Theory of L-Functions II.- E. Kowalski - Classical Automorphic Forms.- E. DeShalit - Artin L-Functions.- E. DeShalit - L-Functions of Elliptic Curves and Modular Forms.- S. Kudla - Tates Thesis.- S. Kudla - From Modular Forms to Automorphic Representations.- D. Bump - Spectral Theory and the Trace Formula.- J. Cogdell - Analytic Theory of L-Functions for GLn.- J. Cogdell - Langlands Conjectures for GLn.- J. Cogdell - Dual Groups and Langlands Functoriality.- D. Gaitsgory - Informal Introduction to Geometric Langlands.

On the Averages of Characteristic Polynomials From Classical Groups

We provide an elementary and self-contained derivation of formulae for averages of products and ratios of characteristic polynomials of random matrices from classical groups using classical results due to Weyl and Littlewood.

On some applications of automorphic forms to number theory

A basic idea of Dirichlet is to study a collection of interesting quantities {an}n≥1 by means of its Dirichlet series in a complex variable w: ∑ n≥1 ann −w. In this paper we examine this construction when the quantities an are themselves infinite series in a second complex variable s, arising from number theory or representation theory. We survey a body of recent work on such series and present a new conjecture concerning them.

Explicit formulas for the waldspurger and bessel models

This paper studies certain models of irreducible admissible representations of the split special orthogonal group SO(2n+1) over a nonarchimedean local field. Ifn=1, these models were considered by Waldspurger. Ifn=2, they were considered by Novodvorsky and Piatetski-Shapiro, who called them Bessel models. In the works of these authors, uniqueness of the models is established; in this paper functional equations and explicit formulas for them are obtained. As a global application, the Bessel period of the Eisenstein series on SO(2n+1) formed with a cuspidal automorphic representation π on GL(n) is computed—it is shown to be a product of L-series. This generalizes work of Böcherer and Mizumoto forn=2 and base field ℚ, and puts it in a representation-theoretic context. In an appendix by M. Furusawa, a new Rankin-Selberg integral is given for the standardL-function on SO(2n+1)×GL(n). The local analysis of the integral is carried out using the formulas of the paper.

Metaplectic Whittaker Functions and Crystals of Type B

The spherical metaplectic Whittaker function on the double cover of Sp (2r, F), where F is a nonarchimedean local field, is considered from several different points of view. Previously, an expression, similar to the Casselman–Shalika formula, had been given by Bump, Friedberg, and Hoffstein as a sum is over the Weyl group. It is shown that this coincides with the expression for the p-parts of Weyl group multiple Dirichlet series of type B r as defined by the averaging method of Chinta and Gunnells. Two conjectural expressions as sums over crystals of type B are given and another as the partition function of a free-fermionic six-vertex model system.

On Kubota's Dirichlet series

Abstract Kubota [T. Kubota, Some results concerning reciprocity law and real analytic automorphic functions, in: 1969 Number Theory Institute (Proc. Sympos. Pure Math. XX, State Univ. New York, Stony Brook, N.Y., 1969), Amer. Math. Soc., Providence, R.I. (1971), 382–395.] showed how the theory of Eisenstein series on the higher metaplectic covers of SL2 (which he discovered) can be used to study the analytic properties of Dirichlet series formed with n-th order Gauss sums. In this paper we will prove a functional equation for such Dirichlet series in the precise form required by the companion paper [B. Brubaker, D. Bump, G. Chinta, S. Friedberg, and J. Hostein, Weyl group multiple Dirichlet series I, preprint, http://sporadic.stanford.edu/bump/wmd.pdf.]. Closely related results are in Eckhardt and Patterson [C. Eckhardt and S. J. Patterson, On the Fourier coefficients of biquadratic theta series, Proc. London Math. Soc. (3) 64(2) (1992), 225–264.].

Integration on p-adic groups and crystal bases

Let G = GL r+1 over a nonarchimedean local field F. The Kashiwara crystal B(∞) is the quantized enveloping algebra of the lower triangular maximal unipotent subgroup N_. Examples are given where an integral over N_(F) may be replaced by a sum over B(∞). Thus the Gindikin-Karpelevich formula evaluates the integral of the standard spherical vector in the induced model of a principal series representation as a product Π(1 ― q ― z α )/(1 ― z α ) where z is the Langlands parameter and the product is over positive roots. This may also be expressed as a sum over B(∞). The corresponding equivalence over a metaplectic cover of GL r+1 is deduced by using Kashiwaras similarity of crystals.

A nonvanishing theorem for derivatives of automorphic

1. A brief history of nonvanishing theorems. The nonvanishing of a Dirichlet series 2 a(n)n~\ or the existence of a pole, at a particular value of s often has applications to arithmetic. Euler gave the first example of this, showing that the infinitude of the primes follows from the pole of Ç(s) at s = 1. A deep refinement was given by Dirichlet, whose theorem on primes in an arithmetic progression depends in a fundamental way upon the nonvanishing of Dirichlet L-functions at s = 1. Among the many examples of arithmetically significant nonvanishing results in this century, one of the most important is still mostly conjectural. Let E be an elliptic curve defined over Q: the set of all solutions to an equationy = x-ax-b where a, b are rational numbers with 4a-27b ^ 0. Mordell showed that E(Q) may be given the structure of a finitely generated abelian group. The Birch-Swinnerton-Dyer Conjecture asserts that the rank of this group is equal to the order of vanishing of a certain Dirichlet series L(s,E) as s = 1—the center of the critical strip—and that the leading Taylor coefficient of this L-function at s = 1 is determined in an explicit way by the arithmetic of the elliptic curve. We refer to the excellent survey article of Goldfeld [5] for details. In 1977, Coates and Wiles [3] proved the first result towards the BirchSwinnerton-Dyer conjecture. The conjecture implies that if the L-series of E does not vanish at 1, then the group of rational points is finite. Coates and Wiles proved this last claim in the special case that E has complex multiplication (nontrivial endomorphisms). In this note, we announce a nonvanishing theorem which, together with work of Kolyvagin and Gross-Zagier, implies that E(Q) is finite when L(l,E) ^ 0 for any modular elliptic curve E. (A modular elliptic curve is one which may be parametrized by automorphic functions. Deuring proved that all elliptic curves with complex multiplication are modular; Taniyama and Weil have conjectured that indeed all elliptic curves defined over Q are modular.) Before giving details of our theorem, we mention several other nonvanishing theorems and arithmetic applications. The following discussion is necessarily not a complete survey Shimura showed that there is a correspondence between modular forms ƒ of even weight k and modular forms ƒ of half-integral weight (k + l)/2.

L

Weyl group multiple Dirichlet series were associated with a root system Φ and a number field F containing the n-th roots of unity by Brubaker, Bump, Chinta, Friedberg, and Hoffstein [2]. Brubaker, Bump, and Friedberg [4] provided for when n is sufficiently large; the coefficients involve n-th order Gauss sums and reflect the combinatorics of the root system. Conjecturally, these functions coincide with Whittaker coefficients of metaplectic Eisenstein series, but they are studied in these papers by a method that is independent of this fact. The assumption that n is large is called stability and allows a simple description of the Dirichlet series. “Twisted” Dirichet series were introduced in Brubaker, Bump, Friedberg, and Hoffstein [5] without the stability assumption, but only for root systems of type A{inr}. Their description is given differently, in terms of Gauss sums associated to Gelfand-Tsetlin patterns. In this paper, we reimpose the stability assumption and study the twisted multiple Dirichlet series for general Φ by introducing a description of the coefficients in terms of the root system similar to that given in the untwisted case in [4]. We prove the analytic continuation and functional equation of these series, and when Φ = A{inr} we also relate the two different descriptions of multiple Dirichlet series given here and in [5] for the stable case.

-functions with applications to elliptic curves

Suppose that G and H are connected reductive groups over a number field F and that an L-homomorphism ρ : LG −→ LH is given. The Langlands functoriality conjecture predicts the existence of a map from the automorphic representations of G(A) to those of H(A). If the adelic points of the algebraic groups G, H are replaced by their metaplectic covers, one may hope to specify an analogue of the Lgroup (depending on the cover), and then one may hope to construct an analogous correspondence. In this paper we construct such a correspondence for the double cover of the split special orthogonal groups,