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Amilcar Pacheco

Publications and source records attributed to Amilcar Pacheco.

12 recordsLinked to original sources
arXiv2013

Normal subgroups of fundamental groups of affine curves in positive characteristic II

Let $k$ be an algebraically closed field of characteristic $p>0$ and let $C/k$ be a smooth connected affine curve. Denote by $π_1(C)$ its algebraic fundamental group. The goal of this paper is to characterize a certain subset of closed normal subgroups $N$ of $π_1(C)$. In "Normal subgroups of fundamental groups of affine curves in positive characteristic" we proved the same result under the additional hypothesis that $k$ had countable cardinality.

math.AG↗
arXiv2013

Rational points on curves over function fields

We provide in this paper an upper bound for the number of rational points on a curve defined over a one variable function field over a finite field. The bound only depends on the curve and the field, but not on the Jacobian variety of the curve.

math.NT↗
arXiv2011

Normal subgroups of the algebraic fundamental group of affine curves in positive characteristic

Let $π_1(C)$ be the algebraic fundamental group of a smooth connected affine curve, defined over an algebraically closed field of characteristic $p>0$ of countable cardinality. Let $N$ be a normal (resp. characteristic) subgroup of $π_1(C)$. Under the hypothesis that the quotient $π_1(C)/N$ admits an infinitely generated Sylow $p$-subgroup, we prove that $N$ is indeed isomorphic to a normal (resp. characteristic) subgroup of a free profinite group of countable cardinality. As a consequence, every proper open subgroup of $N$ is a free profinite group of countable cardinality.

math.AG↗
arXiv2008

Selmer groups of abelian varieties in extensions of function fields

Let $k$ be a field of characteristic $q$, $\cac$ a smooth geometrically connected curve defined over $k$ with function field $K:=k(\cac)$. Let $A/K$ be a non constant abelian variety defined over $K$ of dimension $d$. We assume that $q=0$ or $>2d+1$. Let $p\ne q$ be a prime number and $\cac'\to\cac$ a finite geometrically \textsc{Galois} and étale cover defined over $k$ with function field $K':=k(\cac')$. Let $(τ',B')$ be the $K'/k$-trace of $A/K$. We give an upper bound for the $\bbz_p$-corank of the \textsc{Selmer} group $\text{Sel}_p(A\times_KK')$, defined in terms of the $p$-descent map. As a consequence, we get an upper bound for the $\bbz$-rank of the \textsc{Lang-Néron} group $A(K')/τ'B'(k)$. In the case of a geometric tower of curves whose \textsc{Galois} group is isomorphic to $\bbz_p$, we give sufficient conditions for the \textsc{Lang-Néron} group of $A$ to be uniformly bounded along the tower.

math.NT↗
arXiv2005

On the rank of abelian varieties over function fields

Let $\cac$ be a smooth projective curve defined over a number field $k$, $A/k(\cac)$ an abelian variety and $(τ,B)$ the $k(\cac)/k$-trace of $A$. We estimate how the rank of $A(k(\cac))/τB(k)$ varies when we take a finite cover $π:\cac'\to\cac$ defined over $k$ geometrically abelian.

math.NT↗
arXiv2004

Fibrations et conjecture de Tate

We describe the behaviour of the rank of the Mordell-Weil group of the Picard variety of the generic fibre of a fibration in terms of local contributions given by averaging traces of Frobenius acting on the fibres. The results give a reinterpretation of Tate's conjecture (for divisors) and generalises previous results of Nagao, Rosen-Silverman and the authors.

math.NT↗
arXiv2003

Analogues of Lehmer's conjecture in positive characteristic

Let $C$ be a smooth projective irreducible curve defined over a finite field $\mathbb{F}_q$ and $K=\mathbb{F}_q(C)$. Let $A\subset K$ be the ring of functions regular outside a fixed place $\infty$ of $K$. Let $ϕ:A\to\text{End}(\mathbb{G}_a)$ be a Drinfeld $A$-module of rank $r$ defined over a finite extension $L$ of $K$ and $\hat{h}_ϕ$ its canonical height. Given a non-torsion point $α$ of $ϕ$ of degree $d$ over $K$, we prove that $\hat{h}_ϕ(α)\ge 1/d$. A similar statement is proved for the canonical height of a point of infinite order of a non-constant semi-stable elliptic curve defined over $K$, with the absolute constant 1 replaced by a constant depending on the elliptic curve.

math.NT↗
arXiv2002

On the distribution of the of Frobenius elements on elliptic curves over function fields

Let $C$ be a smooth projective curve over $\mathbb{F}_q$ with function field $K$, $E/K$ a nonconstant elliptic curve and $ϕ:\mathcal{E}\to C$ its minimal regular model. For each $P\in C$ such that $E$ has good reduction at $P$, i.e., the fiber $\mathcal{E}_P=ϕ^{-1}(P)$ is smooth, the eigenvalues of the zeta-function of $\mathcal{E}_P$ over the residue field $κ_P$ of $P$ are of the form $q_P^{1/2}e^{iθ_P},q_{P}e^{-iθ_P}$, where $q_P=q^{°(P)}$ and $0\leθ_P\leπ$. The goal of this note is to determine given an integer $B\ge 1$, $α,β\in[0,π]$ the number of $P\in C$ where the reduction of $E$ is good and such that $°(P)\le B$ and $α\leθ_P\leβ$.

math.NT↗
arXiv2002

Distribution of the traces of Frobenius on elliptic curves over function fields

Let C be a smooth irreducible projective curve defined over a finite field $\mathbb{F}_{q}$ of q elements of characteristic p>3 and $K=\mathbb{F}_{q}(C)$ its function field and $ϕ_{\mathcal{E}}:\mathcal{E}\to C$ the minimal regular model of $\mathbf{E}/K$. For each $P\in C$ denote $\mathcal{E}_P=ϕ^{-1}_{\mathcal{E}}(P)$. The elliptic curve $E/K$ has good reduction at $P\in C$ if and only if $\mathcal{E}_P$ is an elliptic curve defined over the residue field $κ_P$ of $P$. This field is a finite extension of $\mathbb{F}_q$ of degree $°(P)$. Let $t(\mathcal{E}_P)=q^{°(P)}+1-#\mathcal{E}_P(κ_P)$ be the trace of Frobenius at P. By Hasse-Weil's theorem (cf. [10, Chapter V, Theorem 2.4]), $t(\mathcal{E}_P)$ is the sum of the inverses of the zeros of the zeta function of $\mathcal{E}_P$. In particular, $|t(\mathcal{E}_P)|\le 2q^{°(P)}$. Let $C_0\subset C$ be the set of points of C at which $E/K$ has good reduction and $C_0(\mathbb{F}_{q^k})$ the subset of $\mathbb{F}_{q^k}$-rational points of $C_0$. We discuss the following question. Let $k\ge 1$ and t be integers and suppose $|t|\le 2q^{k/2}$. Let $π(k,t)=#\{P\in C_0(\mathbb{F}_{q^k}) | t(\mathcal{E}_P)=t\}$. How big is $π(k,t)$?

math.NT↗
arXiv2002

A note on Galois modules and the algebraic fundamental group of projective curves

Let $X$ be a smooth projective connected curve of genus $g\ge 2$ defined over an algebraically closed field $k$ of characteristic $p>0$. Let $G$ be a finite group, $P$ a Sylow $p$-subgroup of $G$ and $N_G(P)$ its normalizer in $G$. We show that if there exists an étale Galois cover $Y\to X$ with group $N_G(P)$, then $G$ is the Galois group wan étale Galois cover $\mathcal{Y}\to\mathcal{X}$, where the genus of $\mathcal{X}$ depends on the order of $G$, the number of Sylow $p$-subgroups of $G$ and $g$. Suppose that $G$ is an extension of a group $H$ of order prime to $p$ by a $p$-group $P$ and $X$ is defined over a finite field $\mathbb{F}_q$ large enough to contain the $|H|$-th roots of unity. We show that integral idempotent relations in the group ring $\mathbb{C}[H]$ imply similar relations among the corresponding generalized Hasse-Witt invariants.

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