VT Algebra Seminar — Spring 2026

Isogeny graphs of supersingular elliptic curves have broad application, from the study and computation of modular forms to post-quantum cryptography. This is in part because the family of q-isogeny graphs in characteristic p (with prime p varying, for a fixed prime q) is Ramanujan. One tool for studying a graph is its Ihara zeta function, defined as an Euler product over the primes of the graph. Defining the zeta function formally requires a graph in the sense of Serre and Bass, i.e. a directed graph equipped with a fixed-point free involution on the edge set. In general, isogeny graphs fail to be graphs in this sense. In this talk, I will discuss joint work with Lau, Orvis, Scullard, and Zobernig in which we introduce abstract isogeny graphs along with their zeta functions; these graphs capture the combinatorial structure of supersingular isogeny graphs (with level structure) . I will survey some of our results, including an analogue of Ihara’s determinant formula, showing in particular that the zeta function is rational. We use this formula and the Eichler-Shimura relation to give a formula relating the zeta function of a q-isogeny graph with level-H structure (for certain H, including B0(N) and B1(N)) to the Hasse-Weil zeta functions of two associated modular curves over the finite field Fq, generalizing results of Hashimoto, Sugiyama, and Lei-Muller.

As advocated by Aganagic and Okounkov, mirror symmetry in three dimensions admits an enumerative interpretation in terms of quasimap counts to mirror dual symplectic varieties. Specifically, the generating series of these counts, known as vertex functions, are expected to match, up to a distinguished class in elliptic cohomology known as the elliptic stable envelope. After reviewing this general picture, I will focus on the case of bow varieties, which capture mirror symmetry in affine type A, and discuss what we know and what we are beginning to understand. (joint with subsets of R. Rimanyi, H. Dinkins, and S. Tamagni).

The focus of this talk is on families of curves and their associated Sato-Tate groups - compact Lie groups predicted to determine the limiting distributions of coefficients of the normalized L-polynomials of the curves. Complete classifications of Sato-Tate groups for abelian varieties in low dimension have been given in recent years, but there are obstacles to providing classifications in higher dimension. In this talk, I will give an overview of techniques, results, and challenges in this area of research. Concrete examples will be provided throughout the talk.

A celebrated theorem of Merel states that for any fixed degree \( d \), there are only finitely many groups which can arise as the torsion subgroup of an elliptic curve over a number field of degree \( d \). Merel's theorem followed Mazur's classification of torsion subgroups which arise over the rational numbers (\( d = 1 \)), which has been preceded by classifications in degrees \( d = 2 \) and \( d = 3 \) (with a result in degree \( d = 4 \) recently announced). I will discuss joint work with Pete Clark which allows for a complete classification in any specified degree \( d \) if one restricts to torsion subgroups of elliptic curves with complex multiplication, coming from a study of isogeny volcanoes over \( \overline{\mathbb{Q}} \).

A celebrated result of Białynicki-Birula states that a smooth complete \( \mathbb{G}_m \)-variety with finite fixed locus admits a pair of pavings by so-called stable and unstable \( \mathbb{G}_m \)-subvarieties. Flag varieties and toric varieties provide a rich set of case studies. Inspired by these cases, Buch--Chaput--Perrin have in recent work shown that suitable structural constraints on these decompositions imply an equivariant cohomological rigidity property for the cell closures. I will discuss joint work with Teddy Gonzales in which we make a careful study of both sides of the "geometry \( \leftrightarrow \) cohomology" interplay. In particular, we show that the requisite geometric properties are really combinatorial and that the equivariant rigidity of Białynicki-Birula cells is essentially a feature of the theory. A number of concrete examples will be provided.

The endomorphism ring of a supersingular elliptic curve is a maximal order in the definite quaternion algebra over Q ramified at a prime p, call this quaternion algebra A. Over the past 20 years several results have come out giving methods to take a finite index subring R of a maximal order O in A and produce a particular maximal order containing R marked by some conditions. More recently it has come to light that the problem of generating non-scalar endomorphisms of a supersingular elliptic curve is equivalent to computing the endomorphism ring of a supersingular elliptic curve. In this talk we answer the related quantitative question, how many endomorphisms (of prime power order) does one need to compute a finite index subring of a marked maximal order O in A? Specifically, given a maximal order O in A and a prime l not dividing the reduced discriminant of O, we provide effective bounds on the maximal integer N such that O can be generated by elements of norm <= l^N. Along the way we will discuss structural properties of maximal orders in the quaternion algebra A, the Brandt module of a quaternion order, the relation between Brandt modules and Hecke operators, and how the transition from quaternion data to modular form data can yield arithmetic information about O.

Crystal techniques are a standard tool for proving Schur positivity. Can they also be used to study Demazure atom positivity, where the Demazure atoms are the nonsymmetric pieces of Schur polynomials? A substantial obstacle is that the Demazure atom subcrystals are typically not connected. In this talk, I will describe a crystal structure on multiline queues, which can be viewed as a graded KR crystal with fixed row sums. Multiline queues give combinatorial formulas for $q$-Whittaker polynomials (Macdonald polynomials at $t=0$) and their nonsymmetric pieces, the $t=0$ ASEP polynomials. I will explain how the crystal structure can be used to show the $t=0$ ASEP polynomials are Demazure atom positive. As a consequence, we obtain that the permuted-basement Macdonald polynomials are Demazure atom positive at $t=0$. This is joint work with Harper Niergarth and Kartik Singh.