25Fa UArizona Grad Algebra & Number Theory Seminar

In fall 2025, the UA Grad ANT Seminar is organized by Nick Pilotti. This website is maintained by Napoleon Wang.

Previous Talks

Dec 3 Nick Pilotti

Title: Siegel-Weil formulas for similitude groups

Abstract: In Siegel's work on quadratic forms, he proved an identity relating an Eisenstein series to a weighted average of theta functions. In the 1960s, Weil vastly generalized Siegel's formula in the modern framework of automorphic forms. Rallis later initiated a program to extend Weil's formula, which he observed could be used as a key ingredient in proving certain facts about L-function for the classical groups. In this talk, I will discuss the Siegel-Weil formulas and demonstrate how a Siegel-Weil formula for similitude groups can be deduced from the case of isometries. This is practice for my comprehensive exam talk.

Nov 19 Vishakh Vasu

Title: On the characteristic power series of the Up operator

Abstract: The Up operator acts on the space of overconvergent modular forms, and it has a spectral theory. We study the characteristic power series of this operator, and, as conjectured by Mazur, the coefficients of this series happen to be an Iwasawa function. This result was proven by Coleman for all weights of overconvergent modular forms using rigid analytic geometry. We will discuss this result in much simpler setting for the classical (integer) weight case using the Hijikata's trace formula for Hecke operators.

Nov 12 Tyler Kline

Title: Newton Above Hodge: Polygons of L-functions of Exponential Sums

Abstract: By a result of Dwork, Bombieri, and Grothendieck, an L-function of exponential sums is a rational function. In 1989, Adolphson and Sperber showed that under some nondegeneracy assumption this rational function is a polynomial of known degree. But what is this polynomial? One approach to this problem is to look at the Newton Polygon of the L-function. In this talk we’ll talk about what this object is and what is known about them.

Nov 5 Sandra Nair (Colorado State University)

Title: Ekedahl-Oort, Newton and BT_m stratifications on certain unitary Shimura varieties

Abstract: Unitary Shimura varieties are moduli spaces of abelian varieties in characteristic p with certain extra structures, including CM by an imaginary quadratic field. A fruitful way to understand them is by stratifying these spaces. We first focus on the Ekedahl-Oort stratification (based on p-torsion group schemes up to isomorphism) and the Newton stratification (based on p-divisible groups up to isogeny). We study how they interact, and give  a complete classificaton for the case of signature (3,2). In the second half, we explore  the concept of BTm stratification as a refinement of Newton stratification, of which m=1 recovers Ekedahl-Oort stratification. We establish some numerical bounds for m using the signature of the unitary Shimura variety, following techniques first introduced by Traverso. This is joint work with Emerald Andrews, Deewang Bhamidipati, Maria Fox, Heidi Goodson, Steven R. Groen.

Oct 29 Amin Soofiani (University of British Columbia)

Title: Local-Global Principle for the Norm Principle over Function Fields of Curves

Abstract: Let G be a linear algebraic group defined over a field K. The Norm Principle for G concerns how the base change of G to a finite separable field extension L/K interacts with the norm map associated to L/K. There are two formulations of the norm principle: one in terms of rational points on algebraic groups (due to Merkurjev), and the other in terms of principal homogeneous spaces (due to Gille).

The norm principle remains open in general and is closely related to several other important problems in the theory of linear algebraic

groups, including local-global principles, Serre’s injectivity question, and R-equivalence.

In this talk, we will discuss the norm principle for semisimple groups defined over complete discretely valued fields and function fields, focusing in particular on groups of type Dn.

Oct 22 Xinran Qian

Title: The Selberg trace formula for GL(2)

Abstract: In 1956, the Norwegian mathematician Atle Selberg proposed a vast generalization of the Poisson summation formula, now referred to as the Selberg trace formula. The trace formula and its generalizations have played an increasingly central role in the ambitious program of Robert Langlands. In my talk, I will discuss the trace formula for GL(2) and its application to the study of automorphic forms.

Oct 8 Xinran Qian

Title: Arithmetic theta lifts and the Arithmetic Gan-Gross-Prasad conjecture for SO(3)×SO(4)

Abstract: In an MSRI lecture in 2001, Benedict Gross compared the central value formula of Waldspurger with the central derivative formula of Gross-Zagier in the framework of representation theory. In this framework, the Waldspurger formula concerns the toric periods of automorphic forms on quaternion algebras, while the Gross-Zagier formula may be viewed as a formula for the "periods" of "automorphic forms" on the incoherent quaternion algebras. 

To generalize the work of Waldspurger, Gross and Dipendra Prasad formulated a conjecture relating the central value of certain Rankin-Selberg L-function to SO(n) periods of automorphic forms on SO(n)×SO(n+1) with Waldspurger formula being the case n=2. Their conjecture was further generalized to include all classical groups in the 2012 book of Gan-Gross-Prasad, hence the name GGP conjecture. Parallel to the periods of automorphic forms, there is a conjectural generalization of the Gross-Zagier formula to higher-dimensional Shimura varieties, known as the arithmetic GGP conjecture. In my talk, I will present a case of the conjecture for SO(3)×SO(4), following the work of Hang Xue in the case U(2)×U(3).

Oct 1 Vivien Picard (University of Wuppertal)

Title: Logarithmic Hodge numbers and weakly ordinary varieties

Abstract

Sep 24 Sean Zhu

Title: Symmetric Power L-Functions

Abstract: We follow up Tyler’s talk on L-functions, Newton Polytopes, with another method of studying L-Functions with Symmetric Power L-Functions.

Sep 17 Jake Huryn (Ohio State University)

Title: Geometric properties of the "tautological" local systems on Shimura varieties

Abstract: Some Shimura varieties are moduli spaces of Abelian varieties with extra structure. The Tate module of a universal Abelian variety is a natural source of ℓ-adic local systems on such Shimura varieties. Remarkably, the theory allows one to build these local systems intrinsically from the Shimura variety in an essentially tautological way, and this construction can be carried out in exactly the same way for Shimura varieties whose moduli interpretation remains conjectural.

This suggests the following program: Show that these tautological local systems "look as if" they were arising from the cohomology of geometric objects. In this talk, I will describe some recent progress. It is based on joint work with Kiran Kedlaya, Christian Klevdal, and Stefan Patrikis, as well as joint work with Yifei Zhang.

Sep 10 Tyler Kline

Title: A Vanishing Result for p-adic Homology of Exponential Sums

Abstract: Exponential sums are a powerful tool in number theory. One of the approaches to working with these objects is to pack them into a generating function called an L-function and see what we can do. Over finite fields, you can produce a homological description of the L-function. We are going to explore a result that describes the circumstances under which this homology vanishes, in which case the L-function becomes very simple: essentially a polynomial.

This is the alpha version of the talk I intend to give for my comprehensive exam, so I would really appreciate feedback from everyone as well!

Sep 5 Ashley Roberts 

Title: Sparse families of lattices

Abstract: Lattices are algebraic structures with numerous applications, from number theory and coding theory to physics and cryptography. Extremal lattices, which have maximally long shortest vectors, are rare and difficult to classify. In this talk, I'll introduce a relaxed notion called sparse lattices, and explore a natural candidate for such a family, the Barnes-Wall lattices. Along the way, we will explore various different ways to construct lattices from codes, and their applications to my research.