597 research outputs found
A heuristic for boundedness of ranks of elliptic curves
We present a heuristic that suggests that ranks of elliptic curves over the
rationals are bounded. In fact, it suggests that there are only finitely many
elliptic curves of rank greater than 21. Our heuristic is based on modeling the
ranks and Shafarevich-Tate groups of elliptic curves simultaneously, and relies
on a theorem counting alternating integer matrices of specified rank. We also
discuss analogues for elliptic curves over other global fields.Comment: 41 pages, typos fixed in torsion table in section
F-theory on Genus-One Fibrations
We argue that M-theory compactified on an arbitrary genus-one fibration, that
is, an elliptic fibration which need not have a section, always has an F-theory
limit when the area of the genus-one fiber approaches zero. Such genus-one
fibrations can be easily constructed as toric hypersurfaces, and various
and models are presented as examples. To each
genus-one fibration one can associate a -function on the base as well as
an representation which together define the IIB axio-dilaton
and 7-brane content of the theory. The set of genus-one fibrations with the
same -function and representation, known as the
Tate-Shafarevich group, supplies an important degree of freedom in the
corresponding F-theory model which has not been studied carefully until now.
Six-dimensional anomaly cancellation as well as Witten's zero-mode count on
wrapped branes both imply corrections to the usual F-theory dictionary for some
of these models. In particular, neutral hypermultiplets which are localized at
codimension-two fibers can arise. (All previous known examples of localized
hypermultiplets were charged under the gauge group of the theory.) Finally, in
the absence of a section some novel monodromies of Kodaira fibers are allowed
which lead to new breaking patterns of non-Abelian gauge groups.Comment: 53 pages, 9 figures, 6 tables. v2: references adde
Toward a fundamental groupoid for the stable homotopy category
This very speculative sketch suggests that a theory of fundamental groupoids
for tensor triangulated categories could be used to describe the ring of
integers as the singular fiber in a family of ring-spectra parametrized by a
structure space for the stable homotopy category, and that Bousfield
localization might be part of a theory of `nearby' cycles for stacks or
orbifolds.Comment: This is the version published by Geometry & Topology Monographs on 18
April 200
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