Arakelov computations in genus $3$ curves

Jordi Guàrdia

Displaying similar documents to “Arakelov computations in genus $3$ curves”

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.

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

Everett W. Howe, Kristin E. Lauter (2003)

Annales de l’institut Fourier

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We give new arguments that improve the known upper bounds on the maximal number $N_{q} (g)$ of rational points of a curve of genus $g$ over a finite field $𝔽_{q}$ , for a number of pairs $(q, g)$ . Given a pair $(q, g)$ and an integer $N$ , we determine the possible zeta functions of genus- $g$ curves over $𝔽_{q}$ with $N$ points, and then deduce properties of the curves from their zeta functions. In many cases we can show that a genus- $g$ curve over $𝔽_{q}$ with $N$ points must have a low-degree map to another curve over $𝔽_{q}$ , and often this...

On the group orders of elliptic curves over finite fields

Everett W. Howe (1993)

Compositio Mathematica

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Computing modular degrees using $L$ -functions

Christophe Delaunay (2003)

Journal de théorie des nombres de Bordeaux

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We give an algorithm to compute the modular degree of an elliptic curve defined over $ℚ$ . Our method is based on the computation of the special value at $s = 2$ of the symmetric square of the $L$ -function attached to the elliptic curve. This method is quite efficient and easy to implement.

Constructing elliptic curves over finite fields using double eta-quotients

Andreas Enge, Reinhard Schertz (2004)

Journal de Théorie des Nombres de Bordeaux

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We examine a class of modular functions for $Γ^{0} (N)$ whose values generate ring class fields of imaginary quadratic orders. This fact leads to a new algorithm for constructing elliptic curves with complex multiplication. The difficulties arising when the genus of $X_{0} (N)$ is not zero are overcome by computing certain modular polynomials. Being a product of four $η$ -functions, the proposed modular functions can be viewed as a natural generalisation of the functions examined by Weber and usually...

The analytic order of III for modular elliptic curves

J. E. Cremona (1993)

Journal de théorie des nombres de Bordeaux

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In this note we extend the computations described in [4] by computing the analytic order of the Tate-Shafarevich group III for all the curves in each isogeny class ; in [4] we considered the strong Weil curve only. While no new methods are involved here, the results have some interesting features suggesting ways in which strong Weil curves may be distinguished from other curves in their isogeny class.

Supersingular curves of genus two and class numbers

Tomoyoshi Ibukiyama, Toshiyuki Katsura, Frans Oort (1986)

Compositio Mathematica

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On the automorphisms of surfaces of general type in positive characteristic

Edoardo Ballico (1993)

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

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Here we give an explicit polynomial bound (in term of $K_{X}^{2}$ and not depending on the prime $p$ ) for the order of the automorphism group of a minimal surface $X$ of general type defined over a field of characteristic $p > 0$ .

Displaying similar documents to “Arakelov computations in genus 3 curves”

Displaying similar documents to “Arakelov computations in genus $3$ curves”