Mark Goresky : Real structures on abelian varieties
- Number Theory ( 94 Views )In this talk we describe a partially successful attempt to describe a characteristic p > 0 analog of the locally symmetric spaces for GL(n,R), by interpreting this as a "moduli space" for abelian varieties with real structure.
Junyan Xu : Bounds for certain families of character sums: how to obtain strong bounds with more exceptions from weak bounds with fewer exceptions
- Number Theory ( 209 Views )I will first introduce some generalities about exponential sums, in particular that square-root cancellation is expected for many algebraic character sums over the rational points of an algebraic variety over a finite field. I will then set the stage for my work: we consider a family of exponential sums, which in our case is parameterized by the rational points of a variety (the parameter space). Our task is to obtain a good bound on the number of exceptional ("bad") parameters for which square-root cancellations fail. Following an idea of Michael Larsen, we consider even moments of the family of exponential sums. If the summands are of certain product form, a transformation can be applied to produce another family of exponential sums (of the same type). If the summands are products of multiplicative characters composed with certain polynomial functions, a weak bound can then be applied to the character sums in this family (with few bad parameters), yielding bounds for the moments. We know from the theory of l-adic sheaves that the parameter space for the original family have a stratification by smooth varieties, which is uniform in some sense as long as the degrees of the characters and polynomials are bounded. Moreover, on each stratum the character sum behave in certain uniform way, so that we can talk about good and bad strata. The bounds on moments yield bounds on dimensions of bad strata, which in turn yield bounds on the number of bad parameters (in any box) of the original family. Though not optimal, the bounds already imply nontrivial Burgess bounds for forms, in joint work with Lillian Pierce.
Matthew Litman : Markoff-type K3 Surfaces: Local and Global Finite Orbits
- Number Theory ( 102 Views )Markoff triples were introduced in 1879 and have a rich history spanning many branches of mathematics. In 2016, Bourgain, Gamburd, and Sarnak answered a long standing question by showing there exist infinitely many composite Markoff numbers. Their proof relied on showing the connectivity for an infinite family of graphs associated to Markoff triples modulo p for infinitely many primes p. In this talk we discuss what happens for the projective analogue of Markoff triples, that is surfaces W in P^1 x P^1 x P^1 cut out by the vanishing of a (2,2,2)-form that admit three non-commuting involutions and are fixed under coordinate permutations and double sign changes. Inspired by the work of B-G-S we investigate such surfaces over finite fields, specifically their orbit structure under their automorphism group. For a specific one-parameter subfamily W_k of such surfaces, we construct finite orbits in W_k(C) by studying small orbits that appear in W_k(F_p) for many values of p and k. This talk is based on joint work with E. Fuchs, J. Silverman, and A. Tran.
Edna Jones : The Kloosterman circle method and weighted representation numbers of positive definite quadratic forms
- Number Theory ( 152 Views )We develop a version of the Kloosterman circle method with a bump function that is used to provide asymptotics for weighted representation numbers of positive definite integral quadratic forms. Unlike many applications of the Kloosterman circle method, we explicitly state some constants in the error terms that depend on the quadratic form. This version of the Kloosterman circle method uses Gauss sums, Kloosterman sums, Salié sums, and a principle of nonstationary phase. If time permits, we may discuss a potential application of this version of the Kloosterman circle method to a proof of a strong asymptotic local-global principle for certain Kleinian sphere packings.
Viet Bao Le Hung : Congruences between automorphic forms
- Number Theory ( 120 Views )The theory of congruences between automorphic forms traces back to Ramanujan, who observed various congruence properties between coefficients of generating functions related to the partition function. Since then, the subject has evolved to become a central piece of contemporary number theory, lying at the heart of spectacular achievements such as the proof of Fermat's Last Theorem and the Sato-Tate conjecture. In my talk I will explain how the modern theory gives satisfactory explanations of some concrete phenomena for modular forms (the GL_2 case), and discuss recent progress concerning automorphic forms for higher rank groups.
Jonathan P. Wang : Derived Satake equivalence for Godement-Jacquet monoids
- Number Theory ( 254 Views )Godement-Jacquet use the Schwartz space of n-by-n matrices to construct the standard L-function for GL_n. Ben-Zvi, Sakellaridis and Venkatesh conjecture that the local unramified part of this theory can be categorified to an equivalence between an 'analytic' category of constructible sheaves and a 'spectral' category of dg modules. In this talk I will explain the proof of this equivalence and some of its properties. I will also discuss connections to conjectures of Braverman-Kazhdan on constructions of general automorphic L-functions. This is joint work with Tsao-Hsien Chen (in preparation).
Ma Luo : Algebraic iterated integrals on the modular curve
- Number Theory ( 119 Views )In the previous talk, we discussed the algebraic de Rham theory for unipotent fundamental groups of elliptic curves. In this talk, we generalize it to a Q-de Rham theory for the relative completion of the modular group, the (orbifold) fundamental group of the modular curve. Using Chen's method of power series connections, we construct a connection on the modular curve that generalizes the elliptic KZB connection on an elliptic curve. By Tannaka duality, it can be viewed as a universal relative unipotent connection with a regular singularity at the cusp. This connection enables us to construct iterated integrals of modular forms, possibly 'of the second kind', that provide periods called 'multiple modular values' by Brown. These periods include multiple zeta values and periods of modular forms.
Stephen Kudla : Theta integrals and generalized error functions
- Number Theory ( 115 Views )Recently Alexandrov, Banerjee, Manschot and Pioline [ABMP] constructed generalizations of Zwegers theta functions for lattices of signature (n-2,2). They also suggested a generalization to the case of arbitrary signature (n-q,q) and this case was subsequently proved by Nazaroglu. Their functions, which depend on certain collections $\CC$ of negative vectors, are obtained by `completing' a non-modular holomorphic generating series by means of a non-holomorphic theta type series involving generalized error functions. In joint work with Jens Funke, we show that their completed modular series arises as integrals of the q-form valued theta functions, defined in old joint work of the author and John Millson, over a certain singular $q$-cube determined by the data $\CC$. This gives an alternative construction of such series and a conceptual basis for their modularity. If time permits, I will discuss the simplicial case and a curious `convexity' problem for Grassmannians that arises in this context.
Tony Feng : Steenrod operations and the Artin-Tate pairing
- Number Theory ( 217 Views )In 1966 Artin and Tate constructed a canonical pairing on the Brauer group of a surface over a finite field, and conjectured it to be alternating. This duality has analogous incarnations across arithmetic and topology, namely the Cassels-Tate pairing for a Jacobian variety, and the linking form on a 5-manifold. I will explain a proof of the conjecture, which is based on a surprising connection to Steenrod operations.
Wei Ho : Families of lattice-polarized K3 surfaces
- Number Theory ( 97 Views )There are well-known explicit families of K3 surfaces equipped with a low degree polarization, e.g., quartic surfaces in P^3. What if one specifies multiple line bundles instead of a single one? We will discuss representation-theoretic constructions of such families, i.e., moduli spaces for K3 surfaces whose Neron-Severi groups contain specified lattices. These constructions, inspired by arithmetic considerations, also involve some fun geometry and combinatorics. This is joint work with Manjul Bhargava and Abhinav Kumar.
Efrat Bank : Primes in short intervals on curves over finite fields.
- Number Theory ( 98 Views )We prove an analogue of the Prime Number Theorem for short intervals on a smooth proper curve of arbitrary genus over a finite field. Our main result gives a uniform asymptotic count of those rational functions, inside short intervals defined by a very ample effective divisor E, whose principal divisors are prime away from E. In this talk, I will discuss the setting and definitions we use in order to make sense of such a count, and will give a rough sketch of the proof. This is a joint work with Tyler Foster.
William Chen : Arithmetic monodromy actions on the pro-metabelian fundamental group of punctured elliptic curves
- Number Theory ( 153 Views )For a finite 2-generated group G, one can consider the moduli of elliptic curves equipped with G-structures, which is roughly a G-Galois cover of the elliptic curve, unramified away from the origin. The resulting moduli spaces are quotients of the upper half plane by possibly noncongruence subgroups of SL(2,Z). When G is abelian, it is easy to see that such level structures are equivalent to classical congruence level structures, but in general it is difficult to classify the groups G which yield congruence level structures. In this talk I will focus on a recent joint result with Pierre Deligne, where we show that for any metabelian G, G-structures are congruence in an arithmetic sense. We do this by studying the monodromy action of the fundamental group of the moduli stack of elliptic curves (over Q) on the pro-metabelian fundamental group of a punctured elliptic curve.
Dante Bonolis : 2-torsion in class groups of number fields
- Number Theory ( 81 Views )In 2020, Bhargava, Shankar, Taniguchi, Thorne, Tsimerman, and Zhao established that, for a given number field $K$ with a degree $n\geq 5$, the size of the $2$-torsion is bounded by $h_{2}(K) \ll D^{\frac{1}{2}-\frac{1}{2n}}$, where $D_{K}$ is the discriminant of $K$ over $\mathbb{Q}$. In this presentation, we will introduce new bounds that take into account the geometry of the lattice underlying the ring of integers of $K$. This research is a joint project with Pierre Le Boudec.
Jacek Brodzki : A generalised Julg-Valette complex for CAT(0)-cube complexes.
- Number Theory ( 136 Views )This talk will introduce a very natural and interesting differential complex associated with a CAT(0)-cube complex. The construction builds on ideas first introduced by Pytlik and Szwarc for the free group and extended by Julg and Valette in the case of groups acting on trees. We will extend ideas of Julg-Valette to show how this construction can be used to study K-amenability and K-homology of groups acting on CAT(0)-cube complexes. This talk is based on joint work with Erik Guentner and Nigel Higson.
John Voight : Presentations for rings of modular forms
- Number Theory ( 104 Views )We give an explicit presentation for the ring of modular forms for a Fuchsian group with cofinite area, depending on the signature of the group. Our work can be seen as a generalization of the classical theorem of Petri: we give a presentation for the canonical ring of a stacky curve. This is joint work with David Zureick-Brown.
Spencer Leslie : Whittaker functions and connections to crystal graphs
- Number Theory ( 127 Views )Whittaker functions are certain special functions that play a central role in automorphic representation theory. When dealing with automorphic forms on covering groups, new methods are needed to compute these functions. In this talk, I will outline how the study of this problem has uncovered connections with geometric representation theory and crystal graphs. I also explain my work in making this connection practical, allowing for new computations of spherical Whittaker functions for covering groups.
Ayla Gafni : Extremal primes for elliptic curves without complex multiplication
- Number Theory ( 165 Views )Fix an elliptic curve $E$ over $\mathbb{Q}$. An ''extremal prime'' for $E$ is a prime $p$ of good reduction such that the number of rational points on $E$ modulo $p$ is maximal or minimal in relation to the Hasse bound. In this talk, I will discuss what is known and conjectured about the number of extremal primes $p\le X$, and give the first non-trivial upper bound for the number of such primes when $E$ is a curve without complex multiplication. The result is conditional on the hypothesis that all the symmetric power $L$-functions associated to $E$ are automorphic and satisfy the Generalized Riemann Hypothesis. In order to obtain this bound, we use explicit equidistribution for the Sato-Tate measure as in recent work of Rouse and Thorner, and refine certain intermediate estimates taking advantage of the fact that extremal primes have a very small Sato-Tate measure.
Chung Pang Mok : Introduction to Mochizukis works on inter-universal Teichmuller theory
- Number Theory ( 163 Views )Inter-universal Teichmuller theory, as developed by Mochizuki in the past decade, is an analogue for number fields of the classical Teichmuller theory, and also of the p-adic Teichmuller theory of Mochizuki. In this theory, the ring structure of a number field is subject to non-ring theoretic deformation. Absolute anabelian geometry, a refinement of anabelian geometry, plays a crucial role in inter-universal Teichmuller theory. In this talk, we will try to give an introduction to these ideas.
John Voight : Computing with Hilbert modular surfaces
- Number Theory ( 19 Views )Hilbert modular surfaces are 2-dimensional analogues of modular curves, parametrizing polarized abelian surfaces with endomorphism and level structure. Modular curves are stratified by genus, and canonical equations for modular curves are obtained from the graded ring of modular forms. Similar to how curves are stratified by genus, surfaces are organized by their numerical invariants; the Enriques-Kodaira classification organizes smooth surfaces by Kodaira dimension, Hodge numbers, and Chern numbers. In this talk, we explain how to compute these invariants and equations for certain Hilbert modular surfaces. This is joint work with Eran Assaf, Angie Babei, Ben Breen, Sara Chari, Edgar Costa, Juanita Duque-Rosero, Alex Horawa, Jean Kieffer, Avi Kulkarni, Grant Molnar, Abhijit S. Mudigonda, Michael Musty, Sam Schiavone, Shikhin Sethi, and Samuel Tripp.
Jessica Fintzen : Representations of p-adic groups
- Number Theory ( 190 Views )In the 1990s Moy and Prasad revolutionized p-adic representation theory by showing how to use Bruhat-Tits theory to assign invariants to p-adic representations. The tools they introduced resulted in rapid advancements in both representation theory and harmonic analysis -- areas of central importance in the Langlands program. A crucial ingredient for many results is an explicit construction of (types for) representations of p-adic groups. In this talk I will indicate why, survey what constructions are known (no knowledge about p-adic groups assumed) and present recent developments based on a refinement of Moy and Prasad's invariants.
Cheng Chen : Progresses of the local Gan-Gross-Prasad conjecture
- Number Theory ( 18 Views )The classical branching rules describe the spectrum of an irreducible complex representation of a compact Lie group to its subgroup. The local Gan–Gross–Prasad conjecture generalizes the branching problem to classical groups over local fields of characteristic zero. After the pioneering work of Waldspurger, there has been significant progress on the conjecture using various approaches. In my talk, I will introduce a relatively uniform approach to prove the conjecture, including joint work with Z. Luo and joint work with R. Chen and J. Zou.
Freydoon Shahidi : Local Langlands correspondence and the exterior and symmetric square root numbers for GL(n)
- Number Theory ( 123 Views )We will discuss the notion of Artin root numbers attached to an n-dimensional continuous Frobenius-semisimple complex representation of the Weil-Deligne group and show their equalities with those defined by Langlands-Shahidi method through local Langlands correspondence for GL(n) and the exterior and symmetric square representation of the L-group GL(n,C) of GL(n). The proof is a robust deformation argument using local-global techniques, complemented with suitable asymptotic expansions for partial Bessel functions inspired by certain generalized Shalika germ expansions of Jacquet and Ye. This is a joint work with J. Cogdell and T.-L. Tsai.
Hang Xue : Fourier--Jacobi periods on unitary groups
- Number Theory ( 112 Views )We will formulate a conjectural identity relating the Fourier--Jacobi periods on unitary groups and the central value of certain Rankin--Selberg $L$-functions. This refines the famous Gan--Gross--Prasad conjecture. We will give some examples supporting this conjecture.