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Suppose that E is a Weil curve of analytic conductor N. Then the rank of the Mordell-Weil group of E is odd if and only if the Weil parametrization XO(N) In the case where ~ E factors through XO(N)/WN . N is square-free, there is another way of determining the action of WN on w or f. In the notation of §4 we have WN = IT Wp where the product runs through all the prime factors of N; and hence by Theorem 3 f IT A P 30 But in this case E, the reduction of as in §1, a p = E mod p, has a double point; and 1 if each of the tangent directions at the double point is defined over GF(p), and a p = -1 otherwise.

The case of an el- liptic curVe is particularly simple since we can deduce from (17) that the caSe Ai = Em not occur. This implies with m > 1 and E an elliptic curve defined over Q can- THEOREM 4. The decomposition of the new part of Jac(Xo(N» as a product of simple Abelian varieties, up to isogeny, corresponds to the decomposition of the new part of H~(Q) into subspaces invariant under all the Tp. In particular there is a one-one correspondence between elliptic curves over Q which are images of XO(N) but of no XO(M) with M < N and newforms for rO(N) with rational coefficients.

O(ne) est localement libre de rang n > n 0 . > 0 , le faisceau , et que sa formation commute 55 a tout changement de base. La suite exacte de cohomologie fournit de plus une suite exacte (n Par ailleurs, O(e) > 0). 2), que p (He). O(ne) (1. O(me) pour < m < n definissent une 1 ,de gradue associe Gr. O E n+(E) p S Rip! :eg EIS = gElS EIS reg dualite montre que Rl p. "E/S ---. e nElS et que pour la fleche adjointe Les sections de nreg F/<:>" ~ W, la fleche inverse ainsi associees aux sections de W sur s 56 s'appellent les differentielles invariantes.

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