6,338 research outputs found

    Abelian subvarieties of Drinfeld Jacobians and congruences modulo the characteristic

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    We prove a level lowering result over rational function fields, with the congruence prime being the characteristic of the field. We apply this result to show that semi-stable optimal elliptic curves are not Frobenius conjugates of other curves defined over the same field

    On the transfer congruence between pp-adic Hecke LL-functions

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    We prove the transfer congruence between pp-adic Hecke LL-functions for CM fields over cyclotomic extensions, which is a non-abelian generalization of the Kummer's congruence. The ingredients of the proof include the comparison between Hilbert modular varieties, the qq-expansion principle, and some modification of Hsieh's Whittaker model for Katz' Eisenstein series. As a first application, we prove explicit congruence between special values of Hasse-Weil LL-function of a CM elliptic curve twisted by Artin representations. As a second application, we prove the existence of a non-commutative pp-adic LL-function in the algebraic K1K_1-group of the completed localized Iwasawa algebra.Comment: 59 page

    Elliptic curves of large rank and small conductor

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    For r=6,7,...,11 we find an elliptic curve E/Q of rank at least r and the smallest conductor known, improving on the previous records by factors ranging from 1.0136 (for r=6) to over 100 (for r=10 and r=11). We describe our search methods, and tabulate, for each r=5,6,...,11, the five curves of lowest conductor, and (except for r=11) also the five of lowest absolute discriminant, that we found.Comment: 16 pages, including tables and one .eps figure; to appear in the Proceedings of ANTS-6 (June 2004, Burlington, VT). Revised somewhat after comments by J.Silverman on the previous draft, and again to get the correct page break

    N=2 Gauge Theories: Congruence Subgroups, Coset Graphs and Modular Surfaces

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    We establish a correspondence between generalized quiver gauge theories in four dimensions and congruence subgroups of the modular group, hinging upon the trivalent graphs which arise in both. The gauge theories and the graphs are enumerated and their numbers are compared. The correspondence is particularly striking for genus zero torsion-free congruence subgroups as exemplified by those which arise in Moonshine. We analyze in detail the case of index 24, where modular elliptic K3 surfaces emerge: here, the elliptic j-invariants can be recast as dessins d'enfant which dictate the Seiberg-Witten curves.Comment: 42+1 pages, 5 figures; various helpful comments incorporate

    On Atkin and Swinnerton-Dyer Congruence Relations (2)

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    In this paper we give an example of a noncongruence subgroup whose three-dimensional space of cusp forms of weight 3 has the following properties. For each of the four residue classes of odd primes modulo 8 there is a basis whose Fourier coefficients at infinity satisfy a three-term Atkin and Swinnerton-Dyer congruence relation, which is the pp-adic analogue of the three-term recursion satisfied by the coefficients of classical Hecke eigen forms. We also show that there is an automorphic LL-function over Q\mathbb Q whose local factors agree with those of the ll-adic Scholl representations attached to the space of noncongruence cusp forms.Comment: Last version, to appear on Math Annale
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