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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 concerning 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 $n=4,5$, and Melanie Wood for $n=6$. This work makes use of the theory of the classification 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. |
Thanks for your help. Article has been submitted. |
Thanks for your help. Article has been submitted.