Displaying similar documents to “Involutory elliptic curves over 𝔽 q ( T )

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...

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 . 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...

Counting elliptic curves of bounded Faltings height

Ruthi Hortsch (2016)

Acta Arithmetica

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We give an asymptotic formula for the number of elliptic curves over ℚ with bounded Faltings height. Silverman (1986) showed that the Faltings height for elliptic curves over number fields can be expressed in terms of modular functions and the minimal discriminant of the elliptic curve. We use this to recast the problem as one of counting lattice points in a particular region in ℝ².