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L-functions (attached to modular forms and/or to algebraic varieties and algebraic number fields) are prominent in quite a wide range of number theoretic issues, and our recent growth of understanding of the analytic properties of L-functions has already lead to profound applications regarding ( among other things) the statistics related to arithmetic problems. Much of this exciting development involves a coming-together of {\it arithmetic algebraic geometry}, of {\it automorphic forms and representation theory}, of {\it analytic number theory} and of {\it statistics} (such as the statistical heuristics obtained from the contemporary work regarding Random Matrices).

The semester long program at the MSRI on Arithmetic Statistics emphasizes the statistical aspects of L-functions, modular forms, and associated arithmetic and algebraic objects from several different perspectives- theoretical, algorithmic, and experimental.

The program brings together experts on modular forms, analytic number theory, arithmetic and algebraic geometry, mathematical physics, and computational number theory to investigate several difficult problems in number theory from the point of view of understanding their limiting behaviour.

Several factors make this an appropriate time for a program devoted to statistical questions in number theory.

In recent years, progress has been made on the problem of obtaining the asymptotic number, N_{K,n}(X), of extensions of a fixed number field K of given degree n and discriminant \leq X. The case of K={\mathbb Q} and n=3 is a classical result of Heilbronn and Davenport. Bhargava has made significant progress on this problem, obtaining the asympotics, for \mathbb Q and n=4,5. This work makes use of the theory of the classification of prehomogenous vector spaces.

Very recently, Bhargava and Shankar have managed to apply the methods used for number fields to study the asymptotic distribution of ranks of elliptic curves of a number field. ELABORATE..., AND CONNECT UP WITH THE PREDICTIONS FROM RMT, AND SAY SOMETHING ABOUT ADJUSTING DELAUNAY's COHEN-LENSTRA HEURISTICS FOR QUADRATIC TWISTS.

The Sato-Tate conjecture for the case of a modular form associated to an elliptic curve over {\mathbb Q} with somewhere multiplicative reduction was recently proven by Clozel, Harris, Shepherd-Barron, and Taylor. We would like to consider more specific questions such as the rate of convergence to the Sato-Tate distribution.

The Katz and Sarnak philosophy on the spectral nature of L-functions, specifically through function field analogues and connections to the classical compact groups, has opened up the world of L-functions to detailed probabilistic modeling. This has yielded tremendous success in providing detailed models for the statistics of the zeros and values of L-functions, but much work remains to be done, for example, to incorporate subtle arithmetic information into these models. For instance, to understand the distribution of ranks of elliptic curves, statistics of the Tate-Shafarevich groups and their interaction with regulators must be studied and incorporated.

We also wish to understand the lower order terms and uniform asymptotics in these statistics so as to be able to make precise predictions, for example, regarding the maximum size of the Riemann zeta function.

During our program we are also extending work to cover higher degree L-functions. While the theory of degree 1 and 2 L-functions has reached a more mature form, much more attention is needed in higher degree. Furthermore, many crucial and central conjectures have received scant, if any, testing in the case of higher degree, and limited testing in the case of degrees 1 and 2. For example, the models for the moments of L-functions have not yet been applied or tested for degree three or higher L-functions.

Higher degree L-functions arise in many natural ways. Sometimes they arise from operations performed on lower degree L-functions (e.g., symmetric squares) and sometimes they arise from objects that are, in a sense, more complicated. For example Siegel modular and paramodular forms have natural degree 4 and degree 5 L-functions. In order to study these L-functions numerically, one needs to be able to compute their Dirichlet series coefficients; in order to compute their Dirichlet series coefficients, one needs to be able to compute data about the underlying modular form. A group of us will be developing and implementing algorithms to compute these modular forms and their associated L-functions. The computation of Siegel modular forms, for example, is much hard that computing modular forms on {\rm SL}(2,\mathbb{Z}) and is much more {\it ad hoc}.

One might want to evaluate a higher degree L-function in order to, for example, compute the first few zeros on the critical line or central values of quadratic twists of L-functions. Doing such an evaluation is typically limited by the number of Dirichlet series coefficients, a number which, for Siegel modular forms, at least, is usually pretty small. So, another group of us is going to be investigating methods to evaluate L-functions optimally.

Central values of twists of L-functions is a question that has been well studied for degree 2 L-functions and is mostly untouched for higher degree L-functions. A project being carried out by our group is to compute a great number of twists of degree 4 L-functions to verify interesting conjectures that arise from Random Matrix Theory and others that arise as generalizations of results for the modular forms associated to lower degree L-functions.

{\bf Profile: Melanie Wood and Manjul Bhargava}

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