Study group in number theory
Coordinator Karen Taylor

Spring 2020
Meeting in GC 5417, Fridays 11:45 - 1:00

 Feb 28 Edna Jones, Rutgers University Local Densities of Diagonal Integral Ternary Quadratic Forms at Odd Primes Abstract: We give formulas for local densities of diagonal integral ternary quadratic forms at odd primes. Exponential sums and quadratic Gauss sums are used to obtain these formulas. These formulas (along with 2-adic densities and Siegel's mass formula) can be used to compute the representation numbers of certain ternary quadratic forms. Mar 6 Alexandra Florea, Columbia University Non-vanishing for cubic L-functions Abstract: Chowla conjectured that L(1/2,\chi) never vanishes, for $\chi$ any Dirichlet character. Soundararajan showed that more than 87.5% of the values L(1/2,\chi_d), for \chi_d a quadratic character, do not vanish. Much less is known about cubic characters. Baier and Young showed that more than X^{6/7-\epsilon} of L(1/2,\chi) are non-vanishing, for \chi a primitive, cubic character of conductor of size up to X. In joint work with C. David and M. Lalin, we show that a positive proportion of these central L-values are non-vanishing in the function field setting. The same techniques can be used to prove the analogous result in the number field setting, conditional on the Generalized Riemann Hypothesis. Mar 13 Robert Donley, Queensborough Community College Algebraic aspects of stochastic matrices and semi-magic squares Abstract: A square matrix is called a semi-magic matrix if each row and column have a common sum. For the case of size three and nonnegative integer coefficients, counting such matrices is a problem of combinatorial number theory going back to MacMahon (1916), and these matrices play a central role in the theory of angular momentum coupling. The goal of this talk is to explore the multiplicative properties of these matrices as complex algebras using the framework of elementary representation theory of finite groups. Examples include permutation matrices, circulant matrices and dihedral groups. This is joint work with M. S. Ravi.​ Mar 20 No meeting Mar 27 No meeting Apr 3 Cihan Karabulut, William Paterson University Apr 10 Spring Break Apr 17 Spring Break Apr 24 Huy Dang, University of Virginia May 1 Daniel Garbin, Queensborough Commmunity College May 8

Fall 2019
Meeting in GC 3307, Fridays 11:45 - 1:00

 Sept 13 Kevin O’Bryant, College of Staten Island and the GC Techniques in Explicit Number Theory Abstract: The speaker has recently been engaged in some “large” projects related to the distribution of primes, with the aim of producing explicit versions of classically asymptotic theorems, such as the Prime Number Theorem in Arithmetic Progressions, or Liouville/Roth type theorems in Diophantine Approximation. This talk will outline how that work proceeds, and then focus on a familiar technical problem: how to prove that $f(\vec x) \leq g(vec x)$, when $f$ is a hideously complicated (but explicitly computable) function and $g$ is a pleasantly “quotable" function. Sept 20 Heidi Goodson, Brooklyn College Sato-Tate Groups of Trinomial Hyperelliptic Curves Abstract: Let C_m: y^2=x^m+c be a smooth projective curve defined over Q. We would like to study the limiting distributions of the coefficients of the normalized L-polynomial for C_m. To determine the distributions, we study the Sato-Tate groups of the Jacobians of the curves. In this talk we give both general results and explicit examples of Sato-Tate groups for certain curves C_m. We will use these groups to determine the limiting distributions of the coefficients of the normalized L-polynomial. Sept 27 L​arry Rolen, Vanderbilt University Periodicities for Taylor coefficients of half-integral weight modular forms Abstract: Congruences of Fourier coefficients of modular forms have long been an object of central study. By comparison, the arithmetic of other expansions of modular forms, in particular Taylor expansions around points in the upper-half plane, has been much less studied. Recently, Romik made a conjecture about the periodicity of coefficients around $\tau=i$ of the classical Jacobi theta function. Here, in joint work with Michael Mertens and Pavel Guerzhoy, we prove this conjecture and generalize the phenomenon observed by Romik to a general class of modular forms of half-integral weight. Oct 4 No seminar Oct 11 Manami Roy, Fordham​ University Paramodular forms coming from elliptic curves Abstract: We will discuss a lifting from a non-CM elliptic curve over rationals to a paramodular form (a Siegel modular form with respect to the paramodular group) of degree 2 and weight 3 via the symmetric cube map. We find a description of the paramodular form in terms of the coefficients of the Weierstrass equation of the given elliptic curve. In order to understand this lifting, we consider the underlying representation-theoretic mechanism. Oct 18 Karen Taylor, Bronx Community College Coincidence of Theta Functions Attached to ​Quadratic Fields Abstract: We will discuss the coincidence of theta functions via an example of Cohen. We will discuss the shape of a generalization of Cohen's example. Oct 25 Joel Nagloo, Bronx Community College The Ax-Lindemann-Weierstrass theorem for automorphic functions Abstract: In this talk, I will discuss recent work, joint with G. Casale and J. Freitag, around the problem of transcendence of Fuchsian automorphic functions. These are the uniformizing functions for compact Riemann surfaces that are obtained from quotients of the upper half plane by certain Fuchsian groups (a well-known example is the modular j-function). I will place particular emphasis on the so called Ax-Lindemann-Weierstrass theorem. If time permits, I will also explain how our work can be applied to answer questions in arithmetic geometry (or more precisely to a special case of the conjectures on unlikely intersections). Nov 1 Cormac O'Sullivan, Bronx Community College and the GC Expansions of 1/((1-q)(1-q^2)...(1-q^N)) at roots of unity and Rademacher's conjecture Abstract: The generating function for restricted partitions is a finite product and has a Laurent expansion at each root of unity. The question of the behavior of these Laurent coefficients as the size of the product increases goes back to Rademacher and his work on partitions. Building on the methods of Drmota, Gerhold and previous results of mine, we complete this description and give the full asymptotic expansion of each coefficient at every root of unity. The proofs require Mellin transforms, polylogarithms and the saddle-point method. Nov 8 No meeting/MathFest Nov 15 Robert Sczech, Rutgers University A refinement of Stark's conjecture in the Abelian first order zero case Abstract: According to a classical conjecture of Harold Stark from 1980, the derivative of a partial zeta function at s=0, associated to an ideal class in a number field F, yields in certain cases the absolute value of a unit in an abelian extension of F. In my talk, I will present a refinement of that conjecture by giving a formula for that unit itself (and not only its absolute value) in terms of multiple Gamma functions. Nov 22 William Chen, Columbia University Nonabelian level structures for elliptic curves and noncongruence modular forms Abstract: Modular forms and modular curves have played a crucial role in modern number theory, but one almost always eventually restricts to considering only forms for congruence subgroups. In this talk I will try to explain how noncongruence subgroups (of SL(2,Z)) fit into the picture. Specifically, for a finite 2-generated group G, I will begin by defining the moduli space of elliptic curves with "G-structures", which will be a congruence modular curve if G is abelian, and noncongruence if G is sufficiently nonabelian. I will describe how this relates to the unbounded denominators conjecture for noncongruence modular forms, and using the work of Scholl, I will then describe a connection between the Fourier coefficients of noncongruence modular forms and Galois actions on the (nonabelian) fundamental groups of punctured elliptic curves. As time allows I will describe a joint work with Deligne showing that metabelian level structures are congruence, and some partial progress towards understanding Hecke operators on nonabelian level structures. Nov 29 No meeting/Thanksgiving Dec 6 Djordje Milicevic, Bryn Mawr College Extreme values of twisted L-functions Abstract: Distribution of values of L-functions on the critical line, or more generally central values in families of L-functions, has striking arithmetic implications. One aspect of this problem are upper bounds and the rate of extremal growth. The Lindelof Hypothesis states that zeta(1/2+it)<<(1+|t|)^eps for every eps>0 ; however neither this statement nor the celebrated Riemann Hypothesis (which implies it) by themselves do not provide even a conjecture for the precise extremal sub-power rate of growth. Soundararajan's method of resonators and its recent improvement due to Bondarenko-Seip are flexible first moment methods that unconditionally show that zeta(1/2+it), or central values of other degree one L-functions, achieve very large values. In this talk, we address large central values L(1/2, f x chi) of a fixed GL(2) L-function twisted by Dirichlet characters chi to a large prime modulus q. We show that many of these twisted L-functions achieve very high central values, not only in modulus but in arbitrary angular sectors modulo pi*Z, and that in fact given any two modular forms f and g, the product L(1/2, f x chi) * L(1/2, g x chi) achieves very high values. To obtain these results, we develop a flexible, ready-to-use variant of Soundararajan's method that uses only a limited amount of information about the arithmetic coefficients in the family. In turn, these conditions involve small moments of various combinations of Hecke eigenvalues over primes, for which we develop the corresponding Prime Number Theorems using functorial lifts of GL(2) forms. This is part of joint work on moments of twisted L-functions with Blomer, Fouvry, Kowalski, Michel, and Sawin.

Spring 2019
Meeting in GC 3307, Fridays 11:45 - 1:00