arXiv subjects

Jonathan R. Love

Publications and source records attributed to Jonathan R. Love.

4 recordsLinked to original sources

arXiv2025

Supersingular elliptic curves, quaternion algebras and applications to cryptography

This paper contains a survey of supersingular isogeny graphs associated to supersingular elliptic curves and their various applications to cryptography. Within limitation of space, we attempt to address a broad audience and make this part widely accessible. For those graphs we also present three recent results and sketch their proofs. We then discuss a generalization to superspecial isogeny graphs associated to superspecial abelian varieties with real multiplication. These graphs were introduced by Charles, Goren and Lauter and so our discussion is brief. Motivated by their cryptographic applications, we prove a general theorem concerning generation of lattices over totally real fields by elements of specified norm. Throughout the paper we have attempted to clarify certain considerations that are either vaguely stated in the literature, or are folklore. We hope this paper will be useful both to a novice wishing to familiarize themselves with this very active area, and to the expert who may enjoy some vignettes and an overview of some new results.

math.NT↗

arXiv2023

Rational configuration problems and a family of curves

Given $η=\begin{pmatrix} a&b\\c&d \end{pmatrix}\in \text{GL}_2(\mathbb{Q})$, we consider the number of rational points on the genus one curve \[H_η:y^2=(a(1-x^2)+b(2x))^2+(c(1-x^2)+d(2x))^2.\] We prove that the set of $η$ for which $H_η(\mathbb{Q})\neq\emptyset$ has density zero, and that if a rational point $(x_0,y_0)\in H_η(\mathbb{Q})$ exists, then $H_η(\mathbb{Q})$ is infinite unless a certain explicit polynomial in $a,b,c,d,x_0,y_0$ vanishes. Curves of the form $H_η$ naturally occur in the study of configurations of points in $\mathbb{R}^n$ with rational distances between them. As one example demonstrating this framework, we prove that if a line through the origin in $\mathbb{R}^2$ passes through a rational point on the unit circle, then it contains a dense set of points $P$ such that the distances from $P$ to each of the three points $(0,0)$, $(0,1)$, and $(1,1)$ are all rational. We also prove some results regarding whether a rational number can be expressed as a sum or product of slopes of rational right triangles.

math.NT↗

arXiv2023

Hyperelliptic curves mapping to abelian varieties and applications to Beilinson's conjecture for zero-cycles

Let $A$ be an abelian surface over an algebraically closed field $\overline{k}$ with an embedding $\overline{k}\hookrightarrow\mathbb{C}$. When $A$ is isogenous to a product of elliptic curves, we describe a large collection of pairwise non-isomorphic hyperelliptic curves mapping birationally into $A$. For infinitely many integers $g\geq 2$, this collection has infinitely many curves of genus $g$, and no two curves in the collection have the same image under any isogeny from $A$. Using these hyperelliptic curves, we find many rational equivalences in the Chow group of zero-cycles $\text{CH}_0(A)$. We use these results to give some progress towards Beilinson's conjecture for zero-cycles, which predicts that for a smooth projective variety $X$ over $\overline{\mathbb{Q}}$ the kernel of the Albanese map of $X$ is zero.

math.AG↗

arXiv2023

On elements of prescribed norm in maximal orders of a quaternion algebra

Let $\mathcal{O}$ be a maximal order in the quaternion algebra over $\mathbb{Q}$ ramified at $p$ and $\infty$. We prove two theorems that allow us to recover the structure of $\mathcal{O}$ from limited information. The first says that for any infinite set $S$ of integers coprime to $p$, $\mathcal{O}$ is spanned as a $\mathbb{Z}$-module by elements with norm in $S$. The second says that $\mathcal{O}$ is determined up to isomorphism by its theta function.

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