Dinakar Ramakrishnan - Academia.edu (original) (raw)

Papers by Dinakar Ramakrishnan

arXiv (Cornell University), Jan 8, 2014

Journal of Number Theory, 2015

Proceedings of symposia in pure mathematics, Oct 14, 2021

arXiv (Cornell University), Jun 3, 2019

Documenta Mathematica, 2015

We prove that various arithmetic quotients of the unit ball in C n are Mordellic, in the sense th... more We prove that various arithmetic quotients of the unit ball in C n are Mordellic, in the sense that they have only finitely many rational points over any finitely generated field extension of Q. In the previously known case of compact hyperbolic complex surfaces, we give a new proof using their Albanese in conjunction with some key results of Faltings, but without appealing to the Shafarevich conjecture. In higher dimension, our methods allow us to solve an alternative of Ullmo and Yafaev. Our strongest result uses in addition Rogawski's theory and establishes the Mordellicity of the Baily-Borel compactifications of Picard modular surfaces of some precise levels related to the discriminant of the imaginary quadratic fields.

Pure and Applied Mathematics Quarterly, 2005

In memory of Armand Borel Contents 1. Introduction 2. The geometric side: Isolation of the main t... more In memory of Armand Borel Contents 1. Introduction 2. The geometric side: Isolation of the main term 2.1. Regularization and the main integral 2.2. Coset representatives and centralizers 2.3. The test function 2.4. The dominant terms 2.5. The remaining singular terms 2.6. The regular terms 3. A bound for the sum of regular terms 4. The spectral side

arXiv (Cornell University), Apr 4, 2014

arXiv (Cornell University), Oct 6, 2005

First we reprove, using representation theory and the relative trace formula of Jacquet, an avera... more First we reprove, using representation theory and the relative trace formula of Jacquet, an average value result of Duke for modular L-series at the critical center. We also establish a refinement. To be precise, the L-value which appears is L(1/2, f)L(1/2,f,\chi) (divided by the Petersson norm of f), and the average is over newforms f of prime level N and coefficients a_p(f), with \chi being an odd quadratic Dirichlet character of conductor -D and associated quadratic field K. For any prime p not dividing ND, the asymptotic as N goes to infinity is governed by a measure \mu_p, which is the Plancherel measure at p when \chi(p)=-1, but is new if \chi(p)=1; as p goes to infinity both measures approach the Sato-Tate measure. A particular consequence of our refinement is that for any non-empty interval J in [-2,2], there are infinitely many primes N, which are inert in K, such that for some f of level N, a_p(f) is in J and L(1/2, f)L(1/2,f,\chi) is non-zero.

arXiv (Cornell University), Jun 13, 2018

For any elliptic curve E over k ⊂ R with E(C) = C × /q Z , q = e 2πiz , Im(z) > 0, we study the q... more For any elliptic curve E over k ⊂ R with E(C) = C × /q Z , q = e 2πiz , Im(z) > 0, we study the q-average D 0,q , defined on E(C), of the function D 0 (z) = Im(z/(1 − z)). Let Ω + (E) denote the real period of E. We show that there is a rational function R ∈ Q(X 1 (N)) such that for any non-cuspidal real point s ∈ X 1 (N) (which defines an elliptic curve E(s) over R together with a point P (s) of order N), πD 0,q (P (s)) equals Ω + (E(s))R(s). In particular, if s is Q-rational point of X 1 (N), a rare occurrence according to Mazur, R(s) is a rational number.

arXiv (Cornell University), Jul 1, 2008

Proceedings of symposia in pure mathematics, 1999

In Contributions to Automorphic Forms, Geometry, and Number Theory, Haruzo Hida, Dinakar Ramakris... more In Contributions to Automorphic Forms, Geometry, and Number Theory, Haruzo Hida, Dinakar Ramakrishnan, and Freydoon Shahidi bring together a distinguished group of experts to explore automorphic forms, principally via the associated L-functions, representation theory, and geometry. Because these themes are at the cutting edge of a central area of modern mathematics, and are related to the philosophical base of Wiles' proof of Fermat's last theorem, this book will be of interest to working mathematicians and students alike. Never previously published, the contributions to this volume expose the reader to a host of difficult and thought-provoking problems. Each of the extraordinary and noteworthy mathematicians in this volume makes a unique contribution to a field that is currently seeing explosive growth. New and powerful results are being proved, radically and continually changing the field's make up. Contributions to Automorphic Forms, Geometry, and Number Theory will likely lead to vital interaction among researchers and also help prepare students and other young mathematicians to enter this exciting area of pure mathematics. Contributors: Jeffrey Adams, Jeffrey D. Adler, James Arthur, Don Blasius, Siegfried Boecherer, Daniel Bump, William Casselmann, Laurent Clozel, James Cogdell, Laurence Corwin, Solomon Friedberg, Masaaki Furusawa, Benedict Gross, Thomas Hales, Joseph Harris, Michael Harris, Jeffrey Hoffstein, Herve Jacquet, Dihua Jiang, Nicholas Katz, Henry Kim, Victor Kreiman, Stephen Kudla, Philip Kutzko, V. Lakshmibai, Robert Langlands, Erez Lapid, Ilya Piatetski-Shapiro, Dipendra Prasad, Stephen Rallis, Dinakar Ramakrishnan, Paul Sally, Freydoon Shahidi, Peter Sarnak, Rainer Schulze-Pillot, Joseph Shalika, David Soudry, Ramin Takloo-Bigash, Yuri Tschinkel, Emmanuel Ullmo, Marie-France Vigneras, Jean-Loup Waldspurger.

Contemporary Mathematics, 1986

Abstract. Irreducible selfdual representations of any group fall into two classes: those which ca... more Abstract. Irreducible selfdual representations of any group fall into two classes: those which carry a symmetric bilinear form, and the others which carry an alternating bilinear form. The Langlands correspondence, which matches the irreducible representations σ of the Weil group of a local field k of dimension n with the irreducible representations π of the invertible elements of a division algebra D over k of index n, takes selfdual representations to selfdual representations. In this paper we use global methods to study how the Langlands correspondence behaves relative to this distinction among selfdual representations. We prove in particular that for n even, σ is symplectic if and only if π is orthogonal. More generally, we treat the case of GLm(B), for B a division algebra over k of index r, and n=mr. Introduction. Let ρ be a selfdual representation of a group G on a vector space V over C. We will say that ρ is orthogonal, resp. symplectic, if G leaves a nondegenerate symmetric...

This is the first pat of our sequence of papers on enzyme Kinetics giving an algebro-geometric vi... more This is the first pat of our sequence of papers on enzyme Kinetics giving an algebro-geometric view, by making use of the geometry of surfaces in space which arise from our point of view. The new geometry will be exposed in the second part.

A Number Theory Conference honoring Jeff Hoffstein on the occasion of his 61st birthday will be h... more A Number Theory Conference honoring Jeff Hoffstein on the occasion of his 61st birthday will be held at Perrotis College in the American Farm School (AFS), Thessaloniki, July 14 18, 2014. The conference aims to enable exchange of ideas and information among workers on Dirichlet series and automorphic forms, areas to which Jeff Hoffstein has made fundamental contributions. The structure of the conference will consist of research lectures, discussion and problem sessions.

This talk is a survey of the contributions of Shalika to the theory of integral representations o... more This talk is a survey of the contributions of Shalika to the theory of integral representations of L-functions. Of course, other than his first papers, all of Shalika’s work on this topic, which was spread over more that 15 years, was joint with Jacquet or with Jacquet and Piatetski-Shapiro. It is impossible for me to separate out their individual contributions and there is no reason to do so. I will try to lay out the flow of their collaboration and place it in the context of a natural paradigm for thinking about integral representations. If time permits, I will then talk more about three significant papers: the paper on GL(3), the classification paper, and the paper on the exterior square L-function. 2. “On the extension of the fundamental lemma for a certain relative trace formula to the full Hecke algebra” by Masaaki Furusawa (Osaka) Abstract: (This is a joint work with Kimball Martin and Joseph Shalika.) The speaker and Martin formulated a certain relative trace formula for the...

A co-publication of the AMS and Clay Mathematics Institute. This volume constitutes the proceedin... more A co-publication of the AMS and Clay Mathematics Institute. This volume constitutes the proceedings of a conference, “On Certain L-functions”, held July 23–27, 2007 at Purdue University, West Lafayette, Indiana. The conference was organized in honor of the 60th birthday of Freydoon Shahidi, widely recognized as having made groundbreaking contributions to the Langlands program. The articles in this volume represent a snapshot of the state of the field from several viewpoints. Contributions illuminate various areas of the study of geometric, analytic, and number theoretic aspects of automorphic forms and their L-functions, and both local and global theory are addressed. Topics discussed in the articles include Langlands functoriality, the Rankin–Selberg method, the Langlands–Shahidi method, motivic Galois groups, Shimura varieties, orbital integrals, representations of p-adic groups, Plancherel formula and its consequences, the Gross–Prasad conjecture, and more. The volume also includes an expository article on Shahidi's contributions to the field, which serves as an introduction to the subject. Experts will find this book a useful reference, and beginning researchers will be able to use it to survey major results in the Langlands program

Canonical models of Picard modular surfaces Arithmetic compactification of some Shimura surfaces ... more Canonical models of Picard modular surfaces Arithmetic compactification of some Shimura surfaces 2-cohomology is intersection cohomology Analytic expression for the number of points mod p Contribution of the points at the boundary The points on a Shimura variety modulo a prime of good reduction The description of the theorem Orbital integrals of U(3) Remarks on Igusa theory and real orbital integrals Calculation of some orbital integrals Fundamental lemmas for U(3) and related groups The multiplicity formula for A-packets Tate classes and arithmetic quotients of the two-ball The Albanese of unitary Shimura varieties Lefschetz numbers of Hecke correspondences On the shape of the contribution of a fixed point on the boundary: The case of Q-rank one Appendix