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- To characterize the field of elliptic modular functions of level , and especially to decompose the Jacobian variety of this function field into simple factors up to isogeny. Also it is well known that if , a prime, and , then contains elliptic curves with complex multiplication. What can one say for general ? (en)
- Let be a totally real number field, and be a Hilbert modular form to the field . Then, choosing in a suitable manner, we can obtain a system of Erich Hecke's L-series with Größencharakter , which corresponds one-to-one to this by the process of Mellin transformation. This can be proved by a generalization of the theory of operator of Hecke to Hilbert modular functions . (en)
- Let be an elliptic curve defined over an algebraic number field , and the L-function of over in the sense that is the zeta function of over . If the Hasse–Weil conjecture is true for , then the Fourier series obtained from by the inverse Mellin transformation must be an automorphic form of dimension −2 of a special type . If so, it is very plausible that this form is an elliptic differential of the field of associated automorphic functions. Now, going through these observations backward, is it possible to prove the Hasse–Weil conjecture by finding a suitable automorphic form from which can be obtained? (en)
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