6,338 research outputs found
Abelian subvarieties of Drinfeld Jacobians and congruences modulo the characteristic
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 -adic Hecke -functions
We prove the transfer congruence between -adic Hecke -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 -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
-function of a CM elliptic curve twisted by Artin representations. As a
second application, we prove the existence of a non-commutative -adic
-function in the algebraic -group of the completed localized Iwasawa
algebra.Comment: 59 page
Elliptic curves of large rank and small conductor
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
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)
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 -adic analogue of the
three-term recursion satisfied by the coefficients of classical Hecke eigen
forms. We also show that there is an automorphic -function over
whose local factors agree with those of the -adic Scholl representations
attached to the space of noncongruence cusp forms.Comment: Last version, to appear on Math Annale
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