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Register a new userHypergeometric Integrals Modulo $p$ and Hasse--Witt Matrices
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We consider the KZ differential equations over $mathbb C$ in the case, when the hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field $mathbb F_p$. We study the space of polynomial solutions of these differential equations over $mathbb F_p$, constructed in a previous work by V. Schechtman and the second author. Using Hasse-Witt matrices we identify the space of these polynomial solutions over $mathbb F_p$ with the space dual to a certain subspace of regular differentials on an associated curve. We also relate these polynomial solutions over $mathbb F_p$ and the hypergeometric solutions over $mathbb C$.
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We prove general Dwork-type congruences for Hasse--Witt matrices attached to tuples of Laurent polynomials. We apply this result to establishing arithmetic and $p$-adic analytic properties of functions originating from polynomial solutions modulo $p^s$ of Knizhnik--Zamolodchikov (KZ) equations, solutions which come as coefficients of master polynomials and whose coefficients are integers. As an application we show that the $p$-adic KZ connection associated with the family of hyperelliptic curves $y^2=(t-z_1)dots (t-z_{2g+1})$ has an invariant subbundle of rank $g$. Notice that the corresponding complex KZ connection has no nontrivial subbundles due to the irreducibility of its monodromy representation.
The relations in the tautological ring of the moduli space M_g of nonsingular curves conjectured by Faber-Zagier in 2000 and extended to the moduli space of stable pointed curves by Pixton in 2012 are based upon two hypergeometric series A and B. The question of the geometric origins of these series has been solved in at least two ways (via the Frobenius structures associated to 3-spin curves and to CP1). The series A and B also appear in the study of descendent integration on the moduli spaces of open and closed curves. We survey here the various occurrences of A and B starting from their appearance in the asymptotic expansion of the Airy function (calculated by Stokes in the 19th century). Several open questions are proposed.
We study integral representations of the Gevrey series solutions of irregular hypergeometric systems. In this paper we consider the case of the systems associated with a one row matrix, for which the integration domains are one dimensional. We prove that any Gevrey series solution along the singular support of the system is the asymptotic expansion of a holomorphic solution given by a carefully chosen integral representation.
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 implementation for hyperelliptic curves of genus 2 and 3 is more than an order of magnitude faster than alternative methods for N = 2^26.
We consider the KZ differential equations over $mathbb C$ in the case, when the hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field $mathbb F_p$. We study the polynomial solutions of these differential equations over $mathbb F_p$, constructed in a previous work joint with V.,Schechtman and called the $mathbb F_p$-hypergeometric solutions. The dimension of the space of $mathbb F_p$-hypergeometric solutions depends on the prime number $p$. We say that the KZ equations have ample reduction for a prime $p$, if the dimension of the space of $mathbb F_p$-hypergeometric solutions is maximal possible, that is, equal to the dimension of the space of solutions of the corresponding KZ equations over $mathbb C$. Under the assumption of ample reduction, we prove a determinant formula for the matrix of coordinates of basis $mathbb F_p$-hypergeometric solutions. The formula is analogous to the corresponding formula for the determinant of the matrix of coordinates of basis complex hypergeometric solutions, in which binomials $(z_i-z_j)^{M_i+M_j}$ are replaced with $(z_i-z_j)^{M_i+M_j-p}$ and the Euler gamma function $Gamma(x)$ is replaced with a suitable $mathbb F_p$-analog $Gamma_{mathbb F_p}(x)$ defined on $mathbb F_p$.
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