747 research outputs found
Moduli of abelian varieties and p-divisible groups
Lecture notes at a conference on Arithmetic Geometry, Goettingen, July/August
2006: Density of ordinary Hecke orbits and a conjecture by Grothendieck on
deformations of p-divisible groups.Comment: 92 page
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
Noncommutative geometry and motives: the thermodynamics of endomotives
We combine aspects of the theory of motives in algebraic geometry with
noncommutative geometry and the classification of factors to obtain a
cohomological interpretation of the spectral realization of zeros of
-functions. The analogue in characteristic zero of the action of the
Frobenius on l-adic cohomology is the action of the scaling group on the cyclic
homology of the cokernel (in a suitable category of motives) of a restriction
map of noncommutative spaces. The latter is obtained through the thermodynamics
of the quantum statistical system associated to an endomotive (a noncommutative
generalization of Artin motives). Semigroups of endomorphisms of algebraic
varieties give rise canonically to such endomotives, with an action of the
absolute Galois group. The semigroup of endomorphisms of the multiplicative
group yields the Bost-Connes system, from which one obtains, through the above
procedure, the desired cohomological interpretation of the zeros of the Riemann
zeta function. In the last section we also give a Lefschetz formula for the
archimedean local L-factors of arithmetic varieties.Comment: 52 pages, amslatex, 1 eps figure, v2: final version to appea
Abelian Surfaces over totally real fields are Potentially Modular
We show that abelian surfaces (and consequently curves of genus 2) over
totally real fields are potentially modular. As a consequence, we obtain the
expected meromorphic continuation and functional equations of their Hasse--Weil
zeta functions. We furthermore show the modularity of infinitely many abelian
surfaces A over Q with End_C(A)=Z. We also deduce modularity and potential
modularity results for genus one curves over (not necessarily CM) quadratic
extensions of totally real fields.Comment: 285 page
Non-critical equivariant L-values of modular abelian varieties
We prove an equivariant version of Beilinson's conjecture on non-critical
-values of strongly modular abelian varieties over number fields. As an
application, we prove a weak version of Zagier's conjecture on and
Deninger's conjecture on for non-CM strongly modular
-curves.Comment: 18 page
CM cycles on Kuga–Sato varieties over Shimura curves and Selmer groups
Given a modular form f of even weight larger than two and an imaginary quadratic field K
satisfying a relaxed Heegner hypothesis, we construct a collection of CM cycles on a Kuga–Sato
variety over a suitable Shimura curve which gives rise to a system of Galois cohomology classes
attached to f enjoying the compatibility properties of an Euler system. Then we use Kolyvagin’s
method [21], as adapted by Nekova´¿r [28] to higher weight modular forms, to bound the size of the relevant Selmer group associated to f and K and prove the finiteness of the (primary part) of the Shafarevich–Tate group, provided that a suitable cohomology class does not vanish.Peer ReviewedPostprint (author's final draft
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