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Andrew Kobin

Publications and source records attributed to Andrew Kobin.

11 recordsLinked to original sources
arXiv2026

On $p$-adic solubility of $Ax^\ell + By^m + Cz^n = 0$

We study $p$-adic solubility of generalized Fermat equations $Ax^\ell + By^m + Cz^n = 0$ for positive integers $\ell,m,n$. For all but finitely many primes $p$, the probability of having a $p$-adic solution is described by a rational function in $p$ depending only on $\gcd(p-1,\ell,m)$, $\gcd(p-1,\ell,n)$, and $\gcd(p-1,m,n)$. When $\ell,m,n$ are pairwise coprime, we deduce that the proportion of these equations which are everywhere locally soluble is positive, given by a product of these local probabilities; when $\ell,m,n$ are not pairwise coprime, the proportion is 0\%. We then give several detailed examples demonstrating the explicit nature of the results.

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arXiv2025

Wild Stacky Curves and Rings of Mod p Modular Forms

We extend work of Voight and the second author to compute the log canonical ring of a wild stacky curve over a field of characteristic $p > 0$, which allows us to compute rings of mod $p$ modular forms of level $\Gamma_{0}(N)$. Our approach also reveals that in characteristics $2$ and $3$, there are infinitely many levels $N$ for which there are weight $2$ modular forms of level $\Gamma_{0}(N)$ that do not lift to characteristic $0$.

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arXiv2025

Arithmetic Functions and Geometry

In this expository note, we revisit several classical arithmetic functions - namely Euler's totient function, the divisor sum functions and Dedekind's $\psi$-function - within a unifying algebraic framework that highlights their connections to geometry. This framework builds on prior work involving zeta functions and M\"obius inversion. While our main goal is to provide a clear context for similar constructions in the future, we also make an original observation regarding Dedekind's $\psi$-function.

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arXiv2023

Artin-Schreier-Witt Theory for Stacky Curves

We extend our previous classification of stacky curves in positive characteristic using higher ramification data and Artin-Schreier-Witt theory. The main new technical tool introduced is the Artin-Schreier-Witt root stack, a generalization of root stacks to the wildly ramified setting. We then apply our wild Riemann-Hurwitz theorem for stacks to compute the canonical rings of some wild stacky curves.

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arXiv2023

Categorifying Zeta Functions for Quadratic Covers

In various contexts, the zeta function of an object splits into a product of $L$-functions. We categorify this product formula for quadratic covers of objects in the following contexts: quadratic extensions of number fields, ramified double covers of algebraic curves, ramified double covers of topological spaces and Galois double covers of graphs. Our unified approach utilizes objective linear algebra in the abstract incidence algebra of each object, interpreted appropriately. We also provide several applications: for a hyperelliptic curve $C$ over a finite field, we prove a collection of combinatorial formulas relating the number of ramified, split and inert points on $C$ to the overall point count of $C$; and for a graph $G$, we deduce analogous combinatorial formulas for the numbers of split and inert primes in a Galois double cover $\widetilde{G}\rightarrow G$. We then use the formulas for graphs to deduce asymptotic counts of cycles in supersingular isogeny graphs and certain associated dual graphs of special fibers of Shimura curves. Finally, we analyze quadratic reciprocity from the perspective of zeta functions.

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arXiv2022

Categorifying quadratic zeta functions

The Dedekind zeta function of a quadratic number field factors as a product of the Riemann zeta function and the $L$-function of a quadratic Dirichlet character. We categorify this formula using objective linear algebra in the abstract incidence algebra of the division poset.

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arXiv2020

A Primer on Zeta Functions and Decomposition Spaces

Many examples of zeta functions in number theory and combinatorics are special cases of a construction in homotopy theory known as a decomposition space. This article aims to introduce number theorists to the relevant concepts in homotopy theory and lays some foundations for future applications of decomposition spaces in the theory of zeta functions.

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arXiv2019

$\mathbb{A}^{1}$-Local Degree via Stacks

We extend results of Kass--Wickelgren to define an Euler class for a non-orientable (or non-relatively orientable) vector bundle on a smooth scheme, valued in the Grothendieck--Witt group of the ground field. We use a root stack construction to produce this Euler class and discuss its relation to other versions of an Euler class in $\mathbb{A}^{1}$-homotopy theory. This allows one to apply Kass--Wickelgren's technique for arithmetic enrichments of enumerative geometry to a larger class of problems; as an example, we use our construction to give an arithmetic count of the number of lines meeting $6$ planes in $\mathbb{P}^4$.

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arXiv2019

Artin-Schreier Root Stacks

We classify stacky curves in characteristic $p > 0$ with cyclic stabilizers of order $p$ using higher ramification data. This approach replaces the local root stack structure of a tame stacky curve, similar to the local structure of a complex orbifold curve, with a more sensitive structure called an Artin-Schreier root stack, allowing us to incorporate this ramification data directly into the stack. As an application, we compute dimensions of Riemann-Roch spaces for some examples of stacky curves in positive characteristic and suggest a program for computing spaces of modular forms in this setting.

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arXiv2014

Crossing Number Bound in Knot Mosaics

Knot mosaics are used to model physical quantum states. The mosaic number of a knot is the smallest integer $m$ such that the knot can be represented as a knot $m$-mosaic. In this paper we establish an upper bound for the crossing number of a knot in terms of the mosaic number. Given an $m$-mosaic and any knot $K$ that is represented on the mosaic, its crossing number $c$ is bounded above by $(m - 2)^{2} - 2$ if $m$ is odd, and $(m - 2)^{2} - (m - 3)$ if $m$ is even.

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