## Collaborative Number Theory Seminar at the CUNY Graduate Center

Co-organizers: Gautam Chinta, Clayton Petsche, Maria Sabitova, Lucien Szpiro.

## Fall 2010 Schedule:

September 24: No meeting this week.

October 1: Jonas B. Rasmussen (City College of New York).

Title: Higher congruences between modular forms.

Abstract: Given a prime number p and two modular forms on a congruence subgroup that have coefficients in the ring of integers of some number field K, we will discuss the problem of deciding whether these forms have coefficients that are congruent mod P^m (where P is a fixed prime ideal of K over p and m is a positive integer). The main focus will be on eigenforms on Gamma_1(N), and determining necessary and sufficient computable criteria for two such forms to be congruent mod P^m (possibly outside the prime divisors of Np). One of the primary applications of determining such criteria is the ability to determine if the associated mod P^m Galois representations are isomorphic. Using these computable criteria, we have systematically computed a wealth of examples of such higher congruences, and based on these examples we discuss conjectures on maximal congruences between newforms on Gamma_0(N). This is based on joint work with I. Chen and I. Kiming.

October 8: No meeting this week.

October 15: Jeffrey Hoffstein (Brown University).

Title: Shifted multiple Dirichlet Series and moments of L-series.

Abstract: I'll explain what shifted multiple Dirichlet Series are and show how to obtain their meromorphic continuation. I'll give a sample application to mean values of products of two GL(2) L-series averaged over character twists.

October 22: Oliver Lorscheid (City College of New York).

Title: Toroidal automorphic forms.

Abstract: A formula of Erich Hecke in an article from 1917 laid a connection between a sum of values of an Eisenstein series E(-,s) with the value zeta(s) of the zeta function. We call an automorphic form toroidal if the corresponding sum (or integral in its adelic formulation) vanishes for all right translates. The importance of this definition lies in a reformulation of the Riemann hypothesis in terms of the space of toroidal automorphic forms as observed by Don Zagier. Namely, the Eisenstein series E(-,s) lies in a tempered representation if and only if s has real part 1/2, and by Hecke's formula, E(-,s) is toroidal if s is a zero of the zeta functions. In order to reverse the latter statement, non-vanishing results has to be shown for the factors occuring in Hecke's formula. In a joint work with Gunther Cornelissen, double Dirichlet series are used for this purpose. In this talk, we will introduce into the theory of toroidal automorphic forms and give an overview of results in this direction. Then we will explain how to use double Dirichlet series to show non-vanishing results.

October 29: Phil Williams (CUNY Graduate Center).

Title: Resultant and conductor: minimality and semi-stability.

Abstract: The minimal discriminant is a divisor that has support at the points of bad reduction of an elliptic curve over a function field or number field. Szpiro's Theorem bounds the degree of the minimal discriminant divisor of an elliptic curve over a function field in terms of the genus of the function field and the number of points of bad reduction. The corresponding statement for number fields is equivalent to one formulation of the ABC-conjecture. For self-maps of the projective line over a function field, there is an analogous concept of bad reduction. One can construct a divisor which measures this: the minimal resultant. We construct a condition for measuring the minimality (at a point on the given curve) of a given presentation of a dynamical system, and use this to show a counterexample to the natural dynamical analogue to Szpiro's Theorem in this context. We also show that a semi-stable presentation of a dynamical system over a function field realizes the conductor (the set of points of bad reduction). This is joint work with Lucien Szpiro and Michael Tepper.

November 5: Takashi Taniguchi (Kobe University and Princeton University).

Title: The secondary term in the counting function of cubic field discriminants, and their distributions in arithmetic progressions.

Abstract: In this joint work with Frank Thorne, we present a proof of Roberts' conjecture that gives the secondary term in the counting function for cubic fields. Our work is independent of another proof of the conjecture by Bhargava, Shankar and Tsimerman, and uses the Shintani zeta functions. With the twists of the Shintani zeta function by Dirichlet characters, we can also study the distributions of cubic field discriminants in arithmetic progressions. We show a couple of non-equidistribution results in the secondary term.

November 12: Cormac O'Sullivan (Bronx Community College).

Title: Taylor coefficients of modular forms.

Abstract: The Fourier coefficients of modular forms are well-known to contain useful arithmetic information. These Fourier coefficients may be thought of as Taylor coefficients at infinity. In joint work with Morten Risager, we begin to study the Taylor expansions of modular forms at points in the upper half plane, building on earlier work of Rodriguez Villegas and Zagier. At CM points we show these Taylor coefficients are non-zero and that they also have interesting arithmetic properties, many still unexplained.

November 19: Krzysztof Klosin (Queens College).

Title: Modularity of residually reducible Galois representations.

Abstract: Proving that nice'' Galois representations arise from automorphic forms has been a major theme in number theory for well over a decade. We will discuss an approach based on studying the ideal of reducibility of a universal deformation that allows us to prove new cases of modularity. Examples include some 2-dimensional Galois representations over imaginary quadratic fields (unconditional) and 4-dimensional Galois representations over the rationals (conditional on some assumptions). This is joint work with T. Berger.

November 26: Thanksgiving, the University is closed.

December 3: Amanda Folsom (Yale University)

Title: Kac-Wakimoto characters, asymptotics, and mock modular forms.

Abstract: Recently, Kac and Wakimoto established specialized character formulas for irreducible highest weight sl(m,1)^ modules, and later works of the author and Bringmann-Ono show that these characters may be realized as parts of certain non-holomorphic modular functions. We will describe this, and show in joint work with Bringmann, how the modularity" of these characters can be exploited to obtain detailed asymptotics.

December 10: Rafe Jones (College of the Holy Cross).

Title: Galois theory of rational maps with non-trivial automorphisms.

Abstract: I'll describe recent work investigating the dynamical Galois representation attached to a degree-2 rational function that commutes with a non-trivial Mobius transformation. This is in some sense the equivalent of the Galois representation attached to a CM elliptic curve. The image of this dynamical Galois representation must lie in a certain subgroup C of the automorphism group of the complete infinite binary rooted tree. In joint work with Michelle Manes, we show that the image has finite index in C under fairly weak hypotheses.

## Spring 2010 Schedule:

February 5: Clayton Petsche (Hunter College).

Title: A non-Archimedean Weyl equidistribution theorem.

Abstract: Weyl proved that if an N-dimensional real vector v has linearly independent coordinates over Q, then its integer multiples v, 2v, 3v, .... are uniformly distributed modulo 1. Stated multiplicatively (via the exponential map), this can be viewed as a Haar-equidistribution result for the cyclic group generated by a point on the N-dimensional complex unit torus. I will discuss an analogue of this result over a non-Archimedean field K, in which the equidistribution takes place on the N-dimensional Berkovich unit torus over K. The proof uses a general criterion for non-Archimedean equidistribution, along with a theorem of Mordell-Lang type for the group variety G_m^N over the residue field of K, which is due to Laurent.

February 12: Lincoln's Birthday, the University is closed.

February 19: Maria Sabitova (Queens College).

Title: Local root numbers of abelian varieties.

Abstract: Root numbers are important fundamental invariants that arise in connection with several influential conjectures of number theory and representation theory, such as Birch--Swinnerton-Dyer conjecture, conjectural functional equations for L-functions, and the Langlands program. In this talk I will discuss recent progress on root numbers attached to abelian varieties.

February 26: Benjamin Hutz (Amherst College). *Canceled due to snow*

March 5: Andrew Obus (Columbia University).

Title: Ramification in Fields of Moduli.

Abstract: If G is a finite group, then the field of moduli of a branched G-cover of the Riemann sphere is the intersection of all fields of definition of the G-cover. A result of Beckmann says that for any 3-point G-Galois cover of the Riemann sphere, if a prime p does not divide the order of G, then p is unramified in the field of moduli of the G-cover. Wewers generalized this: if p exactly divides the order of G, then p is tamely ramified in the field of moduli. We will discuss extensions of this result, contained in the speaker's thesis, involving more general groups G with cyclic p-Sylow groups.

March 12: Anupam Bhatnagar (CUNY Graduate Center)

Title: Points of Canonical Height Zero.

Abstract: We give a parametrization of points of canonical height zero of an algebraic dynamical system defined over the function field of a curve. The main theorem describes the relation between points of canonical height zero and preperiodic isotrivial subvarieties. This is joint work with Lucien Szpiro.

March 19: Kiryl Tsishchanka (NYU)

Title: On approximation of real numbers by algebraic numbers of bounded degree.

March 26: Matt Papanikolas (Texas A&M)

Title: Special values of Goss L-functions for function fields.

Abstract: Values of Dirichlet L-functions at positive integers are expressible in terms of powers of pi and values of polylogarithms at algebraic numbers. In this talk we will focus on finding analogies of these results over function fields of positive characteristic. In particular, we will consider special values of Goss L-functions for Dirichlet characters, which take values in the completion of the rational function field in one variable over a finite field. Building on work of Anderson for the case of L(1,chi), we deduce various power series identities on tensor powers of the Carlitz module that are "log-algebraic" and in turn use these formulas to determine exact values of L(n,chi) for arbitrary n > 0. Moreover, we relate these L-series values to powers of the Carlitz period and values of Carlitz polylogarithms at algebraic points.

April 2: Spring Break, the University is closed.

April 9: Aaron Levin (Institute for Advanced Study)

Title: Towards Schmidt's Theorem for Algebraic Points of Bounded Degree.

Abstract: The Schmidt subspace theorem is a deep generalization of Roth's theorem in Diophantine approximation to the setting of hyperplanes in projective space. Another well-known generalization of Roth's theorem is the theorem of Wirsing, which extends Roth's theorem from rational points to algebraic points of bounded degree. In a similar way, I will discuss some results giving a version of Schmidt's theorem for algebraic points of bounded degree.

April 16: Yiannis Petridis (University College London)

Title: Dissolving cusp forms into resonances: Higher order Fermi's Golden Rules.

Abstract: For a hyperbolic surface embedded eigenvalues are unstable and tend to become resonances. The sufficient dissolving condition was identified by Phillips-Sarnak and is elegantly expressed in Fermi's Golden Rule. We prove formulas for higher approximations and obtain necessary and sufficient conditions for dissolving a cusp form into a resonance. We relate the result to the special values of L-series involving a Rankin-Selberg convolution of the cusp form with higher order automorphic forms. This is joint work with Morten S. Risager.

April 23: Michael Tepper (Penn State Abington)

Title: Isotrivial is equivalent to potential good reduction.

Abstract: Let K = k(C) be the function field of a complete non-singular curve C over an arbitrary field k. We prove, in two different ways, an endomorphism of projective n-space over K is isotrivial if and only if it has potential good reduction at all places v of K. The first proof uses algebraic geometry and geometric invariant theory; the second uses non-archimedean analysis and dynamics. While both proofs use dynamics, the second more directly, generalize results of Benedetto for polynomial maps and Baker for arbitrary rational maps when n=1. This is joint work with Clayton Petsche and Lucien Szpiro.

April 30: No speaker this week.

May 7: No speaker this week.

Abstract: The talk shall illustrate the proof, joint with David Masser, of the following conjecture of his: Consider the Legendre family of elliptic curves $E_\lambda: y^2=x(x-1)(x-\lambda)$, and pick two points $P_\lambda,Q_\lambda$ on it, with abscissas resp. $2,3$. There are only finitely many complex values of $\lambda$ such that both points become torsion. I shall also discuss variations on this problem, which turns out to be a case of very general conjectures of Pink.