Criterion for the integrality of the Taylor coefficients of mirror maps in several variables

We give a necessary and sufficient condition for the integrality of the Taylor coefficients at the origin of formal power series $q_i({\mathbf z})=z_i\exp(G_i({\mathbf z})/F({\mathbf z}))$, with ${\mathbf z}=(z_1,...,z_d)$ and where $F({\mathbf z})$ and $G_i({\mathbf z})+\log(z_i)F({\mathbf z})$, $i=1,...,d$ are particular solutions of certain A-systems of differential equations. This criterion is based on the analytical properties of Landau's function (which is classically associated with the sequences of factorial ratios) and it generalizes the criterion in the case of one variable presented in "Crit\`ere pour l'int\'egralit\'e des coefficients de Taylor des applications miroir" [J. Reine Angew. Math.]. One of the techniques used to prove this criterion is a generalization of a version of a theorem of Dwork on the formal congruences between formal series, proved by Krattenthaler and Rivoal in "Multivariate $p$-adic formal congruences and integrality of Taylor coefficients of mirror maps" [arXiv:0804.3049v3, math.NT]. This criterion involves the integrality of the Taylor coefficients of new univariate mirror maps listed in "Tables of Calabi--Yau equations" [arXiv:math/0507430v2, math.AG] by Almkvist, van Enckevort, van Straten and Zudilin

Int\'egralit\'e des coefficients de Taylor de racines d'applications miroir

We prove the integrality of the Taylor coefficients of roots of mirror maps at the origin. By mirror maps, we mean formal power series z.exp(G(z)/F(z)), where F(z) and G(z)+log(z)F(z) are particular solutions of certain generalized hypergeometric differential equations. This enables us to prove a conjecture stated by Zhou in "Integrality properties of variations of Mahler measures" [arXiv:1006.2428v1 math.AG]. The proof of these results is an adaptation of the techniques used in our article "Crit\`ere pour l'int\'egralit\'e des coefficients de Taylor des applications miroir", [J. Reine Angew. Math. (to appear)]

Crit\`ere pour l'int\'egralit\'e des coefficients de Taylor des applications miroir

We give a necessary and sufficient condition for the integrality of the Taylor coefficients of mirror maps at the origin. By mirror maps, we mean formal power series z.exp(G(z)/F(z)), where F(z) and G(z)+log(z)F(z) are particular solutions of certain generalized hypergeometric differential equations. This criterion is based on the analytical properties of Landau's function (which is classically associated to the sequences of factorial ratios) and it generalizes results proved by Krattenthaler-Rivoal in "On the integrality of the Taylor coefficients of mirror maps" (to appear in Duke Math. J.). One of the techniques used to prove this criterion is a generalization of a theorem of Dwork on the formal congruences between formal series, which proved to be insufficient for our purposes

On Abel's problem and Gauss congruences

A classical problem due to Abel is to determine if a differential equation $y'=\eta y$ admits a non-trivial solution $y$ algebraic over $\mathbb C(x)$ when $\eta$ is a given algebraic function over $\mathbb C(x)$. Risch designed an algorithm that, given $\eta$, determines whether there exists an algebraic solution or not. In this paper, we adopt a different point of view when $\eta$ admits a Puiseux expansion with rational coefficients at some point in $\mathbb C\cup \{\infty\}$, which can be assumed to be 0 without loss of generality. We prove the following arithmetic characterization: there exists a non-trivial algebraic solution of $y'=\eta y$ if and only if the coefficients of the Puiseux expansion of $\eta$ at $0$ satisfy Gauss congruences for almost all prime numbers. We then apply our criterion to hypergeometric series: we completely determine the equations $y'=\eta y$ with an algebraic solution when $x\eta(x)$ is an algebraic hypergeometric series with rational parameters, and this enables us to prove a prediction Golyshev made using the theory of motives. We also present three other applications, in particular to diagonals of rational fractions and to directed two-dimensional walks

On Dwork's p-adic formal congruences theorem and hypergeometric mirror maps

International audienceUsing Dwork's theory, we prove a broad generalisation of his famous p-adic formal congruences theorem. This enables us to prove certain p-adic congruences for the generalized hypergeometric series with rational parameters; in particular, they hold for any prime number p and not only for almost all primes. Along the way, using Christol's functions, we provide an explicit formula for the ''Eisenstein constant'' of any globally bounded hypergeometric series with rational parameters. As an application of these results, we obtain an arithmetic statement of a new type concerning the integrality of Taylor coefficients of the associated mirror maps. It essentially contains all the similar univariate integrality results in the litterature

Cyclotomic valuation of qq-Pochhammer symbols and qq-integrality of basic hypergeometric series

We give a formula for the cyclotomic valuation of $q$-Pochhammer symbols in terms of (generalized) Dwork maps. We also obtain a criterion for the $q$-integrality of basic hypergeometric series in terms of certain step functions, which generalize Christol step functions. This provides suitable $q$-analogs of two results proved by Christol: a formula for the $p$-adic valuation of Pochhammer symbols and a criterion for the $N$-integrality of hypergeometric series

Sombra, silencio y haiku en la poética de Eric Schierloh

El artículo analiza tres libros de la producción poética de Eric Schierloh: "Costamarina (seguido de diario de Costamarina)", "Frío en las regiones equinocciales" y "Los Cueros": relevamiento topográfico de los últimos 1500 metros del arroyo antes de que vaya a morir al mar (el primero de 2012 y los otros dos de 2014). La poesía de Schierloh plantea un retorno a la Naturaleza encriptado en la observación del paisaje y la reflexión sobre el mismo. La escritura como notación parte de una alianza entre voz y visión. La lectura transita estos libros como un recorrido: los considera tres instancias de una misma búsqueda que parte de una fuerte crítica a una civilización en decadencia y una retirada de esta en pos del retorno hacia lo natural. El sujeto avanza hacia un paulatino borramiento, perdiéndose en el silencio del paisaje. El poema alcanza su voz más poderosa cuando el sujeto es acallado por las sombras del paisaje.This article explores three books from Eric Schierloh's poetic works - "Costamarina (seguido de diario de Costamarina)", "Frío en las regiones equinocciales" and "Los Cueros": relevamiento topográfico de los últimos 1500 metros del arroyo antes de que vaya a morir al mar (the first from 2012 and the other two from 2014). Schierloh's poetry presents a return to the Wilderness encrypted in the observation of and meditation on the landscape. The act of writing as a notation comes from an alliance between voice and vision. The act of reading in this article moves along these books like in a journey: it considers them three levels of one same search that departs from a strong criticism of a decadent civilization and a return to the wilderness. The subject moves forward towards its progressive erasure, losing itself in the silence of the landscape. The poem reaches its most powerful voice when the subject is silenced by the shades of the landscape.Facultad de Humanidades y Ciencias de la Educació

Some supercongruences of arbitrary length

We prove supercongruences modulo p2 for values of truncated hypergeometric series at some special points. The parameters of the hypergeometric series are d copies of 1/2 and d copies of 1 for any integer d≥2. In addition we describe their relation to hypergeometric motives