NTS ABSTRACTFall2024

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Sep 12

Non-reductive special cycles and arithmetic fundamental lemmas

Zhiyu Zhang (Stanford)

We care about arithmetic invariants of polynomial equations e.g. L-functions, which (conjecturally) are often automorphic and related to special cycles on Shimura varieties (or Shimura sets) based on the relative Langlands program. Arithmetic fundamental lemmas reveal such relations in the p-adic local world. In this talk, I will study certain ``universal'’ non-reductive special cycles on local GL_n Shimura varieties, and give applications e.g. the proof of twisted arithmetic fundamental lemma for the tuple (U_n, GL_n, U_n). Time permitting, I will explain some global analogs where at least the (Betti) cohomology class of special cycles could be defined. It turns out that algebraic special cycles are often pullbacks of ``universal’’ non-algebraic cycles (e.g. from Kudla-Millson theory on non-Hermitian symmetric spaces).

Sep 19

Random walks on group extensions

Alireza Golsefidy (UCSD)

Lindenstrauss and Varju asked the following questions: for every prime p, let S_p be a symmetric generating set of G_p:=SL(2,F_p)xSL(2,F_p). Suppose the family of Cayley graphs {Cay(SL(2,F_p), pr_i(S_p))} is a family of expanders, where pr_i is the projection to the i-th component. Is it true that the family of Cayley graphs {Cay(G_p, S_p)} is a family of expanders? We answer this question and go beyond that by describing random-walks in various group extensions. (This is a joint work with Srivatsa Srinivas.)

Sep 26

Rational points on/near homogeneous hyper-surfaces

Niclas Technau (U Bonn)

How many rational points are on/near a compact hyper-surface? This question is related to Serre's Dimension Growth Conjecture. We survey the state of the art, and explain a standard random model. Furthermore, we report on recent joint work with Rajula Srivastava (Uni/MPIM Bonn). Our arguments are rooted in Fourier analysis and, in particular, clarify the role of curvature in the random model.

Oct 3

Distributions of how the p-adic Galois group acts in geometric local systems.

Asvin G (IPAM)

(Joint work with John Yin). We analyze potential generalizations of the Sato-Tate question to fibrations (of elliptic curves and other curves) over $\mathbb Z_p$. This builds on our earlier work on analogues of the Chebotarev density theorem in this setting (with Yifan Wei).

Oct 10

Computing Galois images of abelian threefolds with extra endomorphisms

Shiva Chidambaram (UW-Madison)

Let $C$ be a genus $3$ curve whose Jacobian is geometrically simple and has geometric endomorphism algebra equal to an imaginary quadratic field. In particular, consider Picard curves $y^3 = f_4(x)$ which have a natural order-3 automorphism. We study the associated mod-$\ell$ Galois representations and their images. I will discuss an algorithm, developed in ongoing joint work with Pip Goodman, to compute the set of primes $\ell$ for which the images are not maximal. By running it on several datasets of curves, the largest prime with non-maximal image we find is $13$. This may be compared with genus 1, where Serre's uniformity question asks if the mod-$\ell$ Galois image of non-CM elliptic curves over $\Q$ is maximal for all primes $\ell > 37$.

Oct 17

Quartic Gauss sums over primes and metaplectic theta functions

Alex Dunn (Georgia Tech)

We improve 1987 estimates of Patterson for sums of quartic Gauss sums over primes. Our Type-I and Type-II estimates feature new ideas, including use of the quadratic large sieve over the Gaussian quadratic field, and Suzuki's evaluation of the Fourier-Whittaker coefficients of quartic theta functions at squares. We also conjecture asymptotics for certain moments of quartic Gauss sums over primes. This is a joint work with C.David, A.Hamieh, and H.Lin.

Oct 24

Rigid meromorphic cocycles for orthogonal groups

Michael Lipnowski (OSU)

Rigid meromorphic cocycles are defined in the setting of orthogonal groups of arbitrary real signature and constructed in some instances via a p-adic analogue of Borcherds' singular theta lift. The values of rigid meromorphic cocycles at special points of an associated p-adic symmetric space are then conjectured to belong to class fields of suitable global reflex fields, suggesting an eventual framework for explicit class field theory beyond the setting of CM fields explored in the treatise of Shimura and Taniyama.

Oct 31

Faster isomorphism testing of p-groups of Frattini class-2

Chuanqi Zhang (University of Technology Sydney)

The finite group isomorphism problem asks to decide whether two finite groups of order N are isomorphic. Improving the classical $N^{O(\log N)}$-time algorithm for group isomorphism is a long-standing open problem. It is generally regarded that p-groups of class 2 and exponent p form a bottleneck case for group isomorphism in general. The recent breakthrough by Sun (STOC '23) presents an $N^{O((\log N)^{5/6})}$-time algorithm for this group class. Our work sharpens the key technical ingredients in Sun's algorithm and further improves Sun's result by presenting an $N^{\tilde O((\log N)^{1/2})}$-time algorithm for this group class. Besides, we also extend the result to the more general p-groups of Frattini class-2, which includes non-abelian 2-groups. In this talk, I will present the problem background and our main algorithm in detail, and introduce some connections with other research topics. For example, one intriguing connection is with the maximal and non-commutative ranks of matrix spaces, which have recently received considerable attention in algebraic complexity and computational invariant theory. Results from the theory of Tensor Isomorphism complexity class (Grochow--Qiao, SIAM J. Comput. '23) are utilized to simplify the algorithm and achieve the extension to p-groups of Frattini class-2.

Nov 7

Random Links and Arithmetic statistics

Tomer Schlank (University of Chicago)

Étalé homotopy is a theory that enables the study of algebraic and arithmetic concepts through geometric perspectives. Barry Mazur observed that, from this viewpoint, square-free integers are analogous to links in three-dimensional space. We will explore this analogy and propose a way to give it a statistical flavor. Informally, we assert that a random square-free integer of size approximately X resembles the closure of a random braid on roughly logX strands. We will make this statement more precise by introducing a number-theoretical analog for a family of numerical link invariants (called Kei's) and use this analogy to propose a conjecture regarding their asymptotic behavior. We will also present a few cases where this conjecture has been proven. This is a joint work with Ariel Davis.

Nov 14

Algebraic points on curves

Isabel Vogt (Brown University)

Faltings' famous theorem guarantees that a curve of genus at least 2 defined over a number field has only finitely many rational points.  Thus to get interesting infinite sets of closed points, we must allow points defined over higher degree extensions.  In this talk I'll introduce the framework we have for understanding these algebraic points (using the Abel--Jacobi map and Mordell--Lang Conjecture (a theorem of Faltings)), what infinite sets of algebraic points of bounded degree are known to say about the geometry of the curve in question, and (time-permitting) some interesting directions I see this going in.  This will include results from joint works with Geoff Smith, Borys Kadets and Bianca Viray.

Nov 19

The Decomposable Locus on the Gauss-Picard Modular Surface

Anton Hilado (University of Vermont)

The Igusa cusp form chi_10 is a cusp form on the Siegel modular threefold that parametrizes abelian surfaces. Among its many interesting properties is that its divisor is supported on the decomposable locus, which parametrizes the abelian surfaces which decompose as the product of two elliptic curves. The intersection theory of the decomposable locus and the complex multiplication locus is very interesting and may be seen as an analogue of the theory of Gross and Zagier on singular moduli. In this talk we will discuss an analogue of chi_10 for the Gauss-Picard modular surface which parametrizes abelian threefolds with complex multiplication by Z[i]. This makes use of the theory of Borcherds products for this modular surface developed by Yang and Ye. We will discuss properties of this cusp form, such as the integrality of its Fourier coefficients.

Nov 21

Families of n-smooth Dieudonné modules

Joshua Mundinger (University of Wisconsin-Madison)

Classical Dieudonné theory describes group schemes over a perfect field in terms of modules over a certain noncommutative ring. In this talk, we focus on n-smooth group schemes, e.g. the kernel of F^n on an abelian variety. For n-smooth group schemes, we give a Dieudonné theory in elementary terms over an arbitrary base ring; as a result, we prove conjectures of Drinfeld. This is joint work with C. Kothari.

Dec 5

Tschirnhausen bundles of covers of P^1 and Steinitz classes of number fields

Sameera Vemulapalli (Harvard University)

Given a number field extension L/K of fixed degree, one may consider O_L as an O_K-module. Which modules arise this way? Similarly, in the geometric setting, a cover of the complex projective line by a smooth curve yields a vector bundle on the projective line by pushforward of the structure sheaf; which bundles arise this way? (Equivalently, which rational normal scrolls contain degree d genus g covers of P^1?) In this talk, I'll describe joint work with Ravi Vakil in which we use remarkably simple tools (in particular, binary forms) to address both these questions.

 

Dec 12

Modularity of special cycles in orthogonal and unitary Shimura varieties

Salim Tayou (Dartmouth College)

Since the work of Jacobi and Siegel, it is well known that Theta series of quadratic lattices produce modular forms. In a vast generalization, Kudla and Millson have proved that the generating series of special cycles in orthogonal and unitary Shimura varieties are modular forms. In this talk, I will explain an extension of these results to toroidal compactifications where we prove that when these cycles are corrected by certain boundary cycles, the resulting generating series is still a modular form in the case of divisors in orthogonal Shimura varieties and cycles of codimension 2 in unitary Shimura varieties. The results of this talk are joint work with Philip Engel and François Greer, and joint work in progress with François Greer.