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Theses

Étude de la valeur en s=2 de la fonction L d'une courbe elliptique

Abstract : We study in this thesis the special value at 2 of L-functions of elliptic curves and modular forms of weight 2. We prove an explicit version of Beilinson's theorem for this special value. For any newform f of weight 2, level N ≥ 1 and character \psi, and for any Dirichlet character \chi modulo N (\chi even, primitive and different from the conjugate of \psi), we give an expression for L(f,2) L(f,\chi,1) as the regulator of an explicit Milnor symbol associated to modular units of X_1(N). At level \Gamma_1(p), with p prime, we deduce that Milnor symbols associated to modular units of X_1(p) generate the target vector space of Beilinson's regulator. Using the appendix by Merel, we give an explicit and universal formula for L(E,2), where E is an elliptic curve of prime conductor p, in terms of the twisted values L(E,\chi,1), where \chi is a character of conductor p. We also suggest a reformulation of Zagier's conjecture on L(E,2) for the jacobian J_1(N) of X_1(N), where N is the conductor of E. In this direction we propose an analogue of the elliptic dilogarithm for the jacobian J of an algebraic curve : it is a function R_J from the complex points of J to the dual space of holomorphic 1-forms on J. We show that L(f,2) L(f,\chi,1) is an explicit linear combination of values of R_{J_1(N)}, applied to f, at \Q-rational points of the cuspidal subgroup of J_1(N).
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Submitted on : Friday, February 3, 2006 - 2:17:17 PM
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  • HAL Id : tel-00011533, version 1

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François Brunault. Étude de la valeur en s=2 de la fonction L d'une courbe elliptique. Mathématiques [math]. Université Paris-Diderot - Paris VII, 2005. Français. ⟨tel-00011533⟩

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