# Conference on Modular Forms and Related Topics

## American University of BeirutMay 28-May 31, 2018

### Siegfried Boecherer (Universität Mannheim-Germany)

Title: Arithmetic Properties of Vector-Valued Siegel Modular Forms
Abstract: Using equivariant holomorphic differential operators (as studied in detail by Ibukiyama) one can construct vector-valued modular forms of degree n from scalar-valued modular forms of degree n’. If the scalar weight is large, all modular forms arise in such a way (for quite general groups Γ) from Siegel Eisenstein series. This allows one to get generators of spaces of modular forms with algebraic Fourier coefficients and bounded denominators for the vector-valued case. A main motivation comes from an application of this procedure to vector-valued p-adic modular forms.

### François Brunault (École normale supérieure de Lyon-France)

Title:Mahler Measures of Elliptic Surfaces
Abstract:The Mahler measure M(P) of a polynomial P is defined as the geometric mean of P on the unit circle. This definition extends naturally to multivariate polynomials. After the pioneering result of Smyth identifying the Mahler measure of 1+X+Y in terms of a Dirichlet L-value, Boyd and Deninger investigated Mahler measures of 2-variable polynomials defining elliptic curves, leading to many remarkable identities relating M(P) to L(E,2), where E is the elliptic curve defined by the equation P(x,y)=0. These identities were checked numerically to high accuracy but, as of today, only finitely many of them are proven rigorously. Bertin established similar identities for 3-variable polynomials defining K3 surfaces. In this talk, I will explain a new method to compute the Mahler measure of a 3-variable polynomial defining a modular elliptic surface. The proof is based on an extension of the Rogers-Zudilin method to modular forms of weight 3. This is joint work with Michael Neururer. .

### Ahmad El-Guindy (Texas A&M University-Qatar)

Title:Evaluating Certain Coefficients of Drinfeld-Goss Hecke Eigenforms
Abstract:Unlike the classical case, there are no general formulas in the Drinfeld setting for the coefficients of a Hecke eigenform, in terms of the eigenvalues. In this work we obtain a closed form polynomial expression for certain coefficients of Drinfeld-Goss double-cuspidal modular forms which are eigenforms for the degree one Hecke operators, and we use those formulas to prove some vanishing and non-vanishing results for an infinite family of those coefficients.

### Michael Griffin (Brigham Young University-USA)

Title: The Modular Parameterization of Elliptic Curves
Abstract:In this talk I will discuss recent work with Jonathan Hales, an undergraduate at BYU. The modular parameterization of an elliptic curve defined over the rationals gives two modular functions X(z) and Y (z) which satisfy the defining equation of the curve, and which parameterize the points of the curve over the complex numbers. The motivating idea of this project is to understand the pre-images of rational points on the curve. The theory of complex multiplication shows that traces of Heegner points give rational points on quadratic twists of the elliptic curve. If the rank of the twisted curve is 1, then Gross and Zagier showed the point constructed in this manner has infinite order. We are particularly interested in understanding the pre-images of points which do not arise from Heegner points.

### Alia Hamieh (University of Northern British Columbia-Canada)

Title:Value-Distribution of Cubic L-Functions
Abstract:A significant part of the research in number theory is centered around the values of L-functions in the critical strip 0 ‹ Re(s)<1 . The L-functions in their value-distribution encode important information about the underlying structures. In this talk, we review the history of this subject and survey some recent value-distribution results. We also describe a value-distribution theorem for the logarithms and logarithmic derivatives of a family of L-functions attached to cubic Hecke characters. This is joint work with Amir Akbary. .

### Bernhard Heim (German University of Technology-Oman)

Title:Powers of the Dedekind Eta Function and Hurwitz Polynomials
Abstract:In this talk, we study the vanishing properties of Fourier coefficients of powers of the Dedekind eta function. We give a certain type of classification of this property. Further we extend the results of Atkin, Cohen, and Newman for odd powers and a list Serre presented in 1985. The topic is intimately related with Hurwitz polynomials. We also indicate possible generalization of the Lehmer conjecture. This talk contains joint work with Florian Luca, Atsushi Murase, Markus Neuhauser, Florian Rupp and Alexander Weisse

### Ghaith Hiary (Ohio State University-USA)

Title:Locating Zeros of L-Functions Using the Euler Product
Abstract:A standard provable method to locate zeros of a given L-function is the approximate functional equation, which is based on the Dirichlet series. I report on a heuristic method based on the Euler product, and motivated by several numerical observations

### Tomoyoshi Ibukiyama (Osaka University-Japan)

Title:Explicit Construction of Theta Series for any Vector-Valued Weight and Applications.
Abstract:It is well known that Siegel modular forms are constructed by theta series with pluri harmonic polynomials with good properties. We give here a general method to construct such polynomials explicitly for any vector valued weight of any degree, and give concrete examples of degree 3 which are applied to lifts and congruences. This is a joint work with Sho Takemori. If time allows, we also review some explicit way to construct automorphic differential operators where underlying scheme is very close to theta series.

### Paul Jenkins (Brigham Young University-USA)

Title:The Arithmetic of Modular Grids.
Abstract:A modular grid is a pair of sequences $f_m$, $g_n$ of weakly holomorphic modular forms where for all $m$ and $n$, the $n$th coefficient of $f_m$ is the negative of the $m$th coefficient of $g_n$. These grids were first noted by Zagier in weights $1/2$ and $3/2$ in the Kohnen plus space, and such grids have appeared for Poincare' series, for modular forms of integral weight, and in many other situations. We give a general proof of Zagier duality for canonical bases of spaces of weakly holomorphic modular forms of integral or half-integral weight and arbitrary level. This is joint work with Michael Griffin and Grant Molnar.

### Winfried Kohnen (Heidelberg University-Germany)

Title:On the Ramanujan-Petersson Conjecture for Modular Forms of Half-Integral Weight.
Abstract:We will discuss the so-called Ramanujan-Petersson conjecture for the Fourier coefficients of cusp forms of half-integral weight. We shall prove that the bound predicted is best possible (joint work with S. Gun, 2018)..

### Robert J. Lemke Oliver (Tufts University-USA)

Title:Selmer groups, Tate-Shafarevich groups, and ranks of abelian varieties in quadratic twist families.
Abstract:We determine the average size of the $\phi$-Selmer group in any quadratic twist family of abelian varieties having an isogeny $\phi$ of degree 3 over any number field. This has several applications towards the rank statistics in such families of quadratic twists. For example, it yields the first known quadratic twist families of absolutely simple abelian varieties over $\mathbb{Q}$, of dimension greater than one, for which the average rank is bounded; in fact, we obtain such twist families in arbitrarily large dimension. In the case that $E/F$ is an elliptic curve admitting a 3-isogeny, we prove that the average rank of its quadratic twists is bounded; if $F$ is totally real, we moreover show that a positive proportion of these twists have rank 0 and a positive proportion have $3$-Selmer rank 1. We also obtain consequences for Tate-Shafarevich groups of quadratic twists of a given elliptic curve. This is joint work with Manjul Bhargava, Zev Klagsbrun, and Ari Shnidman.

### Karl Mahlburg (Louisiana State University-USA)

Title:Quantum dimensions and characters for vertex operator algebras.
Abstract:: I will discuss recent work on the characters of irreducible (highest weight) modules for certain vertex operator algebra (in particular, $L_{\frak{sl}_\ell}(-\Lambda_0)$). The main question of study is the asymptotic behavior of characters for the irreducible modules, which is obtained using asymptotic analysis of quasimodular forms and partial theta functions. As a consequence, the quantum dimensions are one, as predicted by representation theory. Further results include a decomposition formula for the full characters in terms of unary theta and false theta functions, which gives their complete modularity properties.

### Steven J Miller (Williams College-USA)

Title:Finite 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 discrepancy 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.

### Larry Rolen (Vanderbilt University-USA)

Title:Jensen-Pólya Criterion for the Riemann Hypothesis and Related Problems.
Abstract::In this talk, I will summarize forthcoming work with Griffin, Ono, and Zagier. In 1927 Pólya proved that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for Riemann's Xi-function. This hyperbolicity has been proved for degrees $d\leq3$. We obtain an arbitrary precision asymptotic formula for the derivatives $\Xi^{(2n)}(0)$, which allows us to prove thehyperbolicity of $100\%$ of the Jensen polynomials of each degree. We obtain a general theorem which models such polynomials by Hermite polynomials. In the case of Riemann's Xi-function, this proves the GUE random matrix model prediction for the distribution of zeros in derivative aspect. This general condition also confirms a conjecture of Chen, Jia, and Wang on the partition function.

### Frank Thorne (University of South Carolina-USA)

Title:Number Field Counting, L-functions, and Automorphic Forms
Abstract::How many number fields are there of bounded discriminant? This is a very active research question, and I'll give a brief overview of what is known. I will highlight some connections of number field counting to questions involving L-functions and automorphic forms. These include (1) an application to a conjecture of Colmez on the Faltings heights of CM abelian varieties (joint with Adrian Barquero-Sanchez and Riad Masri), and (2) a connection between Eisenstein series and shapes of number fields, established by Bob Hough.

### Lynne Walling (Bristol University-UK)

Title:Explicit action of Hecke operators on half-integral weight Siegel Eisenstein series.
Abstract::While classical Eisenstein series of integral weight are well-understood (for any level and character), there are many gaps in our knowledge regarding Siegel Eisenstein series, especially those with half-integral weight. In this talk I will focus primarily on half-integral weight Siegel Eisenstein series of degree n and level 4N where N is odd and square-free (allowing arbitrary character modulo 4N). I will begin by defining such Eisenstein series, then proceed to evaluate the action of the Hecke operators T_j (p 2) (1 ≤ j ≤ n, p an odd prime) on those Eisenstein series attached to the Γ_0(4N )- orbits of elements in Γ_0(4). I will show that the subspace spanned by these Eisenstein series can be simultaneously diagonalised, explicitly computing the eigenvalues, and yielding a multiplicity-one result. Finally, I compare these eigenvalues to those of integral weight Siegel Eisenstein series.