In spring 2026, the UA Grad ANT Seminar is organized by Nick Pilotti. This website is maintained by Napoleon Wang.
We meet Mondays, 3–4 PM Arizona time, in Math 402, unless otherwise noted.
Apr 27 Ameya Borwankar
Title: Computing Endomorphism Rings of Supersingular Elliptic Curves
Abstract: The endomorphism ring of a Supersingular Elliptic Curve E is a maximal order in a certain quaternion algebra over Q. Because of its complex structure, it has gained importance in the field of isogeny-based cryptography. In this talk, we will take a look at an algorithm to compute End(E) using the local-global principle, where End(E) is computed locally over Q_p using the Bruhat-Tits Tree.
Apr 20 Ameya Borwankar
Title: Supersingular Elliptic Curve Cryptography
Abstract: Supersingular Elliptic Curve Cryptography is part of post-quantum Cryptography, which aims at developing quantum-resistant cryptosystems. In this talk we shall go over two post-quantum protocols : SIDH and SIH which are based on supersingular isogeny graphs. We will also take a look at recent attacks against SIDH and SIH.
Apr 13 Xinran Wian
Title: Automorphic forms on GL(2) from the analytic point of view
Abstract: Given a reductive group G over a number field F and a well-behaved test function \phi on G(\mathbb{A}), the Hecke operator associated with \phi is of trace-class, and the goal of the trace formula is to give an explicit formula for this trace. In my talk, I will briefly describe the analytic theory of the trace formula for forms of GL(2) and use it to establish the Jacquet-Langlands correspondence between GL(2) and D^\times for a division quaternion algebra D over F.
Apr 6 Roberto Castillo
Title: Hilbert's tenth problem
Abstract: During 1900, David Hilbert published his famous list of problems. In modern terms, Hilbert's tenth problem asks whether there exists an algorithm that decides if a given Diophantine equation has an integer solution. That is, whether there exists a Alan Turing machine that, given any Diophantine equation, determines in a finite number of steps whether it has a solution.
During the first half of the twentieth century, computability theory and mathematical logic were developed by mathematicians such as Alan Turing, Alfred Tarski, and Kurt Gödel, whose work allowed for a precise formalization of Hilbert's tenth problem and eventually led to a negative solution through the cumulative work of Martin Davis, Hilary Putnam, Julia Robinson, and Yuri Matiyasevich.
Since then, vast generalizations have been developed, revealing surprising connections with different areas of number theory. In this talk, we present a brief survey of these results and discuss some related conjectures.
Mar 30 Jonathan Vittore
Title: Number Theoretic Methods for Hyperbolic 3 - Manifolds
Abstract: Geodesics are an important concept in differential geometry. Generally speaking, it is possible to have closed geodesics, and naturally a question arises. Does the geodesic intersect itself before it completes its path? Such a geodesic is called non-simple. In this talk, we will explore a construction of real hyperbolic 3-manifolds all of whose closed geodesics are simple. Almost all of the techniques used are number theoretic in nature, utilizing quaternion algebras formed over number fields or (characteristic 0) number fields.
Mar 23 Noah Smith
Title: Locally Recoverable Codes: A Paradoxical Approach
Abstract: The theory of error correcting and erasure recovery codes plays a vital role in the modern technological age: every message sent, document saved, or Wi-Fi network joined is quietly made possible through the use of these codes. It turns out that by using techniques from number theory and arithmetic geometry, one can create codes with highly desirable properties. After a brief discussion of the general theory of codes, we will see how number theoretic and algebro-geometric ideas can be applied to this theory, particularly in the construction of locally recoverable codes (LRCs) with availability. We will conclude by presenting a construction of LRCs generated from an algebraic structure known as a paradoxical family which have predictable and high availability.
Mar 16 Matt Wicks
Title: A Brief Introduction to Iwasawa Theory
Abstract: In the late 19th century cyclotomic fields were studied deeply due to their relationship with Fermat’s Last Theorem. In the 1950’s, Kenkichi Iwasawa initiated his own study of cyclotomic fields armed with the recently developed tools of class field theory. In this process, he found himself studying infinite towers of fields and was naturally led to study what are now called /mathbb{Z}_p extensions. By looking at the whole tower all at once, Iwasawa was able to obtain precise results on class numbers and how they grow in these towers. His students, Bruce Ferrero, Ralph Greenberg, and Lawerence Washington honed these results further and the field known as Iwasawa Theory was born. In this talk we will discuss more of the history and background before stating three of the main theorems of classical Iwasawa theory.
Feb 23 Vishakh Vasu
Title: Fredholm coefficients of the Up-operator
Abstract: In the last semester, I discussed the characteristic power series of the Up operator acting on the space of overconvergent modular forms. In this talk, we compute the Fredholm coefficients in the case of integer weights and compare it with the Coleman's result for the overconvergent modular forms of arbitrary weight.
Feb 16 Xinyu Zhou (Boston University)
Title: Dualities between integral local Shimura varieties
Abstract: Generalizing earlier results of Faltings and Fargues on Lubin–Tate spaces and Drinfeld symmetric spaces, Scholze and Weinstein show in the "Berkeley lecture notes" that there is a "duality" between local Shimura varieties associated to a reductive group G and those associated to an inner twist G′, by proving that at the infinite level, they admit isomorphic moduli interpretations in terms of p-adic shtukas.
In this talk, I will review Scholze-Weinstein's approach and show how to generalize this duality result further to the integral models of local Shimura varieties. If time permits, we also discuss an application to cohomology of p-adic groups with some natural mod-p coefficients.
Feb 9 Haochen Cheng (Northwestern)
Title: Non-liftability of abelian schemes and beyond
Abstract: In the complex setting, the embedding of a subvariety into a Shimura variety is largely governed by its monodromy group. In particular, subvarieties with small monodromy groups do not occur inside Shimura varieties. This rigidity phenomenon breaks down in characteristic p: for instance, Moret–Bailly constructed a nontrivial family of supersingular abelian surfaces over the projective line. In this talk, we study the moduli of abelian varieties in characteristic p. We show that any subvariety with trivial l-adic monodromy group is necessarily contained in an isogeny leaf in the sense of Oort. Moreover, we prove that such subvarieties do not even admit a lift to W2(k) by using non-abelian Hodge theory in positive characteristic. Finally, I will discuss ongoing ideas toward extending these results to (exceptional) Shimura varieties, building on recent works of Patrikis, Bakker–Tsimerman–Shankar, et al.
Notes
Feb 2 Chapman Howard
Title: Parabolic Induction, Jacquet Modules, and Mackey Theory, oh my!
Abstract: The central object in the local Langland's Correspondence (for GLn) is the “irreducible admissible representation” (of GLn). LLC for GLn is now a theorem (Harris-Taylor, Henniart), so it makes sense to study the associated functorial lifts. Two examples are the base change induced from the inclusion GLn(F) ↪ GLn(E) for some E/F, and the Jacquet-Langlands transfer induced from the inclusion GLn(F) ↪ GLn(D) for some central division algebra D/F. In this talk, we’ll break down the tools that allow one to study these instances of functoriality, and maybe present an argument from our thesis project.
Jan 26 Napoleon Wang
Title: Vector bundles on Fargues-Fontaine curves
Abstract: The Fargues–Fontaine curve XFF is a p-adic analogy to the projective line ℙ1, constructed by Fargues-Fontaine in a way that p-adic Hodge theoretic objects can be expressed via vector bundles with additional structure at a distinguished point x∞. In analogy with the Beauville-Laszlo gluing description of Hecke modifications via the affine Grassmannian, modifications of G-bundles at x∞ are described by the B+dR-affine Grassmannian (Fargues-Scholze). By Fargues-Fontaine and Anschütz, vector bundles on XFF are classified and formulated as an equivalence with isocrystals. More recent work by Birkbeck et al. and Hong refines this by studying extensions, subbundles, and quotient bundles using the Harder-Narasimhan polygons. In the end, we will briefly discuss the relative versions of the Fargues-Fontaine curve over a perfectoid base following Kedlaya-Liu and Fargues-Scholze.
