# Collaborative Number Theory Seminar

Co-organizers: Gautam Chinta, Brooke Feigon, Maria Sabitova, Lucien Szpiro.

The seminar currently meets Fridays 4:00 to 5:30 PM in Room 4422. The CUNY Graduate Center is located on Fifth Avenue, on the east side of the street, between 34th and 35th Streets in midtown Manhattan. For further information, please contact Maria Sabitova.

## Fall 2011 Schedule:

**September 2: No meeting this week.
**

**September 9: ****No meeting this week.**

**September 16: **Liang-Chung Hsia (National Taiwan Normal University)

Title: Preperiodic points for family of rational maps

September 23: **Marc Masdeu Sabaté** (Columbia University)

**Title:** A p-adic approach to explicit automorphic forms on Shimura curves

**Abstract:** The explicit computation of spaces of automorphic forms is a very active and useful area of research in computational number theory. Several authors have described algorithms to compute these Hecke modules: to name a few, John Voight, Matt Greenberg and Lassina Dembélé have described algorithms that work in different levels of generality. In joint work with Cameron Franc, we propose a p-adic approach to this problem, via understanding quotients of the Bruhat-Tits tree.

In this talk we will give an introduction to this subject and explain our approach to these explicit computations, as well as possible applications of our approach.

September 30: **No meeting this week.**

**October 7: ****No meeting this week.**

**October 14: ****No meeting this week.**

**October 21: ** ***Please note the special time*
2-3:30: Zhengyu Mao (**Rutgers University, Newark

**)**

**Title:** Fourier-Whittaker coefficients of automorphic forms on Mp(2n)

**
October 28: ** Gerard Freixas i Montplet (C.N.R.S. -- Institut de Mathématiques de Jussieu)

**Title:** On the height of the fixed point set of Atkin-Lehner involutions

**Abstract:** The arithmetic Riemann-Roch theorem of Gillet-Soule in Arakelov geometry does not apply to natural examples as modular curves and bundles of modular forms with their Petersson metrics, due to the singularities of the metrics near cusps and elliptic fixed points. I will present a variant of the theorem which applies to this situation, and I will explain how this can be used to derive expressions for heights of fixed point sets of Atkin-Lehner involutions in terms of special values of twisted Selberg zeta functions. In genus 0, these formulas together with the Chowla-Selberg formula provide new Selberg zeta values. Part of this talk will be based on joint work with Anna von Pippich (Humboldt University, Berlin).

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November 4: **Adam Towsley (University of Rochester)

**Title:** Reduction of Orbits

**Abstract:** We will look at several results about the behavior of orbits of rational maps when reduced modulo a prime ideal. For instance, if we look at a rational map taking a number field to itself we will consider for what proportion of primes p will a wandering point become periodic modulo p. Additionally we will look at the function field analogs of these results.

November 11: ***Please note the double header and special times*
**

**4:00-5:00:** Thomas Tucker (University of Rochester)

**Title:** Towards a dynamical relative Manin-Mumford conjecture

**Abstract:** Question: Given two points a, b in C and a family F of rational maps, when can there be infinitely many members f of F such that a and b are both preperiodic for f? Masser-Zannier answered this question or Lattes maps and Baker-DeMarco have answered it for the family x^d + c (where c varies over C). Here, we adapt Baker-DeMarco's method to a more general situation, using recent results of Yuan and Zhang. This work is joint with L.-C. Hsia and D. Ghioca.

**5:15-6:15:** Daniel Fiorilli (IAS)

**Title:** On how the first term of an arithmetic progression can influence the distribution of an arithmetic sequence

**Abstract:** In this talk we will show that many arithmetic sequences have asymetries in their distribution amongst the progressions mod q. The general phenomenon is that if we fix an integer a having some arithmetic properties (these properties depend on the sequence), then the progressions a mod q will tend to contain fewer elements of the arithmetic sequence than other progressions a mod q, on average over q. The observed phenomenon is for quite small arithmetic progressions, and the maximal size of the progressions is fixed by the nature of the sequence. Examples of sequences falling in our range of application are the sequence of primes, the sequence of integers which can be written as the sum of two squares (with or without multiplicity), the sequence of twin primes (under Hardy-Littlewood) and the sequence of integers free of small prime factors. We will focus on these examples as they are quite fun and enlightening.** **

November 18: ***Please note the double header and special times***

**4:00-5:00:** Reinier Bröker (Brown University)

**Title:** Computing Gauss sums and general theta series

**Abstract:** It is a classical problem to compute Gauss sums efficiently. Besides Gauss' result for the quadratic sum, no easy formula is known. In this talk we view Gauss sums as coefficients of a general theta series and we explain a method to efficiently approximate the residue of the theta series. Many examples will be given.

**5:15-6:15:** John T. Cullinan (Bard College)

**Title:** Divisibility properties of torsion subgroups of abelian surfaces

**Abstract:** Let A be an abelian variety defined over a number field K and suppose m >1 is a positive integer. Suppose further that the number of points on the reduction mod p of A is divisible by m for almost all primes p of K. Does there exist a K-isogenous abelian variety A' whose torsion subgroup (over K) has order divisible by m? This question was asked by Lang and answered in the affirmative by Katz for elliptic curves and for abelian surfaces when m is a prime number. We will discuss counterexamples in higher dimensions as well as recent work on abelian surfaces when m is composite.

November 25: **Thanksgiving Break, the University is closed.**

**
December 2: **Jens Funke (University of Durham)

**Title:** Regularized Theta Liftings and periods of modular functions

**Abstract:** In this talk, we discuss regularized theta liftings to construct weak Maass forms weight 1/2 as lifts of weak Maass forms of weight 0. As a special case we give a new proof of some of recent results of Duke, Toth and Imamoglu on cycle integrals of the modular j-invariant and extend these to any congruence subgroup. Moreover, our methods allow us to settle the open question of a geometric interpretation for periods of j along infinite geodesics in the upper half plane. In particular, we give the `central value' of the (non-existing) `L-function' for j.

**
December 9: ** Anna Haensch (Wesleyan University)

**Title:** Almost Universal Ternary Mixed Sums of Squares and Polygonal Numbers

**Abstract:** An inhomogeneous quadratic form is a sum of a quadratic form and a linear form; it is called almost universal if it represents all but finitely many positive integers. Examples of inhomogeneous quadratic forms are mixed sums of squares and generalized polygonal numbers. In this talk we will present a characterization of positive definite almost universal ternary mixed sums of squares and triangular numbers, and extend this idea to mixed sums of squares and m-gonal numbers, where m - 2 = 2p, for a prime p. This generalizes the recent work by Chan and Oh on almost universal ternary sums of triangular numbers. If time permits, we will also discuss a characterization of positive definite almost universal ternary inhomogeneous quadratic forms which satisfy some mild arithmetic conditions.