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Module supersingulier et points rationnels des courbes modulaires

Abstract : We study here the free group generated by isomorphism classes of supersingular elliptic curves in positive characteristic $p$, called the supersingular module. We compare it with others Hecke modules: the homology of modular curve $X_0(p)$ and the set of modular forms of weight $2$ and level $p$. We give several interpretations and applications of Gross and Gross-Kudla's formulas about $L$-functions of modular forms. Using the links between supersingular module and geometry of X_0(p)$ we apply these results in order to study the rational points on certain modular curves. Following the method of Momose and Parent, we determinate an infinite set of primes $p$ for which the quotient of $X_0(p^r)$ ($r\geq 2$) by Atkin-Lehner operator has no rational points other than cusps and CM points.
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Contributor : Marusia Rebolledo <>
Submitted on : Wednesday, January 12, 2005 - 2:45:51 PM
Last modification on : Wednesday, December 9, 2020 - 3:04:58 PM
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  • HAL Id : tel-00008022, version 1


Marusia Rebolledo. Module supersingulier et points rationnels des courbes modulaires. Mathématiques [math]. Université Pierre et Marie Curie - Paris VI, 2004. Français. ⟨tel-00008022⟩



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