José Felipe Voloch
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View article: The Brauer–Manin obstruction for nonisotrivialcurves over global function fields
The Brauer–Manin obstruction for nonisotrivialcurves over global function fields Open
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.
View article: On the Waring problem with Dickson polynomials modulo a prime
On the Waring problem with Dickson polynomials modulo a prime Open
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 Kloost…
View article: Ordinary isogeny graphs with level structure
Ordinary isogeny graphs with level structure Open
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 $\Gamma _0(N)$ , $\Gamma _1(N),$…
View article: Ordinary Isogeny Graphs with Level Structure
Ordinary Isogeny Graphs with Level Structure Open
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 $Γ(N)$-l…
View article: On the Waring problem with Dickson polynomials modulo a prime
On the Waring problem with Dickson polynomials modulo a prime Open
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 Kloost…
View article: Galois invariants of finite abelian descent and Brauer sets
Galois invariants of finite abelian descent and Brauer sets Open
For a variety over a global field, one can consider subsets of the set of adelic points of the variety cut out by finite abelian descent or Brauer–Manin obstructions. Given a Galois extension of the ground field, one can consider similar s…
View article: Etale descent obstruction and anabelian geometry of curves over finite fields
Etale descent obstruction and anabelian geometry of curves over finite fields Open
Let $C$ and $D$ be smooth, proper and geometrically integral curves over a finite field $F$. Any morphism from $D$ to $C$ induces a morphism of their \'etale fundamental groups. The anabelian philosophy proposed by Grothendieck suggests th…
View article: Failing to Hash Into Supersingular Isogeny Graphs
Failing to Hash Into Supersingular Isogeny Graphs Open
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 endomorph…
View article: Doubly isogenous curves of genus two with a rational action of $D_6$
Doubly isogenous curves of genus two with a rational action of $D_6$ Open
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…
View article: Galois invariants of finite abelian descent and Brauer sets
Galois invariants of finite abelian descent and Brauer sets Open
For a variety over a global field, one can consider subsets of the set of adelic points of the variety cut out by finite abelian descent or Brauer-Manin obstructions. Given a Galois extension of the ground field one can consider similar se…
View article: Some lattices from polynomial rings
Some lattices from polynomial rings Open
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 $\mathbb{F}_q^* = \{c_1,\ldots,c_{q-1}\}$.
View article: Etale descent obstruction and anabelian geometry of curves over finite fields
Etale descent obstruction and anabelian geometry of curves over finite fields Open
Let $C$ and $D$ be smooth, proper and geometrically integral curves over a finite field $F$. Any morphism from $D$ to $C$ induces a morphism of their étale fundamental groups. The anabelian philosophy proposed by Grothendieck suggests that…
View article: BCM volume 65 issue 4 Cover and Front matter
BCM volume 65 issue 4 Cover and Front matter Open
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
View article: Weil Sums over Small Subgroups
Weil Sums over Small Subgroups Open
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 alg…
View article: Equations and Character Sums with Matrix Powers, Kloosterman Sums over Small Subgroups, and Quantum Ergodicity
Equations and Character Sums with Matrix Powers, Kloosterman Sums over Small Subgroups, and Quantum Ergodicity Open
We obtain a nontrivial bound on the number of solutions to the equation $$ \begin{align*} &\sum_{i=1}^{\nu} A^{x_i} = \sum_{i=\nu+1}^{2\nu} A^{x_i}, \qquad 1 \leqslant x_i \leqslant \tau, \end{align*}$$with a fixed $n\times n$ matrix $A$ o…
View article: Failing to hash into supersingular isogeny graphs
Failing to hash into supersingular isogeny graphs Open
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 endomorp…
View article: Rational points on symmetric squares of constant algebraic curves over\n function fields
Rational points on symmetric squares of constant algebraic curves over\n function fields Open
We consider smooth projective curves C/$\\mathbb{F}$ over a finite field and\ntheir symmetric squares $C^{(2)}$. For a global function field $K/\\mathbb{F}$,\nwe study the $K$-rational points of $C^{(2)}$. We describe the adelic points of\…
View article: Rational points on symmetric squares of constant algebraic curves over function fields
Rational points on symmetric squares of constant algebraic curves over function fields Open
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^{…
View article: Equations and character sums with matrix powers, Kloosterman sums over small subgroups and quantum ergodicity
Equations and character sums with matrix powers, Kloosterman sums over small subgroups and quantum ergodicity Open
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 $\m…
View article: Recovering affine curves over finite fields fromL-functions
Recovering affine curves over finite fields fromL-functions Open
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 …
View article: Factoring polynomials over function fields
Factoring polynomials over function fields Open
If K/k is a function field in one variable of positive characteristic, we describe a general algorithm to factor one-variable polynomials with coefficients in K. The algorithm is flexible enough to find factors subject to additional restri…
View article: Locally Recoverable Codes on Surfaces
Locally Recoverable Codes on Surfaces Open
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 codewor…
View article: Memorial Article for John Tate
Memorial Article for John Tate Open
Tate was born on March 13, 1925, and died on Oc
View article: Commitment Schemes and Diophantine Equations
Commitment Schemes and Diophantine Equations Open
Motivated by questions in cryptography, we look for diophantine equations that are hard to solve but for which determining the number of solutions is easy.
View article: ON THE PRODUCT OF ELEMENTS WITH PRESCRIBED TRACE
ON THE PRODUCT OF ELEMENTS WITH PRESCRIBED TRACE Open
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 $\text{Tr}$ , for which elements $z$ in $\mathbb{L}$ , and $a$ , $b$ in…
View article: Effect of variation in density on the stability of bilinear shear currents with a free surface
Effect of variation in density on the stability of bilinear shear currents with a free surface Open
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 densit…
View article: Effect of variation in density on the stability of bilinear shear\n currents with a free surface
Effect of variation in density on the stability of bilinear shear\n currents with a free surface Open
We perform the stability analysis for a free surface fluid current modeled as\ntwo finite layers of constant vorticity, under the action of gravity and\nabsence of surface tension. In the same spirit as Taylor ["Effect of variation\nin den…
View article: Tate-Shafarevich groups of constant elliptic curves and isogeny volcanos
Tate-Shafarevich groups of constant elliptic curves and isogeny volcanos Open
We describe the structure of Tate-Shafarevich groups of a constant elliptic curves over function fields by exploiting the volcano structure of isogeny graphs of elliptic curves over finite fields.
View article: Tate-Shafarevich groups of constant elliptic curves and isogeny volcanos
Tate-Shafarevich groups of constant elliptic curves and isogeny volcanos Open
We describe the structure of Tate-Shafarevich groups of a constant elliptic curves over function fields by exploiting the volcano structure of isogeny graphs of elliptic curves over finite fields.