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John Boxall

Publications and source records attributed to John Boxall.

5 recordsLinked to original sources
arXiv2022

Bounds on the number of torsion points of given order on curves embedded in their jacobians

Working over an algebraically closed field of arbitrary characteristic we study, for integers $N\geq 2$ and $g\geq 2$, the set of points of order dividing $N$ lying on an irreducible smooth proper curve of genus $g$ embedded in its jacobian using a fixed base point. We discuss bounds on its cardinality and describe an efficient method for computing the set. Our method uses wronskians similar to those used in the study of Weierstrass points and the strength of our bounds is related to whether or not a certain multiple of the curve is contained in the negative of the theta divisor. Several examples are discussed. This generalizes our previous work [https://doi.org/10.1216/rmj.2023.53.357] dealing with the case of hyperelliptic curves embedded in their jacobian using a Weierstrass point as base point.

math.AG↗
arXiv2020

On the number of point of given order on odd degree hyperelliptic curves

For integers $N\geq 3$ and $g\geq 1$, we study bounds on the cardinality of the set of points of order dividing $N$ lying on a hyperelliptic curve of genus $g$ embedded in its jacobian using a Weierstrass point as base point. This leads us to revisit division polynomials introduced by Cantor in 1995 and strengthen a divisibility result proved by him. Several examples are discussed.

math.NT↗
arXiv2017

Croissance asymptotique de nombres de Weil appartenant \`a un corps de nombres fix\'e

We prove an asymptotic formula as $x\to +\infty$ for the number of algebraic integers $\alpha$ belonging to a fixed CM number field and satisfying $\alpha\overline{\alpha}\leq x$. This problem is related to the height zeta function $Z_h(X^K,s)$ associated to the anticanonical class of a certain toric variety $X^K$ over $\mathbb{Q}$ and we show that $Z_h(X^K,s)$ has a meromorphic continuation to the half-plane $\{\Re(s)>\frac{1}{2}\}$ where it is holomorphic except at $s=1$. Along the way we obtain a new proof of Manin's conjecture on the asymptotic growth of points on $X^K(\mathbb{Q})$ of bounded height.

math.NT↗
arXiv2011

Heuristics on pairing-friendly elliptic curves

We present a heuristic asymptotic formula as $x\to \infty$ for the number of isogeny classes of pairing-friendly elliptic curves with fixed embedding degree $k\geq 3$, with fixed discriminant, with rho-value bounded by a fixed $\rho_0$ such that $1<\rho_0<2$, and with prime subgroup order at most $x$.

math.NT↗