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Zbl 0824.14039
Nart, Enric
Formal group laws for certain formal groups arising from modular curves. (English)
[J] Compos. Math. 85, No.1, 109-119 (1993). ISSN 0010-437X; ISSN 1570-5846/e

Let $N \ge 5$ be an odd, square-free integer and denote by $J\sb 0 (N)\sp{\text {new}}$ (resp. ${\cal J}\sp{\text {new}})$ the new part of the Jacobian of the modular curve $X\sb 0 (N)\sb{\vert \bbfQ}$ (resp. the Néron model of $J\sb 0 (N)\sp{\text {new}})$. Let $S\sb 2 (\Gamma\sb 0 (N))$ be the space of the cusp forms of weight 2, with respect to $\Gamma\sb 0 (N)$. The author proves that the Shimura-Taniyama-Weil conjecture can be deduced from its formal analogous. More precisely, if $E\sb{\vert \bbfQ}$ is an elliptic curve with conductor $N$ and ${\cal E}\sb{\vert \bbfZ}$ denotes the Néron model of $E$, he proves the equivalence of the following conditions:\par (i) There exists a nonzero homomorphism of formal groups over $\bbfZ$, $({\cal J}\sp{\text {new}})\sp \wedge\to {\cal E}\sp \wedge$, where the $\sp wedge$-exponent means the formal completion along the zero-section of the given group.\par (ii) There exists a normalized newform $f \in S\sb 2 (\Gamma\sb 0 (N))$ such that $L(f,s) = L(E,s)$.\par (iii) There exists a nonzero homomorphism $J\sb 0 (N)\sp{\text {new}} \to E$ defined over $\bbfQ$.\par This result is deduced from an improvement of a previous result of {\it C. Deninger} and {\it E. Nart} [cf. Comment. Math. Helv. 65, No. 2, 318- 333 (1990; Zbl 0741.14026); corollary 2.9], which relates the matrix coefficients of the logarithm of the formal group ${\cal J}\sp{\text {new}}$, and the integral matrices describing the action of the Hecke operators on the $\bbfZ$-module of the cusp forms in $S\sb 2 (\Gamma\sb 0 (N))$ having integral Fourier expansion at infinity.
[M.Candilera (Padova)]

MSC 2000:: *14L05 Formal groups
14H40 Jacobians
14G35 Modular and Shimura varieties
14H25 Arithmetic ground fields (curves)

Keywords: formal groups; modular curves; new part of the Jacobian of the modular curve; Néron model; Shimura-Taniyama-Weil conjecture

Citations: Zbl 0741.14026

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