相关论文: Computing period matrices and the Abel-Jacobi map …
In this talk we discuss Feynman integrals which are related to elliptic curves. We show with the help of an explicit example that in the set of master integrals more than one elliptic curve may occur. The technique of maximal cuts is a…
New method of calculation of master integrals using differential equations and asymptotical expansion is presented. This method leads to the results exact in space-time dimension $D$ having the form of the convergent power series. As an…
In this paper we give an algorithm of how to determine a Weierstrass equation with minimal discriminant for superelliptic curves generalizing work of Tate for elliptic curves and Liu for genus 2 curves.
We study the transcendence of periods of abelian differentials, both at the arithmetic and functional level, from the point of view of the natural bi-algebraic structure on strata of abelian differentials. We characterise geometrically the…
We present an efficient algorithm to compute the Hasse-Witt matrix of a hyperelliptic curve C/Q modulo all primes of good reduction up to a given bound N, based on the average polynomial-time algorithm recently introduced by Harvey. An…
In this work, we investigate hyperelliptic curves of type $C: y^2 = x^{2g+1} + ax^{g+1} + bx$ over the finite field $\mathbb{F}_q, q = p^n, p > 2$. For the case of $g = 3$ and $4$ we propose algorithms to compute the number of points on the…
We present a method for computing the generic degree of a period map defined on a quasi-projective surface. As an application, we explicitly compute the generic degree of three period maps underlying families of Calabi-Yau 3-folds coming…
We discuss the computation of coefficients of the L-series associated to a hyperelliptic curve over Q of genus at most 3, using point counting, generic group algorithms, and p-adic methods.
We give an efficient algorithm to compute equations of twists of hyperelliptic curves of arbitrary genus over any separable field (of characteristic different from 2), and we explicitly describe some interesting examples.
Iteration of the quadratic map produces sequences of polynomials whose degrees {\sl explode} as the orbital period grows more and more. The polynomial mixing all 335 period-12 orbits has degree $4020$, while for the $52,377$ period-20…
We consider a special class of periodic continued fractions (called alpha-fractions) and discuss the related algebraic and geometric problems. A classical description of the Jacobi variety of a hyperelliptic curve due to Jacobi naturally…
We provide an algorithm for computing the Euler characteristic of the curves $S_p$ in the space of cubic polynomials, consisting of all polynomials with a periodic critical point of period $p$. The curves were introduced in [Milnor,…
We present an algorithm which uses Fujiwara's inequality to bound algebraic functions over ellipses of a certain type, allowing us to concretely implement a rigorous Gauss-Legendre integration method for algebraic functions over a line…
A period is a complex number arising as the integral of a rational function with algebraic number coefficients over a rationally-defined region. Although periods are typically transcendental numbers, there is a conjectural Galois theory of…
We construct genus one curves on base extensions of generic Severi--Brauer varieties of a given index and period which are versal objects for families of geometrically elliptic normal curves. We also compute the periods and indices of these…
We present a simple and efficient algorithm to compute the sum of the algebraic conjugates of a point on an elliptic curve.
We study the geometry and cohomology of algebraic super curves, using a new contour integral for holomorphic differentials. For a class of super curves (``generic SKP curves'') we define a period matrix. We show that the odd part of the…
We provide a framework for using elliptic curves with complex multiplication to determine the primality or compositeness of integers that lie in special sequences, in deterministic quasi-quadratic time. We use this to find large primes,…
We represent vector bundles over a regular algebraic curve as pairs of lattices over the maximal orders of its function field and we give polynomial time algorithms for several tasks: computing determinants of vector bundles, kernels and…
A new approach has been recently developed to study the arithmetic of hyperelliptic curves $y^2=f(x)$ over local fields of odd residue characteristic via combinatorial data associated to the roots of $f$. Since its introduction, numerous…