306 research outputs found
A criterion to rule out torsion groups for elliptic curves over number fields
We present a criterion for proving that certain groups of the form do not occur as the torsion subgroup of
any elliptic curve over suitable (families of) number fields. We apply this
criterion to eliminate certain groups as torsion groups of elliptic curves over
cubic and quartic fields. We also use this criterion to give the list of all
torsion groups of elliptic curves occurring over a specific cubic field and
over a specific quartic field.Comment: 13 page
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
Weight enumerators of Reed-Muller codes from cubic curves and their duals
Let be a finite field of characteristic not equal to or
. We compute the weight enumerators of some projective and affine
Reed-Muller codes of order over . These weight enumerators
answer enumerative questions about plane cubic curves. We apply the MacWilliams
theorem to give formulas for coefficients of the weight enumerator of the duals
of these codes. We see how traces of Hecke operators acting on spaces of cusp
forms for play a role in these formulas.Comment: 19 pages. To appear in "Arithmetic, Geometry, Cryptography, and
Coding Theory" (Y. Aubry, E. W. Howe, C. Ritzenthaler, eds.), Contemp. Math.,
201
A table of elliptic curves over the cubic field of discriminant -23
Let F be the cubic field of discriminant -23 and O its ring of integers. Let
Gamma be the arithmetic group GL_2 (O), and for any ideal n subset O let
Gamma_0 (n) be the congruence subgroup of level n. In a previous paper, two of
us (PG and DY) computed the cohomology of various Gamma_0 (n), along with the
action of the Hecke operators. The goal of that paper was to test the
modularity of elliptic curves over F. In the present paper, we complement and
extend this prior work in two ways. First, we tabulate more elliptic curves
than were found in our prior work by using various heuristics ("old and new"
cohomology classes, dimensions of Eisenstein subspaces) to predict the
existence of elliptic curves of various conductors, and then by using more
sophisticated search techniques (for instance, torsion subgroups, twisting, and
the Cremona-Lingham algorithm) to find them. We then compute further invariants
of these curves, such as their rank and representatives of all isogeny classes.
Our enumeration includes conjecturally the first elliptic curves of ranks 1 and
2 over this field, which occur at levels of norm 719 and 9173 respectively
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