UTRGV Algebra & Number Theory Seminar

The Algebra & Number Theory Seminar meets on Wednesdays at 3:00pm (with some exceptions) in EMAGC 3.502 during Fall '24. Talks will be given in a hybrid format. If you would like to give a talk in the seminar please contact Debanjana Kundu or Luigi Ferraro.

Raul Marquez

University of Texas Rio Grande Valley

Upper Bounds on the Lowest First Zero In Families of Cuspidal New Forms

Assuming the Generalized Riemann Hypothesis, the non-trivial zeros of $L$-functions lie on the critical line with the real part $1/2$. We find an upper bound of the lowest first zero in families of even cuspidal newforms of prime level tending to infinity. We obtain explicit bounds using the $n$-level densities and results towards the Katz-Sarnak density conjecture. We prove that as the level tends to infinity, there is at least one form with a normalized zero within $0.218503$ of the average spacing. We also obtain the first-ever bounds on the percentage of forms in these families with a fixed number of zeros within a small distance near the central point.

Emily Payne

University of Texas Rio Grande Valley

Generalized Partition Identities and Fixed Perimeter Analogues

Euler's classic partition identity states that the number of partitions of $n$ into odd parts equals the number of partitions of $n$ into distinct parts. We develop a new generalization of this identity and prove an accompanying Beck-type companion identity. Strikingly, in 2016, Straub proved that Euler's identity holds true for partitions with largest hook (perimeter) $n$. This inspired further study of the relationship between classical partitions and fixed perimeter partitions. Motivated by recent findings in this area, we develop fixed perimeter analogues of several standard partition results as well as a Beck-type identity in the fixed perimeter setting. We also use combinatorial methods to prove analogues of various results related to parity in the fixed perimeter setting.

Debanjana Kundu

University of Texas Rio Grande Valley

On the $p$-ranks of class groups of certain Galois extensions

Let $p$ be an odd prime, let $N$ be a prime with $N \equiv 1 \pmod{p}$, and let $\zeta_p$ be a primitive $p$-th root of unity. We discuss the variation of the $p$-rank of the class group of $\mathbb{Q}(\zeta_p, N^{1/p})$ in two ways. Using Galois cohomological methods we obtain an exact formula for the $p$-rank in terms of the dimensions of certain Selmer groups. Our formula provides a numerical criterion to establish upper and lower bounds for the $p$-rank. When $p=3$, we use simpler techniques (like Redei matrices) to provide a numerical criterion to exactly calculate the $3$-rank, and also study the distribution of the $3$-ranks as $N$ varies through primes which are $4,7 \pmod{9}$.

Brandt Kronholm

University of Texas Rio Grande Valley

Formulas for integer partition functions and the usefulness of a forgotten technique

The function $p(n,m,N)$ enumerates the partitions of $n$ into at most $m$ parts, none of which are larger than $N$. The function $p(n,m)$ indicates the number of partitions of $n$ into at most $m$ parts. In this talk we will establish a host of results including the following:
1) $p(2N-2,4,N)=p(2N-1,4,N)=p(2N+1,4,N)=p(2N+2,4,N)$.
2) $p(60k,5,24k)-p(60k-1,5,24k)=p(6k,3)$.
3) The series expansion of $\displaystyle \frac{1-q+q^5-2q^6+2q^7-q^8+q^{12}-q^{13}}{(1-q)\left(1-q^4\right) \left(1-q^6\right) \left(1-q^8\right)}$ has only non-negative terms.
These results are unified by a common technique in $q$-series noted in a single paper by Hansraj Gupta in the 1970s, but otherwise forgotten except to the field of combinatorial geometry. One of the goals of this presentation is to highlight the usefulness of this technique. We will conclude this talk with at least one conjecture.

Joselyne Aniceto

University of Texas Rio Grande Valley

Congruences in arithmetic progression for coefficients of Gaussian polynomials and crank statistics

We are motivated by a 2007 result of Kronholm, referred to as the Interval Theorem, which establishes infinite families of consecutive partition congruences in arithmetic progression for the restricted partition function $p(n,m)$, which enumerates the partitions of $n$ into at most $m$ parts. A recent result of Eichhorn, Engle, and Kronholm achieves a similar infinite family of partition congruences for $p(n,m,N)$, the function enumerating partitions of $n$ into at most $m$ parts, no part larger than $N$. However, this result is narrow in scope and empirically there are many congruences for $p(n,m,N)$ that are not accounted for. Our main result is much more comprehensive and encompasses parts of the Interval Theorem and the results of Eichhorn, Engle, and Kronholm as special cases. In order to prove this, we extend the techniques used to achieve those previous results. We begin by establishing infinite families of congruences for a two-colored partition function, and in conjunction with the generating functions for $p(n,m,N)$, establish our main result. We conclude by exploring combinatorial witnesses, known as cranks, for the congruences described in our main result.

Luigi Ferraro

University of Texas Rio Grande Valley

Quasi-projective dimension with respect to semidualizing modules

Let $R$ be a commutative Noetherian ring. Gheibi, Jorgensen, and Takahashi recently introduced the quasi-projective dimension of a module, a new homological invariant that generalizes projective dimension. In this talk, we explore the concept of $C$-quasi-projective dimension of a module, where $C$ is a semidualizing module. When $C = R$ this recovers the original quasi-projective dimension. This homological invariant also generalizes Takahashi and White’s
$C$-projective dimension. We establish several results, including versions of the Auslander-Buchsbaum formula, of the depth formula, of the Auslander-Reiten conjecture, and of Jorgensen’s dependency formula.

Melissa Emory

Oklahoma State University

On determining Sato-Tate groups

The original Sato-Tate conjecture was posed around 1960 by Mikio Sato and John Tate (independently) and is a statistical conjecture regarding the distribution of the normalized traces of Frobenius on an elliptic curve. In 2012, the conjecture was generalized to higher genus curves by Serre. In recent years classifications of Sato-Tate groups in dimensions 1, 2, and 3 have been given, but there are obstacles to providing classifications in higher dimension. In this talk, I will describe work to prove nondegeneracy and determine Sato-Tate groups for two families of Jacobian varieties. This work is joint with Heidi Goodson.

Dave Jorgensen

University of Texas Arlington

A construction of Shamash revisited

Avramov and Iyengar have recently proven that if $f$ and $g$ and  are regular elements in an ideal $I$ in a commutative local ring $(P,n,k)$ such that $f-g\in nI$, then for any $P/I$-module $M$, $\mathrm{Tor}_i^{P/(f)}(M,k)\cong\mathrm{Tor}_i^{P/(g)}(M,k)$ for all $i\ge 0$. We give an alternate proof of this fact using an embellishment of a construction of Shamash that yields a $P/(f)$-free resolution of $M$ from a $P$-free resolution of $M$. This is part of some joint work with Petter Bergh.

Lea Beneish

University of North Texas

Degree $d$ points on curves

Given a plane curve $C$ defined over $\mathbb{Q}$, when the genus of the curve is greater than one, Faltings' theorem tells us that the set of rational points on the curve is finite. It is then natural to consider higher degree points, that is, points on this curve defined over fields of degree $d$ over $\mathbb{Q}$. We ask for which natural numbers $d$ are there points on the curve in a field of degree $d$. For positive proportions of certain families of curves, we give results about which degrees of points do not occur. This talk is based on joint work with Andrew Granville.

Lars Christensen

Texas Tech University

Acyclic complexes over Gorenstein rings

Recall that a commutative noetherian ring $R$ is Gorenstein if and only if every finitely generated $R$-module has finite Gorenstein projective dimension. In papers from 2011 (Murfet and Salarian), 2017 (Estrada, Fu, and Iacob), and 2018 (Christensen and Kato) Gorenstein rings are characterized in terms of qualitative properties of their complexes, but always under some additional assumption on $R$ such as finite Krull dimension or finite projective dimension of flat $R$-modules. These assumptions are, we now know, superfluous, and I will sketch a proof of the equivalence of the following conditions:
1) $R$ is Gorenstein.
2) Every acyclic complex of injective $R$-modules is totally acyclic.
3) Every acyclic complex of flat $R$-modules is totally acyclic.
4) Every acyclic complex of projective $R$-modules is totally acyclic.
5) For every acyclic complex $P$ of projective $R$-modules the complex
    $\mathrm{Hom}_R(P,R)$ is acyclic.
6) For every acyclic complex $P$ of projective $R$-modules and every injective
    $R$-module $E$ the complex $E \otimes_R P$ is acyclic.

Rusiru Gambheera

University of California Santa Barbara

Equivariant Iwasawa Theory for the Ritter-Weiss Module and Applications

We will discuss a new equivariant main conjecture in Iwasawa theory associated with the cyclotomic $\mathbb{Z}_p$-extension of a CM number field over a totally real number field. Our object of interest $\nabla_T^S(H_\infty)_p$ is the projective limit of certain $p$-adic Ritter-Weiss modules which is class field theoretically significant and has nice cohomological properties. Our main result is a number field analogue of the recent result of Bley and Popescu on a certain Drinfeld modular Iwasawa tower of function fields. As an application, we compute the 0-th Fitting ideal of a naturally arising Iwasawa module over the relevant equivariant Iwasawa algebra. This is joint work with Cristian Popescu.

Andrew Soto Levins

Texas Tech University

The Euler Form

Using homological algebra, Serre defined and studied an intersection multiplicity for finitely generated modules over a regular local ring. He did this by using the Euler characteristic, and showed that it satisfied many properties one would expect with an intersection theory. The goal of this talk is to give an introduction to the partial Euler characteristics and share a result about an Ext analog of the partial Euler characteristics.