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José Felipe Voloch

Publications and source records attributed to José Felipe Voloch.

34 records · Page 2Linked to original sources
arXiv2021

Recovering affine curves over finite fields from $L$-functions

Let $K$ be the function field of a curve over a finite field of odd characteristic. We investigate using $L$-functions of Galois extensions of $K$ to effectively recover $K$. When $K$ is the function field of the projective line with four rational points removed, we show how to use $L$-functions of a ray class field to effectively recover the removed points up to automorphisms of the projective line. When $K$ is the function field of a plane curve, we show how to effectively recover the equation of that curve using $L$-functions of Artin-Schreier extensions of $K$.

math.NT↗
arXiv2021

Locally recoverable codes on surfaces

A linear error correcting code is a subspace of a finite-dimensional space over a finite field with a fixed coordinate system. Such a code is said to be locally recoverable with locality $r$ if, for every coordinate, its value at a codeword can be deduced from the value of (certain) $r$ other coordinates of the codeword. These codes have found many recent applications, e.g., to distributed cloud storage. We will discuss the problem of constructing good locally recoverable codes and present some constructions using algebraic surfaces that improve previous constructions and sometimes provide codes that are optimal in a precise sense. The main conceptual contribution of this paper is to consider surfaces fibered over a curve in such a way that each recovery set is constructed from points in a single fiber. This allows us to use the geometry of the fiber to guarantee the local recoverability and use the global geometry of the surface to get a hold on the standard parameters of our codes. We look in detail at situations where the fibers are rational or elliptic curves and provide many examples applying our methods.

cs.IT↗
arXiv2020

Recovering algebraic curves from L-functions of Hilbert class fields

In this paper, we prove that a smooth hyperbolic projective curve over a finite field can be recovered from L-functions associated to the Hilbert class field of the curve and its constant field extensions. As a consequence, we give a new proof of a result of Mochizuki and Tamagawa that two such curves with isomorphic fundamental groups are themselves isomorphic.

math.NT↗
arXiv2020

Effect of variation in density on the stability of bilinear shear currents with a free surface

We perform the stability analysis for a free surface fluid current modeled as two finite layers of constant vorticity, under the action of gravity and absence of surface tension. In the same spirit as Taylor ["Effect of variation in density on the stability of superposed streams of fluid," Proc. R. Soc. A 132, 499 (1931)], a geometrical approach to the problem is proposed, which allows us to present simple analytical criteria under which the flow is stable. A strong destabilizing effect of stratification in density is revealed by comparison with the physical setting where the vorticity interface is also a density interface separating two immiscible fluids with constant densities.

physics.flu-dyn↗
arXiv2019

On the product of elements with prescribed trace

This paper deals with the following problem. Given a finite extension of fields $\mathbb{L}/\mathbb{K}$ and denoting the trace map from $\mathbb{L}$ to $\mathbb{K}$ by $\mathrm{Tr}$, for which elements $z$ in $\mathbb{L}$, and $a$, $b$ in $\mathbb{K}$, is it possible to write $z$ as a product $x\cdot y$, where $x,y\in \mathbb{L}$ with $\mathrm{Tr}(x)=a, \mathrm{Tr}(y)=b$? We solve most of these problems for finite fields, with a complete solution when the degree of the extension is at least $5$. We also have results for arbitrary fields and extensions of degrees $2,3$ or $4$. We then apply our results to the study of PN functions, semifields, irreducible polynomials with prescribed coefficients, and to a problem from finite geometry concerning the existence of certain disjoint linear sets.

math.CO↗
arXiv2018

The Sato-Tate Distribution in Thin Parametric Families of Elliptic Curves

We obtain new results concerning the Sato-Tate conjecture on the distribution of Frobenius traces over single and double parametric families of elliptic curves. We consider these curves for values of parameters having prescribed arithmetic structure: product sets, geometric progressions, and most significantly prime numbers. In particular, some families are much thinner than the ones previously studied.

math.NT↗
arXiv2017

The d-primary Brauer-Manin obstruction for Curves

For a curve over a global field we consider for which integers d the d-primary part of the Brauer group can obstruct the existence of rational points. We give examples showing it is possible that there is a d-primary obstruction for infinitely many relatively coprime d, and also an example where the odd part of the Brauer group does not obstruct although there is a Brauer-Manin obstruction. These examples demonstrate that a slightly stronger form of a conjecture of Poonen is false, despite being supported by the same heuristic.

math.NT↗
arXiv2016

On the number of rational points on special families of curves over function fields

We construct families of curves which provide counterexamples for a uniform boundedness question. These families generalize those studied previously by several authors. We show, in detail, what fails in the argument of Caporaso, Harris, Mazur that uniform boundedness follows from the Lang conjecture. We also give a direct proof that these curves have finitely many rational points and give explicit bounds for the heights and number of such points.

math.NT↗
arXiv2016

Local-global principles for Weil-Châtelet divisibility in positive characteristic

We extend existing results characterizing Weil-Châtelet divisibility of locally trivial torsors over number fields to global fields of positive characteristic. Building on work of González-Avilés and Tan, we characterize when local-global divisibility holds in such contexts, providing examples showing that these results are optimal. We give an example of an elliptic curve over a global field of characteristic $2$ containing a rational point which is locally divisible by $8$, but is not divisible by $8$ as well as examples showing that the analogous local-global principle for divisibility in the Weil-Châtelet group can also fail.

math.NT↗
arXiv2016

Anabelian geometry and descent obstructions on moduli spaces

We study the section conjecture of anabelian geometry and the sufficiency of the finite descent obstruction to the Hasse principle for the moduli spaces of principally polarized abelian varieties and of curves over number fields. For the former we show that the section conjecture fails and the finite descent obstruction holds for a general class of adelic points, assuming several well-known conjectures. This is done by relating the problem to a local-global principle for Galois representations. For the latter, we prove some partial results that indicate that the finite descent obstruction suffices. We also show how this sufficiency implies the same for all hyperbolic curves.

math.NT↗
arXiv2010

On the Number of Places of Convergence for Newton's Method over Number Fields

Let f be a polynomial of degree at least 2 with coefficients in a number field K, let x_0 be a sufficiently general element of K, and let alpha be a root of f. We give precise conditions under which Newton iteration, started at the point x_0, converges v-adically to the root alpha for infinitely many places v of K. As a corollary we show that if f is irreducible over K of degree at least 3, then Newton iteration converges v-adically to any given root of f for infinitely many places v. We also conjecture that the set of places for which Newton iteration diverges has full density and give some heuristic and numerical evidence.

math.NT↗
arXiv2008

A Local-Global Criterion for Dynamics on P^1

Let K be a number field or a function field, let F:P^1 --> P^1 be a rational map of degree at least two defined over K, let P be a point in P^1(K) having infinite F-orbit, and let Z be a finite subset of Z. We prove a local-global criterion for the intersection of the F-orbit of P and the finite set Z. This is a special case of a dynamical Brauer-Manin criterion suggested by Hsia and Silverman.

math.NT↗
arXiv1999

Diophantine approximation and deformation

We associate certain curves over function fields to given algebraic power series and show that bounds on the rank of Kodaira-Spencer map of this curves imply bounds on the exponents of the power series, with more generic curves giving lower exponents. If we transport Vojta's conjecture on height inequality to finite characteristic by modifying it by adding suitable deformation theoretic condition, then we see that the numbers giving rise to general curves approach Roth's bound. We also prove a hierarchy of exponent bounds for approximation by algebraic quantities of bounded degree.

math.NT↗
arXiv1998

Difference subgroups of commutative algebraic groups over finite fields

The work of Chatzidakis and Hrushovski on the model theory of difference fields in characteristic zero showed that groups defined by difference equations have a very restricted structure. Recent work of Chatzidakis, Hrushovski and Peterzil [CHP] extends the class of difference fields for which this sort of result is known to positive characteristic. In this note, we analyze the subgroups of the torsion points of simple commutative algebraic groups over finite fields that can be constructed by such difference equations. Our results are reasonably complete modulo some well-known conjectures in Number Theory. In one case, we need the $p$-adic version of the four exponentials conjecture and in another we need a version of Artin's conjecture on primitive roots. We recover part of a theorem of Boxall on the intersection of varieties with the group of $m$-power torsion points, but in general this theorem does not follow from the model-theoretic analysis, because there may be no field automorphism $σ$ so that the $m$-power torsion group is contained in a modular group definable with $σ$. On the other hand, some of the groups defined by modular difference equations are much larger than the group of $m$-power torsion points, so our results are stronger in another direction. In some ways, the model theoretic approach extends the approach of Bogomolov and the original one of Lang.

math.NT↗
arXiv1997

Torsion points on y^2=x^6+1

Let C be the curve y^2=x^6+1 of genus 2 over a field of characteristic zero. Consider C embedded in its Jacobian J by sending one of the points at infinity on C to the origin of J. In this brief note we show that the points of C whose image on J are torsion are precisely the two points at infinity and the six points with y=0.

math.NT↗