arXiv subjects

Thomas Jaklitsch

Publications and source records attributed to Thomas Jaklitsch.

4 recordsLinked to original sources

arXiv2025

A Filtration of the Chow Group of Zero-Cycles for a Product of Curves and an Abelian Variety

In this paper we define a descending filtration on the Chow group of zero cycles for varieties of the form $A \times C_1 \times \cdots \times C_d$ where $A$ is an abelian variety and each $C_i$ is a smooth projective curve. We give explicit generators and relations for the successive quotients of this filtration by showing that they can be described by Somekawa K-groups. This extends the work of Raskind and Spiess who proved this result for products of curves and Gazaki who proved this for abelian varieties.

math.AG↗

arXiv2023

On Certain McKay Numbers of Symmetric Groups

For primes $\ell$ and nonnegative integers $a$, we study the partition functions $$p_\ell(a;n):= \#\{λ\vdash n : \text{ord}_\ell(H(λ))=a\},$$ where $H(λ)$ denotes the product of hook lengths of a partition $λ$. These partition values arise as the McKay numbers $m_\ell(\text{ord}_\ell(n!) - a; S_n)$ in the representation theory of the symmetric group. We determine the generating functions for $p_\ell(a;n)$ in terms of $p_\ell(0;n)$ and specializations of specific D'Arcais polynomials. For $\ell = 2$ and $3$, we give an exact formula for the $p_\ell(a;n)$ and prove that these values are zero for almost all $n$. For larger primes $\ell$, the $p_\ell(a;n)$ are positive for sufficiently large $n$. Despite this positivity, we prove that $p_\ell(a;n)$ is almost always divisible by $m$ for any integer $m$. Furthermore, with these results we prove several Ramanujan-type congruences. These include the congruences $$p_\ell(a;\ell^k n - δ(a,\ell)) \equiv 0 \pmod{\ell^{k+1}},$$ for $0<a< \ell$, where $\ell = 5, 7, 11$ and $δ(a,\ell) := (\ell^2 - 1)/24 + a\ell$, which answer a question of Ono.

math.NT↗

arXiv2022

Class Numbers and Pell's Equation $x^2 + 105y^2 = z^2$

Two well-studied Diophantine equations are those of Pythagorean triples and elliptic curves, for the first we have a parametrization through rational points on the unit circle, and for the second we have a structure theorem for the group of rational solutions. Recently, Yekutieli discussed a connection between these two problems, and described the group structure of Pythagorean triples and the number of triples for a given hypotenuse. In arXiv:2112.03663 we generalized these methods and results to Pell's equation. We find a similar group structure and count on the number of solutions for a given $z$ to $x^2 + Dy^2 = z^2$ when $D$ is 1 or 2 modulo 4 and the class group of $\mathbb{Q}[\sqrt{-D}]$ is a free $\mathbb{Z}_2$ module, which always happens if the class number is at most 2. In this paper, we discuss the main results of arXiv:2112.03663 using some concrete examples in the case of $D=105$.

math.NT↗

arXiv2021

Connections of Class Numbers to the Group Structure of Generalized Pythagorean Triples

Two well-studied Diophantine equations are those of Pythagorean triples and elliptic curves; for the first, we have a parametrization through rational points on the unit circle, and for the second we have a structure theorem for the group of rational solutions. Recently Yekutieli discussed a connection between these two problems and described the group structure of Pythagorean triples and the number of triples for a given hypotenuse. We generalize these methods and results to Pell's equation. We find a similar group structure and count on the number of solutions for a given $z$ to $x^2 + Dy^2 = z^2$ when $D$ is 1 or 2 modulo 4 and the class group of $\mathbb{Q}[\sqrt{-D}]$ is a free $\mathbb{Z}_2$ module, which always happens if the class number is at most 2. We give examples of when the results hold for a class number greater than 2, as well as an example with different behavior when the class group does not have this structure.

math.NT↗