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

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dc.creatorPacheco, Amilcar
dc.date2001-11-09
dc.date2002-06-09
dc.date.accessioned2026-07-07T04:44:29Z
dc.date.available2026-07-07T04:44:29Z
dc.descriptionLet 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 $π(k,t)$?
dc.description11 pages, replaced version, minor correction on the degree of the j-map
dc.identifierhttps://arxiv.org/abs/math/0111105
dc.identifierhttp://arxiv.org/abs/math/0111105
dc.identifierActa Arithmetica, 106.3 (2003), 255-263
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/62605
dc.subjectNumber Theory
dc.subjectAlgebraic Geometry
dc.subject11G05
dc.titleDistribution of the traces of Frobenius on elliptic curves over function fields
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