Introduction to Modular Towers
نویسنده
چکیده
منابع مشابه
Explicit modular towers
We give a general recipe for explicitly constructing asymptotically optimal towers of modular curves such as {X0(l)}n>1. We illustrate the method by giving equations for eight towers with various geometric features. We conclude by observing that such towers are all of a specific recursive form and speculate that perhaps every tower of this form which attains the Drinfeld-Vlăduţ bound is modular...
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Modular towers, a notion due to M. Fried, are towers of Hurwitz spaces, with levels corresponding to the characteristic quotients of the p-universal Frattini cover of a fixed finite group G (with p a prime divisor of |G|). The tower of modular curves X1(pn) (n>0) is the original example: the group G is then the dihedral group Dp. There are diophantine conjectures on modular towers, inspired by ...
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We give explicit equations for the simplest towers of Drinfeld modular curves over any finite field, and observe that they coincide with the asymptotically optimal towers of curves constructed by Garcia and Stichtenoth.
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1. The universal p-Frattini cover. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2. The p-Frattini module. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3. Restriction to the normalizer of a p-Sylow. . . . . . . . . . . . . . . . . . . . . . . 8 4. Asymptotics of the p-Frattini modules Mn . . . . . . . . . . . . . . . . . . . . ....
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We give first a simple proof of a generalized Jacobi identity for n-dimensional odd diagonal lattices which specializes to the classical Jacobi identity for the lattice Z2. For Z +√ Z, it recovers a one-parameter family of Jacobi identities discovered recently by Chan, Chua and Solé, used to deduce two quadratically converging algorithms for computing π corresponding to elliptic functions for t...
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