30,662 research outputs found
Functional relations for elliptic polylogarithms
Numerous examples of functional relations for multiple polylogarithms are known. For elliptic polylogarithms, however, tools for the exploration of functional relations are available, but only very few relations are identified. Starting from an approach of Zagier and Gangl, which in turn is based on considerations about an elliptic version of the Bloch group, we explore functional relations between elliptic polylogarithms and link them to the relations which can be derived using the elliptic symbol formalism. The elliptic symbol formalism in turn allows for an alternative proof of the validity of the elliptic Bloch relation. While the five-term identity is the prime example of a functional identity for multiple polylogarithms and implies many dilogarithm identities, the situation in the elliptic setup is more involved: there is no simple elliptic analogue, but rather a whole class of elliptic identities
Renormalization, isogenies and rational symmetries of differential equations
We give an example of infinite order rational transformation that leaves a
linear differential equation covariant. This example can be seen as a
non-trivial but still simple illustration of an exact representation of the
renormalization group.Comment: 36 page
Algebraic theta functions and p-adic interpolation of Eisenstein-Kronecker numbers
We study the properties of Eisenstein-Kronecker numbers, which are related to
special values of Hecke -function of imaginary quadratic fields. We prove
that the generating function of these numbers is a reduced (normalized or
canonical in some literature) theta function associated to the Poincare bundle
of an elliptic curve. We introduce general methods to study the algebraic and
-adic properties of reduced theta functions for CM abelian varieties. As a
corollary, when the prime is ordinary, we give a new construction of the
two-variable -adic measure interpolating special values of Hecke
-functions of imaginary quadratic fields, originally constructed by
Manin-Vishik and Katz. Our method via theta functions also gives insight for
the case when is supersingular. The method of this paper will be used in
subsequent papers to study the precise -divisibility of critical values of
Hecke -functions associated to Hecke characters of quadratic imaginary
fields for supersingular , as well as explicit calculation in two-variables
of the -adic elliptic polylogarithm for CM elliptic curves.Comment: 55 pages, 2 figures. Minor misprints and errors were correcte
Elliptic nets and elliptic curves
An elliptic divisibility sequence is an integer recurrence sequence
associated to an elliptic curve over the rationals together with a rational
point on that curve. In this paper we present a higher-dimensional analogue
over arbitrary base fields. Suppose E is an elliptic curve over a field K, and
P_1, ..., P_n are points on E defined over K. To this information we associate
an n-dimensional array of values in K satisfying a nonlinear recurrence
relation. Arrays satisfying this relation are called elliptic nets. We
demonstrate an explicit bijection between the set of elliptic nets and the set
of elliptic curves with specified points. We also obtain
Laurentness/integrality results for elliptic nets.Comment: 34 pages; several minor errors/typos corrected in v
Spectral Curves for Super-Yang-Mills with Adjoint Hypermultiplet for General Lie Algebras
The Seiberg-Witten curves and differentials for supersymmetric
Yang-Mills theories with one hypermultiplet of mass in the adjoint
representation of the gauge algebra \G, are constructed for arbitrary
classical or exceptional \G (except ). The curves are obtained from the
recently established Lax pairs with spectral parameter for the (twisted)
elliptic Calogero-Moser integrable systems associated with the algebra \G.
Curves and differentials are shown to have the proper group theoretic and
complex analytic structure, and to behave as expected when tends either to
0 or to . By way of example, the prepotential for \G = D_n, evaluated
with these techniques, is shown to agree with standard perturbative results. A
renormalization group type equation relating the prepotential to the
Calogero-Moser Hamiltonian is obtained for arbitrary \G, generalizing a
previous result for \G = SU(N). Duality properties and decoupling to theories
with other representations are briefly discussed.Comment: 27 pages, Plain TeX; minor typos corrected, 5 refs adde
Varieties via their L-functions
We describe a procedure for determining the existence, or non-existence, of
an algebraic variety of a given conductor via an analytic calculation involving
L-functions. The procedure assumes that the Hasse-Weil L-function of the
variety satisfies its conjectured functional equation, but there is no
assumption of an associated automorphic object or Galois representation. We
demonstrate the method by finding the Hasse-Weil L-functions of all
hyperelliptic curves of conductor less than 500.Comment: 14 pages, 2 figure
- …