Amod Agashe

CURRICULUM VITAE

Amod Agashe

ADDRESS 2479 Virginia Street, Apt. A, Berkeley, CA 94709 U.S.A. Phone: (510)666-9220 FAX: (510)642-8204 email: amod@math.berkeley.edu web-page: http://math.berkeley.edu/˜ amod

EDUCATION University of California, Berkeley Ph.D. in Mathematics, expected May 2000. Topic: Number theory. Thesis advisors: L. Merel and K. Ribet.

Stanford University M.S. in Electrical Engineering, June 1993.

Indian Institute of Technology (I.I.T.), Bombay B.S. in Electrical Engineering, May 1991.

DISSERTATION The Birch-Swinnerton-Dyer conjectural formula for certain quotients of J_{0}(N). Committee: K. Ribet (Chair), H. Lenstra, D. Forsyth.

AWARDS Mathematics Department fellowship, Spring 1999, U. C., Berkeley. Raymond H. Sciobereti fellowship, Spring 1998, U. C., Berkeley. President of India Gold medal, being the top I.I.T. Bombay graduate, 1991. I.I.T. entrance exam: ranked 7th in India (out of about 100,000), 1987. Physics Olympiad: ranked first in India (out of 14,000), 1987.

RESEARCH INTERESTS Number theory, especially arithmetic algebraic geometry: My current work revolves around the Birch and Swinnerton-Dyer conjecture, but I am also interested in other areas of number theory and mathematics in general.

PUBLICATIONS On invisible elements of the Tate-Shafarevich group, C. R. Acad. Sci. Paris Ser. I Math. 328 (1999), no. 5, 369–374.

PREPRINTS A generalization of the Manin constant and the relation between congruence primes and modular degree.

With W. Stein: Some Abelian Varieties with Visible Shafarevich-Tate Groups.

On the ratio of the special L-value to the real period when the level is a product of two distinct primes.

On the calculation of |\frac{H_{e}^{+}}{ℑe} |.

(Most of my work can be accessed on my web-page. Also, the abstracts are given at the end of the C.V.)

INVITED TALKS Tata Institute of Fundamental Research, Bombay, India, Aug 1999: On the Birch and Swinnerton-Dyer formula.

NSF sponsored IAS/Park City Mathematics Institute, Utah, July 1999: On the ratio of the special L-value to the real volume.

Number theory seminar, Université de Paris 6, Paris, Jan 1999: On invisible elements of the Shafarevich-Tate group.

Number theory seminar (R. Coleman), U. C., Berkeley, Aug 1998 - Sept 1999: At least 3 talks, including the last two above and also on A generalization of the Manin constant and the relation between congruence primes and the modular degree.

NSF funded Arizona Winter School, U. of Arizona, March 1998: Detecting invisible elements of the Shafarevich-Tate group.

Plus several talks at seminars at U. C. Berkeley, including Modular forms and Galois representations (K. Ribet), Abelian varieties (H. Lenstra), graduate student seminars and a summer program for undergraduates.

CONFERENCES ATTENDED NSF sponsored IAS/Park City Mathematics Institute, Utah, July 1999. Séminaire Bourbaki, Paris, June 1999, 1998. Workshop on Local-to-Global Principles, Univ. of Arizona, Mar 1999. Arithmétique et Formes Automorphes, Paris, Dec 1998. International Congress of Mathematicians, Berlin, Aug 1998. Number theory and Topology: in honor of B. Mazur, Harvard U., May 1998. Arithmetic Geometry: Rational points, Cambridge, U.K., Apr 1998. Elliptic curves and Modular forms, Washington D.C., Mar 15-17, 1996. Modular forms and Fermat’s Last theorem, Boston, Aug 9-15, 1995.

TEACHING EXPERIENCE NSF sponsored IAS/Park City Mathematics Institute, Utah, July 1999 Teaching assistant: Led problem sessions for the Graduate Summer School courses titled “Elliptic curves” (J. Buhler) and “Open problems” (A. Silverberg).

University of California, Berkeley, Aug 1994-present Graduate student instructor: Led discussion sections for at least 8 lowerdivision Mathematics classes, including two head teaching assistantships. Courses include Calculus for social sciences majors, Calculus for science majors and Honors Linear Algebra.

Summer Institute in Mathematical Sciences, Berkeley, Summer 1997 Teaching Assistant: Led discussion sessions for the course “Computational Linear algebra” (N. Mackey), part of a summer program for women undergraduate students in mathematics from all over the United States.

COMPUTING Programmed in C++, Pascal and Fortran; Mathematical software used: MAT-LAB, Maple, PARI, LiDIA.

LANGUAGES English, French, Hindi, Marathi.

MEMBERSHIPS AND SERVICE Reviewer for Mathematical Reviews Member, American Mathematical Society Member, Mathematics Graduate Student Association, U. C., Berkeley REFERENCES Prof. Kenneth A. Ribet

Department of Mathematics University of California Berkeley, CA 94720-3840

(510) 642-0648 ribet@math.berkeley.edu

Prof. Loic Merel UFR de mathématiques Case 7012, Université Paris 7 – Denis Diderot 2, place Jussieu F-75251 Paris cedex 05 France Phone: 33 1 44 27 85 24 merel@math.jussieu.fr

Prof. Barry C. Mazur Department of Mathematics Harvard University One Oxford Street Cambridge, MA 02138

(617) 495-2171 ext. 512 mazur@abel.math.harvard.edu

Prof. Robert F. Coleman Department of Mathematics University of California Berkeley, CA 94720-3840

(510) 642-5101 coleman@math.berkeley.edu

Prof. Niloufer Mackey 6618 Everett Tower Department of Mathematics and Statistics Western Michigan University Kalamazoo, MI 49008-5152

(616) 387-4594 mackey@math-stat.wmich.edu

Prof. Lawrence C. Evans Department of Mathematics University of California Berkeley, CA 94720-3840 evans@math.berkeley.edu ABSTRACTS The Birch-Swinnerton-Dyer conjectural formula for certain quotients of J_{0}(N)

Let N be any positive integer and let A be an optimal quotient of J_{0}(N) such that L_{A}(1) is non-zero. Then the Birch and Swinnerton-Dyer (BSD) conjecture (as generalized by Tate and Gross) asserts the formula:

\frac{L_{A}(1)}{Ω_{A}} = \frac{|� _{A}| ∏_{p|N} |Φ_{p}(A)|}{|A(Q)||Â(Q)|},

where Ω_{A} denotes the volume of A(R) calculated using a set of generators of the invariant differentials on the Néron model model of A, � _{A} is the Shafarevich-Tate group of A, Φ_{p}(A) is the component group of the special fiber (at the prime p) of the Néron model model of A, and Â is the dual abelian variety of A. We give formulae for the ratio L_{A}(1)/Ω_{A} appearing above when A is the winding quotient, and when A is the quotient associated (by Shimura) to a newform (of weight 2 and level N). One would like to prove that the formula for L_{A}(1)/Ω_{A} is equal to the right hand side of the BSD formula above. We prove partial results in that direction and also outline a program to approach the problem. We also give some calculations of the value of L_{A}(1)/Ω_{A} obtained using modular symbols. Using this and assuming the BSD formula, when N = 1091 and A is the winding quotient, we find some elements of � _{A} that are invisible in J_{0}(N), as well as in J_{1}(N). While investigating the BSD formula, we naturally comes across a generalization of the Manin constant. We give its precise definition and state some results about it. Associated to an elliptic curve are three numbers: the modular degree, the congruence number and the Manin constant. We prove a result about primes whose squares do not divide N: if such a prime divides either the Manin constant or the congruence number, then it divides the modular degree also.

On invisible elements of the Tate-Shafarevich group Mazur has introduced the concept of visible elements in the Tate-Shafarevich group of optimal modular elliptic curves. We generalized the notion to arbitrary abelian subvarieties of abelian varieties and found, based on calculations that assume the Birch-Swinnerton-Dyer conjecture, that there are elements of the Tate-Shafarevich group of certain sub-abelian varieties of J_{0}(p) and J_{1}(p) that are not visible.

A generalization of the Manin constant and the relation between congruence primes and modular degree We extend the techniques of a paper of Abbes and Ullmo to show that if E is a strong modular elliptic curve and if p is a prime which divides the Manin constant of E, but whose square does not divide the conductor of E, then p divides the modular degree of E. We also show that except for primes whose squares divide the conductor, the congruence primes associated to the newform corresponding to E divide the modular degree of E. Finally we generalize the notion of the Manin constant to arbitrary quotients of J_{0}(N) under the action of ideals of the Hecke algebra and mention the corresponding results and a conjecture for it.

Some Abelian Varieties with Visible Shafarevich-Tate Groups We give examples of abelian variety quotients of the modular jacobian J_{0}(N) with nontrivial visible Shafarevich-Tate groups. In computing these examples we developed formulas for invariants of higher dimensional modular abelian varieties.

The ratio of the special L-value to the real period when the level is a product of two distinct primes We prove a formula for the ratio of the special L-value to the real volume for the winding quotient of J_{0}(pq), where p and q are two distinct primes.

On the calculation of |\frac{H_{e}^{+}}{ℑe} | In this article, we describe briefly some details of the calculation of the order of the group H_{e}^{+}/ℑe. This result was used in a publication of the author.