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Tanguy Rivoal

Publications and source records attributed to Tanguy Rivoal.

At least 37 records · Page 2Linked to original sources
arXiv2017

Exceptional values of E-functions at algebraic points

E-functions are entire functions with algebraic Taylor coefficients satisfying certain arithmetic conditions, and which are also solutions of linear differential equations with rational functions coefficients. They were introduced by Siegel in 1929 to generalize Diophantine properties of the exponential function, and studied further by Shidlovskii in 1956. The celebrated Siegel-Shidlovskii Theorem deals with the algebraic (in)dependence of values at algebraic points of E-functions solutions of a differential system. However, somewhat paradoxically, this deep result may fail to decide whether a given E-fuction assumes an algebraic or a transcendental value at some given algebraic point. Building upon André's theory of E-operators, Beukers refined in 2006 the Siegel-Shidlovskii Theorem in an optimal way. In this paper, we use Beukers' work to prove the following result: there exists an algorithm which, given a transcendental E-function $f(z)$ as input, outputs the finite list of all exceptional algebraic points $α$ such that $f(α)$ is also algebraic, together with the corresponding list of values $f(α)$. This result solves the problem of deciding whether values of E-functions at algebraic points are transcendental.

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arXiv2017

Linear independence of values of G-functions

Given any non-polynomial $G$-function $F(z)=\sum\_{k=0}^\infty A\_k z^k$ of radius of convergence $R$, we consider the $G$-functions $F\_n^{[s]}(z)=\sum\_{k=0}^\infty \frac{A\_k}{(k+n)^s}z^k$ for any integers $s\geq 0$ and $n\geq 1$. For any fixed algebraic number $α$ such that $0 \textless{} \vert α\vert \textless{} R$ and any number field $\mathbb{K}$ containing $α$ and the $A\_k$'s, we define $Φ\_{α, S}$ as the $\mathbb{K}$-vector space generated by the values $F\_n^{[s]}(α)$, $n\ge 1$ and $0\leq s\leq S$. We prove that $u\_{\mathbb{K},F}\log(S)\leq \dim\_{\mathbb{K}}(Φ\_{α, S})\leq v\_F S$ for any $S$, with effective constants $u\_{\mathbb{K},F}\textgreater{}0$ and $v\_F\textgreater{}0$, and that the family $(F\_n^{[s]}(α))\_{1\le n \le v\_F, s \ge 0}$ contains infinitely many irrational numbers. This theorem applies in particular when $F$ is an hypergeometric series with rational parameters or a multiple polylogarithm, and it encompasses a previous result by the second author and Marcovecchio in the case of polylogarithms. The proof relies on an explicit construction of Padé-type approximants. It makes use of results of André, Chudnovsky and Katz on $G$-operators, of a new linear independence criterion à la Nesterenko over number fields, of singularity analysis as well as of the saddle point method.

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arXiv2015

Microsolutions of differential operators and values of arithmetic Gevrey series

We continue our investigation of $E$-operators, in particular their connection with $G$-operators; these differential operators are fundamental in understanding the diophantine properties of Siegel's $E$ and $G$-functions. We study in detail microsolutions (in Kashiwara's sense) of Fuchsian differential operators, and apply this to the construction of basis of solutions at $0$ and $\infty$ of any $E$-operator from microsolutions of a $G$-operator; this provides a constructive proof of a theorem of André. We also focus on the arithmetic nature of connection constants and Stokes constants between different bases of solutions of $E$-operators. For this, we introduce and study in details an arithmetic (inverse) Laplace transform that enables one to get rid of transcendental numbers inherent to André's original approach. As an application, we define a set of special values of arithmetic Gevrey series, and discuss its conjectural relation with the ring of exponential periods.

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arXiv2014

Arithmetic theory of E-operators

In [Séries Gevrey de type arithmétique I Théorémes de pureté et de dualité, Annals of Math. 151 (2000), 705--740], André has introduced E-operators, a class of differential operators intimately related to E-functions, and constructed local bases of solutions for these operators. In this paper we investigate the arithmetical nature of connexion constants of E-operators at finite distance, and of Stokes constants at infinity. We prove that they involve values at algebraic points of E-functions in the former case, and in the latter one, values of G-functions and of derivatives of the Gamma function at rational points in a very precise way. As an application, we define and study a class of numbers having certain algebraic approximations defined in terms of E-functions. These types of approximations are motivated by the convergents to the number e, as well as by recent constructions of approximations to Euler's constant and values of the Gamma function. Our results and methods are completely different from those in our paper [On the values of G-functions, Commentarii Math. Helv., to appear], where we have studied similar questions for G-functions.

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arXiv2013

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

Using 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.

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arXiv2013

Multiple zeta values, Padé approximation and Vasilyev's conjecture

Sorokin gave in 1996 a new proof that pi is transcendental. It is based on a simultaneous Padé approximation problem involving certain multiple polylogarithms, which evaluated at the point 1 are multiple zeta values equal to powers of pi. In this paper we construct a Padé approximation problem of the same flavour, and prove that it has a unique solution up to proportionality. At the point 1, this provides a rational linear combination of 1 and multiple zeta values in an extended sense that turn out to be values of the Riemann zeta function at odd integers. As an application, we obtain a new proof of Vasilyev's conjecture for any odd weight, concerning the explicit evaluation of certain hypergeometric multiple integrals; it was first proved by Zudilin in 2003.

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arXiv2012

Hardy-Littlewood series and even continued fractions

For any $s\in (1/2,1]$, the series$F_s(x)=\sum_{n=1}^{\infty} e^{iπn^2 x}/n^s$ converges almost everywhere on $[-1,1]$ by a result of Hardy-Littlewood, but not everywhere. However, there does not yet exist an intrinsic description of the set of convergence for $F_s$. In this paper, we define in terms of even or regular continued fractions certain subsets of points of $[-1,1]$ of full measure where the series converges. Our method is based on an approximate function equation for $F_s(x)$. As a by-product, we obtain the convergence of certain series defined in term of the convergents of the even continued fraction of an irrational number.

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arXiv2011

On the values of G-functions

Let f be a G-function (in the sense of Siegel), and x be an algebraic number; assume that the value f(x) is a real number. As a special case of a more general result, we show that f(x) can be written as g(1), where g is a G-function with rational coefficients and arbitrarily large radius of convergence. As an application, we prove that quotients of such values are exactly the numbers which can be written as limits of sequences a(n)/b(n), where the generating series of both sequences are G-functions with rational coefficients. This result provides a general setting for irrationality proofs in the style of Apery for zeta(3), and gives answers to questions asked by T. Rivoal in [Approximations rationnelles des valeurs de la fonction Gamma aux rationnels : le cas des puissances, Acta Arith. 142 (2010), no. 4, 347-365].

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arXiv2011

Analytic properties of mirror maps

We consider a multi-parameter family of canonical coordinates and mirror maps o\ riginally introduced by Zudilin [Math. Notes 71 (2002), 604-616]. This family includes many of the known one-variable mirror maps as special cases, in particular many of modular origin and the celebrated example of Candelas, de la Ossa, Green and\ Parkes [Nucl. Phys. B359 (1991), 21-74] associated to the quintic hypersurface in $\mathbb P^4(\mathbb C)$. In [Duke Math. J. 151 (2010), 175-218], we proved that all coeffi\ cients in the Taylor expansions at 0 of these canonical coordinates (and, hence, of the corresponding mirror maps) are integers. Here we prove that all coefficients in the Taylor expansions at 0 of these canonical coordinates are positive. Furthermore, we provide several results pertaining to the behaviour of the canonical coordinates and mirror maps as complex functions. In particular, we address analytic continuation, points of singularity, and radius of convergence of these functions. We present several very precise conjectures on the radius of convergence of the mirror maps and the sign pattern of the coefficients in their Taylor expansions at 0.

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arXiv2010

Multivariate $p$-adic formal congruences and integrality of Taylor coefficients of mirror maps

We generalise Dwork's theory of $p$-adic formal congruences from the univariate to a multi-variate setting. We apply our results to prove integrality assertions on the Taylor coefficients of (multi-variable) mirror maps. More precisely, with $\mathbf z=(z_1,z_2,...,z_d)$, we show that the Taylor coefficients of the multi-variable series $q(\mathbf z)=z_i\exp(G(\mathbf z)/F(\mathbf z))$ are integers, where $F(\mathbf z)$ and $G(\mathbf z)+\log(z_i) F(\mathbf z)$, $i=1,2,...,d$, are specific solutions of certain GKZ systems. This result implies the integrality of the Taylor coefficients of numerous families of multi-variable mirror maps of Calabi-Yau complete intersections in weighted projective spaces, as well as of many one-variable mirror maps in the "Tables of Calabi-Yau equations" [arXiv:math/0507430] of Almkvist, van Enckevort, van Straten and Zudilin. In particular, our results prove a conjecture of Batyrev and van Straten in [Comm. Math. Phys. 168 (1995), 493-533] on the integrality of the Taylor coefficients of canonical coordinates for a large family of such coordinates in several variables.

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arXiv2009

On the integrality of the Taylor coefficients of mirror maps, II

We continue our study begun in "On the integrality of the Taylor coefficients of mirror maps" (arXiv:0907.2577) of the fine integrality properties of the Taylor coefficients of the series ${\bf q}(z)=z\exp({\bf G}(z)/{\bf F}(z))$, where ${\bf F}(z)$ and ${\bf G}(z)+\log(z) {\bf F}(z)$ are specific solutions of certain hypergeometric differential equations with maximal unipotent monodromy at $z=0$. More precisely, we address the question of finding the largest integer $v$ such that the Taylor coefficients of $(z ^{-1}{\bf q}(z))^{1/v}$ are still integers. In particular, we determine the Dwork-Kontsevich sequence $(u_N)_{N\ge1}$, where $u_N$ is the largest integer such that $q(z)^{1/u_N}$ is a series with integer coefficients, where $q(z)=\exp(G(z)/F(z))$, $F(z)=\sum_{m=0}^{\infty} (Nm)! z^m/m!^N$ and $G(z)=\sum_{m=1}^{\infty} (H_{Nm}-H_m)(Nm)! z^m/m!^N$, with $H_n$ denoting the $n$-th harmonic number, conditional on the conjecture that there are no prime number $p$ and integer $N$ such that the $p$-adic valuation of $H_N-1$ is strictly greater than 3.

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arXiv2009

On the integrality of the Taylor coefficients of mirror maps

We show that the Taylor coefficients of the series ${\bf q}(z)=z\exp({\bf G}(z)/{\bf F}(z))$ are integers, where ${\bf F}(z)$ and ${\bf G}(z)+\log(z) {\bf F}(z)$ are specific solutions of certain hypergeometric differential equations with maximal unipotent monodromy at $z=0$. We also address the question of finding the largest integer $u$ such that the Taylor coefficients of $(z ^{-1}{\bf q}(z))^{1/u}$ are still integers. As consequences, we are able to prove numerous integrality results for the Taylor coefficients of mirror maps of Calabi-Yau complete intersections in weighted projective spaces, which improve and refine previous results by Lian and Yau, and by Zudilin. In particular, we prove the general "integrality" conjecture of Zudilin about these mirror maps.

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arXiv2009

Irrationality exponent and rational approximations with prescribed growth

Let $ξ$ be a real irrational number. We are interested in sequences of linear forms in 1 and $ξ$, with integer coefficients, which tend to 0. Does such a sequence exist such that the linear forms are small (with given rate of decrease) and the coefficients have some given rate of growth? If these rates are essentially geometric, a necessary condition for such a sequence to exist is that the linear forms are not too small, a condition which can be expressed precisely using the irrationality exponent of $ξ$. We prove that this condition is actually sufficient, even for arbitrary rates of growth and decrease. We also make some remarks and ask some questions about multivariate generalizations connected to Fischler-Zudilin's new proof of Nesterenko's linear independence criterion.

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arXiv2009

On the integrality of the Taylor coefficients of mirror maps

We show that the Taylor coefficients of the series ${\bf q}(z)=z\exp({\bf G}(z)/{\bf F}(z))$ are integers, where ${\bf F}(z)$ and ${\bf G}(z)+\log(z) {\bf F}(z)$ are specific solutions of certain hypergeometric differential equations with maximal unipotent monodromy at $z=0$. We also address the question of finding the largest integer $u$ such that the Taylor coefficients of $(z ^{-1}{\bf q}(z))^{1/u}$ are still integers. As consequences, we are able to prove numerous integrality results for the Taylor coefficients of mirror maps of Calabi-Yau complete intersections in weighted projective spaces, which improve and refine previous results by Lian and Yau, and by Zudilin. In particular, we prove the general ``integrality'' conjecture of Zudilin about these mirror maps. A further outcome of the present study is the determination of the Dwork-Kontsevich sequence $(u_N)_{N\ge1}$, where $u_N$ is the largest integer such that $q(z)^{1/u_N}$ is a series with integer coefficients, where $q(z)=\exp(F(z)/G(z))$, $F(z)=\sum_{m=0} ^{\infty} (Nm)! z^m/m!^N$ and $G(z)=\sum_{m=1} ^{\infty} (H_{Nm}-H_m)(Nm)! z^m/m!^N$, with $H_n$ denoting the $n$-th harmonic number, conditional on the conjecture that there are no prime number $p$ and integer $N$ such that the $p$-adic valuation of $H_N-1$ is strictly greater than 3.

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arXiv2008

On a linear form for Catalan's constant

It is shown how Andrews' multidimensional extension of Watson's transformation between a very-well-poised $_8ϕ_7$-series and a balanced $_4ϕ_3$-series can be used to give a straightforward proof of a conjecture of Zudilin and the second author on the arithmetic behaviour of the coefficients of certain linear forms of 1 and Catalan's constant. This proof is considerably simpler and more stream-lined than the first proof, due to the second author.

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arXiv2007

Phénomènes de symétrie dans des formes linéaires en polyzêtas

We give two generalizations, in arbitrary depth, of the symmetry phenomenon used by Ball-Rivoal to prove that infinitely many values of Riemann $ζ$ function at odd integers are irrational. These generalizations concern multiple series of hypergeometric type, which can be written as linear forms in some specific multiple zeta values. The proof makes use of the regularization procedure for multiple zeta values with logarithmic divergence.

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arXiv2006

Séries hypergéométriques multiples et polyzêtas

We describe a theoretical and effective algorithm which enables us to prove that rather general hypergeometric series and integrals can be decomposed as linear combinations of multiple zeta values, with rational coefficients.

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