271 research outputs found
Polynomial-Time Isomorphism Test of Groups that are Tame Extensions
We give new polynomial-time algorithms for testing isomorphism of a class of
groups given by multiplication tables (GpI). Two results (Cannon & Holt, J.
Symb. Comput. 2003; Babai, Codenotti & Qiao, ICALP 2012) imply that GpI reduces
to the following: given groups G, H with characteristic subgroups of the same
type and isomorphic to , and given the coset of isomorphisms
, compute Iso(G, H) in time poly(|G|).
Babai & Qiao (STACS 2012) solved this problem when a Sylow p-subgroup of
is trivial. In this paper, we solve the preceding problem in
the so-called "tame" case, i.e., when a Sylow p-subgroup of
is cyclic, dihedral, semi-dihedral, or generalized quaternion. These cases
correspond exactly to the group algebra
being of tame type, as in the
celebrated tame-wild dichotomy in representation theory. We then solve new
cases of GpI in polynomial time.
Our result relies crucially on the divide-and-conquer strategy proposed
earlier by the authors (CCC 2014), which splits GpI into two problems, one on
group actions (representations), and one on group cohomology. Based on this
strategy, we combine permutation group and representation algorithms with new
mathematical results, including bounds on the number of indecomposable
representations of groups in the tame case, and on the size of their cohomology
groups.
Finally, we note that when a group extension is not tame, the preceding
bounds do not hold. This suggests a precise sense in which the tame-wild
dichotomy from representation theory may also be a dividing line between the
(currently) easy and hard instances of GpI.Comment: 23 page
Geometric non-vanishing
We consider -functions attached to representations of the Galois group of
the function field of a curve over a finite field. Under mild tameness
hypotheses, we prove non-vanishing results for twists of these -functions by
characters of order prime to the characteristic of the ground field and by
certain representations with solvable image. We also allow local restrictions
on the twisting representation at finitely many places. Our methods are
geometric, and include the Riemann-Roch theorem, the cohomological
interpretation of -functions, and some monodromy calculations of Katz. As an
application, we prove a result which allows one to deduce the conjecture of
Birch and Swinnerton-Dyer for non-isotrivial elliptic curves over function
fields whose -function vanishes to order at most 1 from a suitable
Gross-Zagier formula.Comment: 46 pages. New version corrects minor errors. To appear in Inventiones
Mat
- …