655 research outputs found
Computing automorphic forms on Shimura curves over fields with arbitrary class number
We extend methods of Greenberg and the author to compute in the cohomology of
a Shimura curve defined over a totally real field with arbitrary class number.
Via the Jacquet-Langlands correspondence, we thereby compute systems of Hecke
eigenvalues associated to Hilbert modular forms of arbitrary level over a
totally real field of odd degree. We conclude with two examples which
illustrate the effectiveness of our algorithms.Comment: 15 pages; final submission to ANTS I
Computing fundamental domains for the Bruhat-Tits tree for GL2(Qp), p-adic automorphic forms, and the canonical embedding of Shimura curves
We describe an algorithm for computing certain quaternionic quotients of the
Bruhat-Tits tree for GL2(Qp). As an application, we describe an algorithm to
obtain (conjectural) equations for the canonical embedding of Shimura curves.Comment: Accepted for publication in LMS Journal of Computation and
Mathematic
Graded structures and differential operators on nearly holomorphic and quasimodular forms on classical groups
We wish to use graded structures [KrVu87], [Vu01] on dffierential operators
and quasimodular forms on classical groups and show that these structures
provide a tool to construct p-adic measures and p-adic L-functions on the
corresponding non-archimedean weight spaces. An approach to constructions of
automorphic L-functions on uni-tary groups and their p-adic avatars is
presented. For an algebraic group G over a number eld K these L functions are
certain Euler products L(s, , r, ). In particular, our constructions
cover the L-functions in [Shi00] via the doubling method of Piatetski-Shapiro
and Rallis. A p-adic analogue of L(s, , r, ) is a p-adic analytic
function L p (s, , r, ) of p-adic arguments s Z p , mod
p rPresented in a talk for the INTERNATIONAL SCIENTIFIC CONFERENCE "GRADED
STRUCTURES IN ALGEBRA AND THEIR APPLICATIONS" dedicated to the memory of Prof.
Marc Krasner on Friday, September 23, 2016, International University Centre
(IUC), Dubrovnik, Croatia
Emergent spacetime from modular motives
The program of constructing spacetime geometry from string theoretic modular
forms is extended to Calabi-Yau varieties of dimensions two, three, and four,
as well as higher rank motives. Modular forms on the worldsheet can be
constructed from the geometry of spacetime by computing the L-functions
associated to omega motives of Calabi-Yau varieties, generated by their
holomorphic forms via Galois representations. The modular forms that emerge
from the omega motive and other motives of the intermediate cohomology are
related to characters of the underlying rational conformal field theory. The
converse problem of constructing space from string theory proceeds in the class
of diagonal theories by determining the motives associated to modular forms in
the category of motives with complex multiplication. The emerging picture
indicates that the L-function can be interpreted as a map from the geometric
category of motives to the category of conformal field theories on the
worldsheet.Comment: 40 page
- …