arXiv subjects
Publications and source records attributed to José Felipe Voloch.
We propose a conjecture on how to recover an algebraic curve from the isogeny classes of its Prym varieties. Motivated by this conjecture, we analyze constructions of pairs of curves with isogenous Jacobians and several isogenous Pryms.
We improve a bound due to the second author on number of rational points on smooth surfaces in $\mathbb{P}^3$ over finite fields and look at families of surfaces that achieve or nearly achieve this bound, for which we compute their exact number of rational points. These computations may have independent interest.
Let $C$ and $C'$ be curves over a finite field $K$, provided with embeddings $ι$ and $ι'$ into their Jacobian varieties. Let $D\to C$ and $D'\to C'$ be the pullbacks (via these embeddings) of the multiplication-by-$2$ maps on the Jacobians. We say that $(C,ι)$ and $(C',ι')$ are \emph{doubly isogenous} if $\mathrm{Jac}(C)$ and $\mathrm{Jac}(C')$ are isogenous over $K$ and $\mathrm{Jac}(D)$ and $\mathrm{Jac}(D')$ are isogenous over~$K$. When we restrict attention to the case where $C$ and $C'$ are curves of genus $2$ whose groups of $K$-rational automorphisms are isomorphic to the dihedral group $D_6$ of order $12$, we find many more doubly isogenous pairs than one would expect from reasonable heuristics. Our analysis of this overabundance of doubly isogenous curves over finite fields leads to the construction of a pair of doubly isogenous curves over a number field. That such a global example exists seems extremely surprising. We show that the Zilber--Pink conjecture implies that there can only be finitely many such examples. When we exclude reductions of this pair of global curves in our counts, we find that the data for the remaining curves is consistent with our original heuristic. Computationally, we find that doubly isogenous curves in our family of $D_6$ curves can be distinguished from one another by considering the isogeny classes of the Prym varieties of certain unramified covers of exponent $3$ and $4$. We discuss how our family of curves can be potentially be used to obtain a deterministic polynomial-time algorithm to factor univariate polynomials over finite fields via an argument of Kayal and Poonen.
Capsets are subsets of $\mathbb{F}_3^n$ with no three points on a line and a capset is complete if it is not a subset of a larger capset. We study some new constructions of capsets via algebraic equations over extensions of $\mathbb{F}_3$. In particular we construct the smallest known complete capsets with size proportional to the best known lower bound.
We describe a new construction of a subset of P^4 with no four points on a plane over any finite field of order q in which 3 is not a square. This set has size 2q + 1, is maximal with respect to inclusion, and is the largest known such set.
In a paper of Tate and the author, we conjectured a uniform bound for the p-adic distance of torsion points on a semiabelian variety, not lying in a subvariety, to that subvariety. We survey the progress made on that conjecture and on similar statements in analogous situations.
Consider a Jacobian elliptic surface $E \to C$ with a section $P$ of infinite order. Previous work of the first author and Urzúa over the complex numbers gives a bound on the number of tangencies between $P$ and a torsion section of $E$ (an ``unlikely intersection''), and more precisely, an exact formula for the weighted number of tangencies between $P$ and elements of the ``Betti foliation''. This work used analytic techniques that apparently do not generalize to positive characteristic. In this paper, we extend their work to characteristic $p$, and we develop a second approach to tangency properties of algebraic curves on a complex elliptic surface, yielding a new family of unlikely intersections with a strong connection to a famous homomorphism of Manin. We also correct inaccuracies in the literature about this homomorphism.
We study $\ell$-isogeny graphs of ordinary elliptic curves defined over $\mathbb{F}_q$ with an added level structure. Given an integer $N$ coprime to $p$ and $\ell,$ we look at the graphs obtained by adding $Γ_0(N),$ $Γ_1(N),$ and $Γ(N)$-level structures to volcanoes. Given an order $\mathcal{O}$ in an imaginary quadratic field $K,$ we look at the action of generalised ideal class groups of $\mathcal{O}$ on the set of elliptic curves whose endomorphism rings are $\mathcal{O}$ along with a given level structure. We show how the structure of the craters of these graphs is determined by the choice of parameters.
A capset is a subset $C \subset \mathbb{F}_3^n$ with no three points on a line. We characterise the capsets produced by successively removing points from the ambient space such that the removed point has the maximum number of lines contained in the set of remaining points and passing through it until the set of remaining points contains no lines.
We improve recent results of D. Gomez and A. Winterhof (2010) and of A. Ostafe and I. E. Shparlinski (2011) on the Waring problem with Dickson polynomials in the case of prime finite fields. Our approach is based on recent bounds of Kloosterman and Gauss sums due to A. Ostafe, I. E. Shparlinski and J. F. Voloch (2021).
In this paper, we consider how probability theory can be used to determine the survival strategy in two of the ``Squid Game" and ``Squid Game: The Challenge" challenges: the Hopscotch and the Warships. We show how Hopscotch can be easily tackled with the knowledge of the binomial distribution, taught in introductory statistics courses, while Warships is a much more complex problem, which can be tackled at different levels.
An important open problem in supersingular isogeny-based cryptography is to produce, without a trusted authority, concrete examples of "hard supersingular curves" that is, equations for supersingular curves for which computing the endomorphism ring is as difficult as it is for random supersingular curves. A related open problem is to produce a hash function to the vertices of the supersingular $\ell$-isogeny graph which does not reveal the endomorphism ring, or a path to a curve of known endomorphism ring. Such a hash function would open up interesting cryptographic applications. In this paper, we document a number of (thus far) failed attempts to solve this problem, in the hope that we may spur further research, and shed light on the challenges and obstacles to this endeavour. The mathematical approaches contained in this article include: (i) iterative root-finding for the supersingular polynomial; (ii) gcd's of specialized modular polynomials; (iii) using division polynomials to create small systems of equations; (iv) taking random walks in the isogeny graph of abelian surfaces; and (v) using quantum random walks.
We study some properties of the lattices defined as the kernel of the map $(a_1,\ldots, a_{q-1}) \mapsto \prod (1-c_ix)^{a_i} \in (\mathbb{F}_q[x]/(x^e))^*$, where $\mathbb{F}_q^* = \{c_1,\ldots,c_{q-1}\}$.
We prove that the set of rational points on a nonisotrivial curves of genus at least 2 over a global function field is equal to the set of adelic points cut out by the Brauer-Manin obstruction.
We obtain new bounds on short Weil sums over small multiplicative subgroups of prime finite fields which remain nontrivial in the range the classical Weil bound is already trivial. The method we use is a blend of techniques coming from algebraic geometry and additive combinatorics.
Let $\mathbb{F}$ be a finite field and $C,D$ smooth, geometrically irreducible proper curves over $\mathbb{F}$ and set $K = \mathbb{F}(D)$. We consider Brauer-Manin and abelian descent obstructions to the existence of rational points and to weak approximation for the curve $C \otimes_\mathbb{F} K$. In particular, we show that Brauer-Manin is the only obstruction to weak approximation and the Hasse principle in the case that the genus of $D$ is less than that of $C$. We also show that we can identify the points corresponding to non-constant maps $D \to C$ using Frobenius descents.
We consider smooth projective curves C/$\mathbb{F}$ over a finite field and their symmetric squares $C^{(2)}$. For a global function field $K/\mathbb{F}$, we study the $K$-rational points of $C^{(2)}$. We describe the adelic points of $C^{(2)}$ surviving Frobenius descent and how the $K$-rational points fit there. Our methods also lead to an explicit bound on the number of $K$-rational points of $C^{(2)}$ satisfying an additional condition. Some of our results apply to arbitrary constant subvarieties of abelian varieties, however we produce examples which show that not all of our stronger conclusions extend.
We obtain a nontrivial bound on the number of solutions to the equation $$ A^{x_1} + \ldots + A^{x_ν} = A^{x_{ν+1}} + \ldots + A^{x_{2ν}}, \quad 1 \le x_1, \ldots,x_{2ν} \le τ, $$ with a fixed $n\times n$ matrix $A$ over a finite field $\mathbb F_q$ of $q$ elements of multiplicative order $τ$. We give applications of our result to obtaining a new bound of additive character sums with a matrix exponential function, which is nontrivial beyond the square-root threshold. For $n=2$ this equation has been considered by Kurlberg and Rudnick (2001) (for $ν=2$) and Bourgain (2005) (for large $ν$) in their study of quantum ergodicity for linear maps over residue rings. Here we use a new approach to improve their results. We also obtain a bound on Kloosterman sums over small subgroups, of size below the square-root threshold.