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

Feb 1

  Fernando Herrera, University of Chile
A Lipschitz summation formula on Siegel's half space of degree three
Abstract: The Lipschitz summation formula is a useful tool for automorphic forms, it gives a Fourier expansion for certain functions which arise in the theory of modular forms. Furthermore there are several works about generalizations. I will begin this talk showing the Lipschitz summation formula on the Poincare upper half plane (the classical version) and then to show a version on Siegel's half space of degree three.
Feb 8
  No meeting
Feb 15
  No meeting
Feb 22
  Dan White, Bryn Mawr College
Twelfth Power Moment of Dirichlet L-functions
Mar 1

  Fikreab  S. Admasu, The Graduate Center of CUNY
Subgroup growth zeta functions and Hecke algebras
Abstract: The study of subgroup growth zeta functions is a relatively young research area. In my thesis, I consider nilpotent groups and I attempt to generalize the notion of the cotype zeta function of an integer lattice to finitely generated nilpotent groups. This helps in determining the distribution of subgroups of finite index and provides more refined invariants in the analytic number theory of nilpotent groups. Similar attempt on algebraic groups leads to a rederivation of zeta functions of classical algebraic groups using Hecke algebras. A connection with Cohen-Lenstra heuristics will also be discussed.​
Mar 8

  Karen Taylor, Bronx Community College
Dirichlet Series and Generalized Hecke Groups
Abstract: Duke introduce the groups G(A) which naturally correspond to an ideal class, A, of a real quadratic field. These groups generalize the Hecke groups G(lambda), lambda>2. In this talk, I will describe two problems, which have been solved for the Hecke groups, which I am currently working on generalizing. One problem, solved by Rosen, is the description of the elements of G(lambda) by continued fractions. The other is to Knopp and Sheingorn's construction of modular integrals  on G(lambda) with prescribed log-polymomial periods.

Mar 15
  Linda Keen, The Graduate Center of CUNY and Lehman College
Symbolic dynamics, curves on surfaces, Farey sequences and discreteness problems
Abstract: The free group on two generators arises in many contexts. In this talk we will see it as the fundamental group of a punctured torus and a pair of pants, building blocks for Riemann surfaces. We will show how symbolic dynamics and Farey sequences can be used to find generating sets for the group. We will use these to answer some geometric and discreteness questions.

Mar 22
  No meeting
Mar 29
  Robert Donley, Queensborough Community College
Binomial arrays and generalized Vandermonde identities
Abstract: "It is not exaggerated to say the Catalan numbers are the most prominent sequence in combinatorics." (Kauers and Paule, 2013) Associated to the Catalan numbers is an evolving set of numerical triangles, for which we suggest perhaps a final form. These triangles belong to a larger class of arrays with features of both the extended Pascal's triangle and Riordan arrays. We introduce the notions of generalized binomial transform and binomial array, and we describe several hockey stick rules (3 short, 6 long) and a visual interpretation of Vandermonde's identity/convolution for these arrays. Finally, we obtain two families of sequences generalizing the Catalan numbers mostly absent from the OEIS.
Apr 5
  Peter Wear, University of California San Diego
Perfectoid covers of abelian varieties and their tilts
Abstract: We will give an example-based overview of perfectoid spaces, explain a recent construction (joint with Blakestad, Gvirtz, Heuer, Shchedrina, Shimizu, and Yao) which associates a perfectoid space to abelian varieties over perfectoid fields, and an extension of these ideas to describe the tilt of this perfectoid space.​
Apr 12
  Abdul Hassen, Rowan University
Hyperbolic Euler and Bernoulli Numbers and Polynomials

Abstract: In this talk we present a generalization of the Euler and Bernoulli Numbers and polynomials based on generalizations of the hyperbolic functions. We shall discuss some properties and identities satisfied by these polynomials. 
Apr 19
  Spring Break
Apr 26
  Spring Break
May 3

  Andrew Obus, Baruch College
Fun with Mac Lane valuations
Abstract: Mac Lane's technique of "inductive valuations" is over 80 years old, but has only recently been used to attack problems about arithmetic surfaces. We will give an explicit, hands-on introduction to the theory, requiring little background beyond the definition of a valuation on a ring. We will then outline how this theory is helpful for resolving "weak wild" quotient singularities of arithmetic surfaces, as well as for proving conductor-discriminant inequalities for higher genus curves. The first project is joint work with Stefan Wewers, and the second is joint work with Padmavathi Srinivasan.
May 10
  Cihan Karabulut, William Paterson University
Modular Forms Whose Fourier Coefficients Involve Zeta Functions of Binary Hermitian Forms


Fall 2018
Meeting in GC 3307, Fridays 11:45 - 1:00

Sept 14

  Ivan Horozov, Bronx Community College
Multiple zeta values and the adeles
Sept 21

  Ivan Horozov, Bronx Community College
Eisenstein cohomology of SL(3,Z)​
Sept 28

  Karen Taylor, Bronx Community College
Quadratic Identities and Maass Waveforms
Oct 5

  Karen Taylor, Bronx Community College
Quadratic Identities and Maass Waveforms II
Oct 12

  Jorge Pineiro, Bronx Community College
Dirichlet's unit theorem in algebraic dynamics​
Oct 19

  Cormac O'Sullivan, Bronx Community College and The Graduate Center
A generalization of the Riemann-Siegel formula
Abstract: The celebrated Riemann-Siegel formula expresses the difference between the Riemann zeta function on the critical line and its partial sums as an expansion in terms of decreasing powers of the imaginary variable t. Siegel anticipated that this formula could be generalized to include the Hardy-Littlewood approximate functional equation, valid in any vertical strip. We give this generalization for the first time. The asymptotics contain Mordell integrals and an interesting new family of polynomials.
Oct 26
  Karen Taylor, Bronx Community College
Ramanujan's Third Order Mock Theta Function
Abstract: I will be discussing, primarily, Watson's 1935 address to the LMS 
Nov 2
Nov 9

  Lindsay Dever, Bryn Mawr College
The Trace Formula in Hyperbolic 3-Space
Abstract: The trace formula allows us to compare spectral and geometric information (such as automorphic forms, and the lengths of closed geodesics) on hyperbolic surfaces, including arithmetic ones. These ideas have been extended to automorphic forms in many different contexts including three-dimensional hyperbolic space. In this expository talk, I will introduce automorphic forms in hyperbolic 3-space and give an overview of the trace formula.​​
Nov 16
  Daniel Garbin, Queensborough Community College
Effective bounds for Fourier coefficients of certain weakly holomorphic modular forms
Nov 23
Nov 30
  Nelson Carella, Bronx Community College
Density of Primes and Primitive Roots
Dec 7
  Kris Klosin, Queens College and The Graduate Center
The paramodular conjecture for abelian surfaces with rational torsion
Abstract: The Paramodular Conjecture can be viewed as an analog of the Taniyama-Shimura Conjecture for abelian surfaces asserting that they should correspond to certain (paramodular) Siegel modular forms. We will discuss recent progress on the conjecture in the case when the abelian surface has a rational torsion point. This is joint work with T. Berger.