Number Theory seminar

This semester the Number Theory seminar will meet on Tuesday from 3:35--4:30 pm in Hylan 1101, unless otherwise noted. There will be an after seminar dinner from 6pm, and the dinner place will be announced later.

Previous years: 02-03 01-02

Schedule for 2009-2010

DateSpeakerTopic Tuesday, Sept 15, 2009 Nick Rogers, Univ. of Rochester Descent via 2-isogeny and complete 2-descent. (Part 1)

In this talk we'll describe three related methods of so-calle"2-descent": descent via 2-isogeny, complete 2-descent, and second descent. We'll describe the obstruction for each of these methods in giving a sharp upper bound for the rank of an elliptic curve, and the relative size of these obstructions, for a variety of concrete examples. Tuesday, Sept 22, 2009Nick Rogers, Univ of RochesterDescent via 2-isogeny and complete 2-descent. (Part 2)Tuesday, Sept 29, 2009Tom Tucker, Univ. of RochesterTowards a dynamical Manin-Mumford conjecture

Abstract: We begin with a simple question: when can two rational maps have infinitely many preperiodic points in common? We will show that a thorough-going answer to this question leads to counterexamples to a well-known conjecture of Zhang's as well as possible ways to correct the conjecture. The proofs of the main facts prove to be surprisingly simple but draw on work of Lyubich, Thurston-Douady-Hubbard, and others.Thursday, Oct 8, 2009Phong Le, Univ of Caliornia IrvineCoherent Decomposition Newton Polygons of L-functions of Exponential Sums

Abstract: In this talk we investigate the decomposition theory for generic Newton polygons associated to L-functions of n-dimensional exponential sums over finite fields. The main result presented is a new decomposition theorem. This is a generalization of work developed by Daqing Wan.Tuesday, Oct 13, 2009Steven J Miller, Williams CollegeFinite conductor models for zeros near the central point of elliptic curve L-functions.

Abstract: Random Matrix Theory has successfully modeled the behavior of zeros of elliptic curve L-functions in the limit of large conductors. In this talk we explore the behavior of zeros near the central point for one-parameter families of elliptic curves with rank over Q(T) and small conductors. Zeros of L-functions are conjectured to be simple except possibly at the central point for deep arithmetic reasons; these families provide a fascinating laboratory to explore the effect of multiple zeros on nearby zeros. Though theory suggests the family zeros (those we believe exist due to the Birch and Swinnerton-Dyer Conjecture) should not interact with the remaining zeros, numerical calculations show this is not the case; however, the discrepency is likely due to small conductors, and unlike excess rank is observed to noticeably decrease as we increase the conductors. We shall mix theory and experiment and see some surprisingly results, which lead us to conjecture that a new random matrix ensemble correctly models the small conductor behavior.Tuesday, Oct 20, 2009No seminarTuesday, Oct 27, 2009Sheldon Joyner, Univ. of Western OntarioUnipotent connections and functional equations of zeta functions

Abstract: By generalizing Chen's iterated integral, one may exhibit arithmetic L-functions as integral transforms of certain functions with natural domain of definition the upper half plane H. For the Riemann zeta function and the Dirichlet L-functions, the associated functions on H are rational (in coordinates on P^{1}\{0,1,\infty}); and one may ask what properties of the latter reflect properties of the former. Data coming from the universal unipotent bundle with connection on P^{1}\{0,1,\infty}, constructed from the rational function associated to the Riemann zeta function, may be used to give a family of proofs of the analytic continuation and functional equation of the zeta function. Wednesday, Nov 18, 2009
1pm - 2pm in Hylan 1106a Steve Gonek, Univ. of Rochester The First 150 Years of the Riemann Zeta-Function

Schedule for 2008-2009

Date	Speaker	Topic
Tuesday, Sept. 23	Tom Tucker, U of R	Orbits of points under groups of matrices [abstract]
Tuesday, Sept. 30	Nick Rogers U of R	Heuristics for Aliquot Sequences [abstract]
Tuesday, Oct. 7	Andrew Ledoan, U of R	The index of a Farey sequence [abstract]
Tuesday, Oct. 14	C. Douglas Haessig, U of R	L-functions of Galois representations arising from families of exponential sums [abstract]
Tuesday, Oct. 21	Tom Tucker, U of R	The dynamical Bogomolov conjecture for lines in the plane [abstract]
Tuesday, Oct. 28	David McKinnon University of Waterloo	K3 surfaces, rational curves, and rational points [abstract]
Tuesday, Nov. 4	Doug Haessig, U of R	An introduction to Bombieri's proof of an upper bound for the degree of an L-function of exponential sums [abstract]
Tuesday, Nov. 11	Vijay Sookdeo, U of R	Orbits of Polynomial Modulo p [abstract]
Tuesday, Nov. 18	Andrew Ledoan, U of R	The complex zeros of random polynomials [abstract]
Tuesday, Dec. 9	Justin Sukiennik, U of R	Symmerty and assymetry of height functions