VT Algebra Seminar — Fall 2022

An endomorphism of an elliptic curve satisfies a degree two charateristic polynomial with integer coefficients, and the degree-one coefficient is its trace. An algorithm for computing the trace of the Frobenius endomorphism of an elliptic curve defined over a finite field yields an algorithm for point counting on elliptic curves. Schoof gave the first such polynomial-time algorithm, which was later improved by Elkies and Atkin resulting in the celebrated SEA algorithm. When the elliptic curve is supersingular, computing the traces of its endomorphisms is a critical subroutine in algorithms for computing the structure of the endomorphism ring as a quaternion algebra. In this talk, we will give an exposition of Schoof's algorithm, a taste of Elkies' improvements, and discuss novel, asymptotic speedups to a generalization of Schoof's algorithm for computing the trace of an endomorphism of a supersingular elliptic cuve. In particular, our algorithm achieves a runtime which is quartic in its input size assuming only the generalized Riemann hypothesis -- without additional hypotheses, the SEA algorithm is no faster (asymptotically) than Schoof's algorithm, even under GRH. This is joint work with Jana Sotakova and Michael Wills.

In 1971, Dowling introduced a class of geometric lattices in connection with coding theory. The elements of these lattices are the ${𝔽}_q$-linear subspaces of ${𝔽}_q^n$ having a basis of vectors with Hamming weight bounded from above, ordered by inclusion. The theory of Dowling lattices and their extensions (called Higher-Weight Dowling Lattices, HWDL in short) have been brought forward by Bonin, Kung, and more recently by Ravagnani.

In this talk, we introduce the $q$-analogue of HWDLs, which we call rank-metric lattices (RML in short). Their elements are the ${𝔽}_{q^m}$-linear subspaces of ${𝔽}_{q^m}^n$ having a basis of vectors with rank weight bounded from above, ordered by inclusion. We determine which RMLs are supersolvable, computing their characteristic polynomials. We also investigate other structural properties of these lattices. In the second part of the talk, we establish a connection between RMLs and the problem of distinguishing between inequivalent rank-metric codes.

The new results in this talk are joint work with A. Ravagnani.

In the first part of this talk, we will consider the flag variety X=Fl(1,2,3) parametrizing pairs (p,l), where l is a line in the complex projective plane, and p is a point on l. In particular, we will discuss maps from the projective line to X satisfying natural constraints. Such maps come in families which are varieties themselves. Moreover, these varieties have arithmetic genus 0. This example fits into the broader study of the quantum K-theory of flag varieties. In the second part of the talk, I will give a gentle introduction to this subject and outline my past and current work on the quantum K-theory of the incidence variety Fl(1,n-1,n).

Givental and Lee introduced quantum K-theory, a K-theoretic generalization of Gromov--Witten theory. It studies holomorphic Euler characteristics of coherent sheaves on moduli spaces of stable maps to given target spaces. In this talk, I will introduce the quantum K-theory for orbifold target spaces which generalizes the work of Tonita-Tseng. In genus zero, I will define a quantum K-ring which specializes to the full orbifold K-ring introduced by Jarvis-Kaufmann-Kimura. As an application, I will give a detailed description of the quantum K-ring of weighted projective spaces, which generalizes a result by Goldin-Harada-Holm-Kimura. This talk is based on joint work in progress with Yang Zhou.

Key establishment is a cryptographic primitive which allows two users to establish a shared secret piece of information (the “key”) in the presence of eavesdroppers. I will discuss key establishment from a classical perspective, discuss the need for post-quantum key establishment, and introduce the isogeny-based key establishment protocols SIDH (which was recently broken) and CSIDH from a high level. Then I will discuss algorithmic aspects of these protocols, and develop a systematic approach to optimizing algorithms to compute the isogenies required in SIDH and CSIDH. Finally, I’ll present the results of these optimizations and discuss some open questions in the area.

An elliptic curve is a curve in projective space whose points can be given the structure of an abelian group. In this talk, we will focus on torsion points, which are points having finite order under this group law. While we can generally determine the torsion points of a fixed elliptic curve defined over a number field, there are several open problems which require controlling the existence of torsion points within infinite families of elliptic curves. Success stories include Merel's Uniform Boundedness Theorem, which states that the order of a torsion point can be bounded by the degree of its field of definition. On the other hand, a proof of Serre's Uniformity Conjecture---which has been open for 50 years---would in particular imply that for sufficiently large primes $p$, there do not exist points of order $p^2$ arising on elliptic curves defined over field extensions of ``unusually low degree." In this talk, I will give a brief introduction to the arithmetic of elliptic curves before addressing the problem of identifying elliptic curves producing a point of large order in usually low degree, i.e., those possessing a sporadic torsion point. More precisely, let $E$ be an elliptic curve defined over a field extension $F/\mathbb{Q}$ of degree $d$, and let $P$ be a point of order $N$ with coordinates in $F$. Such a point is called ``rational" since it is defined over the same field as $E$. We say $P$ is sporadic if, as one ranges over all fields $F/\mathbb{Q}$ of degree at most $d$ and all elliptic curves $E/F$, there are only finitely many elliptic curves which possess a rational point of order $N$. Sporadic pairs $(E,P)$ correspond to exceptional points on modular curves, which are points whose existence is not explained by standard geometric constructions.

Quot schemes parameterize subsheaves of a fixed vector bundle on a smooth projective curve C. In the first part of the talk, I will present formulas for holomorphic Euler characteristics of tautological bundles over Quot schemes (based on joint work with Prof. Dragos Oprea). We note a surprising analogy with the Hilbert scheme of points on surfaces. In the second part of the talk, I will describe the celebrated Vafa-Intriligator formula in the context of Quot schemes, and I will report on a Vafa-Intriligator type formula for (rank 2) isotropic Quot schemes. Furthermore, I will compare this formula with Gromov-Ruan-Witten invariants of (rank 2) isotropic Grassmannians.

One of the most important results in coding theory is the MacWilliams Theorem (1962), which is known as the "MacWilliams identities", and relates the weight enumerator of a linear code and the weight enumerator of its dual code. Coding theory has also been developed with respect to alternative metrics. One of the most studied of those metrics is the NRT metric, which was introduced to study array codes. The NRT metric was introduced by Rosenbloom and Tsfasman in [1]. In this same paper the authors pointed out that the NRT metric models transmission over a set of parallel channels subject to fading. In this talk, we will see how we can apply polynomial invariant theory to describe the shape enumer- ator of self-dual NRT metric codes in Mn,2(F2). The new results in this talk are in [2] References [1] M, Yu. Rosenbloom, M. A. Tsfasman, Codes for the m-metric, Problemy Peredachi Informatsii, Vol. 33, 55-63 (Russian), 1997. English translation in Problems infrom. transmission vol. 33, pp. 45-52, 1997. [2] W. Santos and M. M. Alves, Polynomial Invariant Theory and Shape Enumerator of Self-Dual Codes in the NRT-Metric, in IEEE Transactions on Information Theory, vol. 66, no. 7, pp. 4061- 4074, July 2020.

Francisco, Mermin, and Schweig introduced the concept of Q-Borel ideals as a generalization of Borel ideals (also called strongly stable ideals). Using a poset Q and a monomial m, we can build a large monomial ideal and many of its properties can be read by just looking at the pair (Q,m). In this talk, we determine some of these properties, in particular the symbolic powers of principal Q-Borel ideals.

I will investigate certain matrix enumeration problems over a finite field, guided by the phenomenon that many such problems tend to have a generating function with a nice factorization. I then give a uniform and geometric explanation of the phenomenon that works in many cases, using the statistics of finite-length modules (or coherent sheaves) studied by Cohen and Lenstra. However, my recent work on counting pairs of matrices of the form AB=BA=0 (arXiv: 2110.15566) and AB=uBA for a root of unity u (arXiv: 2110.15570), through purely combinatorial methods, gives examples where the phenomenon still holds true in the absence of the above explanation. Time permitting, I will talk about a partial progress on the system of equations AB=BA, A^2=B^3 in a joint work with Ruofan Jiang. In particular, it verifies a pattern that I previously conjectured in an attempt to explain the phenomenon in the AB=BA=0 case geometrically.

TBA

We will begin the talk by describing a central construction in the arithmetic of characteristic-p function fields: Drinfeld modules and t-motives. We will then describe our recent result which allows us to make some progress on an important deficiency in the area, the lack of integration theory. We will describe two pairings between the t-motive and the dual t-motive associated to a t-module which provide an algebraic alternative to integration. Specializations of these pairings allow us to give explicit formulas for the exponential and logarithmic functions of the t-module. We also explain a few applications of these pairings, such as a Mellin transform-like formula for Carlitz zeta values.