Improved upper bounds for the number of points on curves over finite fields

Everett W. Howe; Kristin E. Lauter

Displaying similar documents to “Improved upper bounds for the number of points on curves over finite fields”

Arakelov computations in genus $3$ curves

Jordi Guàrdia (2001)

Journal de théorie des nombres de Bordeaux

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Arakelov invariants of arithmetic surfaces are well known for genus 1 and 2 ([4], [2]). In this note, we study the modular height and the Arakelov self-intersection for a family of curves of genus 3 with many automorphisms: $C_{n} : Y^{4} = X^{4} - (4 n - 2) X^{2} Z^{2} + Z^{4} .$ Arakelov calculus involves both analytic and arithmetic computations. We express the periods of the curve $C_{n}$ in terms of elliptic integrals. The substitutions used in these integrals provide a splitting of the jacobian of $C_{n}$ as a product of...

Fields of definition of $ℚ$ -curves

Jordi Quer (2001)

Journal de théorie des nombres de Bordeaux

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Let $C$ be a $ℚ$ -curve with no complex multiplication. In this note we characterize the number fields $K$ such that there is a curve $C^{'}$ isogenous to $C$ having all the isogenies between its Galois conjugates defined over $K$ , and also the curves $C^{'}$ isogenous to $C$ defined over a number field $K$ such that the abelian variety Res $_{K / ℚ} (C^{'} / K)$ obtained by restriction of scalars is a product of abelian varieties of GL $_{2}$ -type.

Abelian varieties and curves in $W_{d} (C)$

Dan Abramovich, Joe Harris (1991)

Compositio Mathematica

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Reed-Muller codes and supersingular curves. I

Gerard Van der Geer, Marcel Van der Vlugt (1992)

Compositio Mathematica

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Involutory elliptic curves over $𝔽_{q} (T)$

Andreas Schweizer (1998)

Journal de théorie des nombres de Bordeaux

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For $n \in 𝔽_{q} [T]$ let $G$ be a subgroup of the Atkin-Lehner involutions of the Drinfeld modular curve $X_{0} (𝔫)$ . We determine all $𝔫$ and $G$ for which the quotient curve $G ∖ X_{0} (𝔫)$ is rational or elliptic.

Points at rational distance from the corners of a unit square

T. G. Berry (1990)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Euclidean algorithm and Kummer covers with many points.

Garzón, Alvaro (2003)

Revista Colombiana de Matemáticas

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Some remarks on Set-theoretic Intersection Curves in $P^{3}$

Roberto Paoletti (1996)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Motivated by the notion of Seshadri-ampleness introduced in [11], we conjecture that the genus and the degree of a smooth set-theoretic intersection $C \subset P^{3}$ should satisfy a certain inequality. The conjecture is verified for various classes of set-theoretic complete intersections.