We give an overview of some p-adic algorithms for computing with elliptic and hyperelliptic curves, starting with Kedlaya’s algorithm. While the original purpose of Kedlaya’s algorithm was to compute the zeta function of a hyperelliptic curve over a finite field, it has since been used in a number of applications. In particular, we describe how to use Kedlaya’s algorithm to compute Coleman inte...

We review the deformed W -algebra Wq,t(ŝlN ) and its screening currents. We explicitly construct the Local Integrals of Motion In (n = 1, 2, · · ·) for this deformed W -algebra. We explicitly construct the Nonlocal Integrals of Motion Gn (n = 1, 2, · · ·) by means of the screening currents. Our Integrals of Motion commute with each other, and give the elliptic version of those for the Virasoro ...

This talk reviews Feynman integrals, which are associated to elliptic curves. The will give an introduction into the mathematics behind them, covering topics of curves, modular forms and moduli space $n$ marked points on a genus one curve. latter be important, as integrals can expressed iterated ${\mathcal M}_{1,n}$, in same way evaluate multiple polylogarithms M}_{0,n}$. With right language, m...

We report on an implementation within GiNaC to evaluate iterated integrals related elliptic Feynman numerically arbitrary precision the region of convergence series expansion integrand. The includes modular forms as well involving Kronecker coefficient functions g ( k ) z , ? . For in d? and dz are implemented. This multiple polylogarithms. Note: program uploaded CPC Library is full program, or...

We present a rational version of the classical Landen transformation for elliptic integrals. This is employed to obtain explicit closed-form expressions for a large class of integrals of even rational functions and to develop an algorithm for numerical integration of these functions.

Computable lower and upper bounds for the symmetric elliptic integrals and for Legendre’s incomplete integral of the first kind are obtained. New bounds are sharper than those known earlier. Several inequalities involving integrals under discussion are derived. © 2007 Elsevier Inc. All rights reserved.

In recent work on multivariate elliptic hypergeometric integrals, the author generalized a conjectural integral formula of van Diejen and Spiridonov to a ten parameter integral provably invariant under an action of the Weyl group E7. In the present note, we consider the action of the affine Weyl group, or more precisely, the recurrences satisfied by special cases of the integral. These are of t...

We derive a number of local identities of arbitrary rank involving Jacobi elliptic functions and use them to obtain several new results. First, we present an alternative, simpler derivation of the cyclic identities discovered by us recently, along with an extension to several new cyclic identities of arbitrary rank. Second, we obtain a generalization to cyclic identities in which successive ter...