Publications

Ruzsa's conjecture asserts that any sequence a(n) of integers that preserves congruences, i.e., satisfies a(n+k) = a(n) mod k, and has the growth condition limsup |a(n)|^{1/n} < e, must be a polynomial sequence. While previous results by Hall, Ruzsa, Perelli, and Zannier have confirmed this conjecture under stricter growth bounds, the general case remains open. In this paper, we establish a new partial result by proving that if in addition the generating series f of a(n) has at most two singular directions at x = 0, then a(n) is necessarily a polynomial sequence. Our approach is based on an adaptation of Carlson's method, originally developed for the Pólya-Carlson dichotomy, combined with a refined analysis of Hankel determinants. Specifically, we derive an upper bound on these determinants using Pólya's inequality and a transfinite diameter argument of Dubinin, while a non-Archimedean divisibility condition on Hankel determinants yields a lower bound, ultimately leading to the rationality of f. This confirms that counterexamples to Ruzsa's conjecture, if they exist, must exhibit at least three singular directions.

Abel's problem consists in identifying the conditions under which the differential equation y'= ηy, with η an algebraic function in C(x), possesses a non-zero algebraic solution y. This problem has been algorithmically solved by Risch. In a previous paper, we presented an alternative solution in the special arithmetic situation where η has a Puiseux expansion with rational coefficients at the origin: there exists a non-trivial algebraic solution of y'= ηy if and only if the coefficients of the Puiseux expansion of xη(x) at 0 satisfy Gauss congruences for almost all prime numbers. In this paper, we generalize this criterion to arbitrary η algebraic over Qbar(x), by means of a natural generalization to number fields of Gauss congruences and of the weaker Cartier congruences recently introduced in this context by Bostan. We then provide applications of this criterion in the hypergeometric setting and for Artin-Mazur zeta functions.

The aim of this note is to show that any algebraic relation over the field of algebraic numbers between the values of the trigonometric functions sine and cosine at algebraic points can be derived from the Pythagorean identity and the angle addition formulas. This result is obtained as a consequence of the Lindemann-Weierstrass theorem.

E-functions were introduced by Siegel in 1929 to generalize Diophantine properties of the exponential function. After developments of Siegel's methods by Shidlovskii, Nesterenko and André, Beukers proved in 2006 an optimal result on the algebraic independence of the values of E-functions. Since then, it seems that no general result was stated concerning the relations between the values of a single E-function. We prove that André's theory of E-operators and Beukers' result lead to a Lindemann-Weierstrass theorem for E-functions.

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.

A classical problem due to Abel is to determine if a differential equation y'= r y admits a non-trivial solution y algebraic over C(x) when r is a given algebraic function over C(x). Risch designed an algorithm that, given r, determines whether there exists an algebraic solution or not. In this paper, we adopt a different point of view when r admits a Puiseux expansion with  rational coefficients at some point in C or at infinity, 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'=r y if and only if the coefficients of the Puiseux expansion of r at 0 satisfy Gauss congruences for almost all prime numbers. We then apply our criterion to hypergeometric series: we completely determine the equations y'=r y with an algebraic solution when xr(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.

Every polynomial P(X) ∈ Z[X] satisfies the congruences P(n + m) ≡ P(n) mod m for all integers n, m ≥ 0. An integer valued sequence (a_n) is called a pseudo-polynomial when it satisfies these congruences. Hall characterized pseudopolynomials and proved that they are not necessarily polynomials. A long standing conjecture of Ruzsa says that a pseudo-polynomial a_n is a polynomial as soon as lim sup_n |a_n|^(1/n) < e. Under this growth assumption, Perelli and Zannier proved that the generating series of a_n is a G-function. A primary pseudo-polynomial is an integer valued sequence (a_n) such that a_{n+p} ≡ a_n mod p for all integers n ≥ 0 and all prime numbers p. The same conjecture has been formulated for them, which implies Ruzsa’s, and this paper revolves around this conjecture. We obtain a Hall type characterization of primary pseudo-polynomials and draw various consequences from it. We give a new proof and generalize a result due to Zannier that any primary pseudo-polynomial with an algebraic generating series is a polynomial. This leads us to formulate a conjecture on diagonals of rational fractions and primary pseudo-polynomials, which is related to classic conjectures of Christol and van der Poorten. We make the Perelli-Zannier Theorem effective. We prove a Pólya type result: if there exists a function F analytic in a right-half plane with not too large exponential growth (in a precise sense) and such that for all large n the primary pseudo-polynomial a_n = F(n), then a_n is a polynomial. Finally, we show how to construct a non-polynomial primary pseudo-polynomial starting from any primary pseudo-polynomial generated by a G-function different of 1/(1 − x).

We prove supercongruences modulo p 2 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.

We prove congruence relations modulo cyclotomic polynomials for multisums of q-factorial ratios, therefore generalizing many well-known p-Lucas congruences. Such congruences connect various classical generating series to their q-analogs. Using this, we prove a propagation phenomenon: when these generating series are algebraically independent, this is also the case for their q-analogs.

We develop a new method for proving algebraic independence of G-functions. Our approach rests on the following observation: G-functions do not always come with a single linear differential equation, but also sometimes with an infinite family of linear difference equations associated with the Frobenius that are obtained by reduction modulo prime ideals. When these linear difference equations have order one, the coefficients of the G-function satisfy congruences reminiscent of a classical theorem of Lucas on binomial coefficients. We use this to derive a Kolchin-like algebraic independence criterion. We show the relevance of this criterion by proving, using p-adic tools, that many classical families of G-functions turn out to satisfy congruences “à la Lucas”.

We provide lower bounds for p-adic valuations of multisums of factorial ratios which satisfy an Apéry-like recurrence relation: these include Apéry, Domb, Franel numbers, the numbers of abelian squares over a finite alphabet, and constant terms of powers of certain Laurent polynomials. In particular, we prove Beukers' conjectures on the p-adic valuation of Apéry numbers. Furthermore, we give an effective criterion for a sequence of factorial ratios to satisfy the p-Lucas property for almost all primes p.

Using Dwork's theory, we prove a broad generalization 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. Furthermore, using Christol's functions, we provide an explicit formula for the Eisenstein constant of any hypergeometric series with rational parameters.
As an application of these results, we obtain an arithmetic statement on average of a new type concerning the integrality of Taylor coefficients of the associated mirror maps. It contains all the similar univariate integrality results in the literature, with the exception of certain refinements that hold only in very particular cases.

We give a necessary and sufficient condition for the integrality of the Taylor coefficients at the origin of formal power series qi (z) = zi exp(Gi (z)/F(z)), with z = (z1, . . . ,zd ) and where F(z) and Gi (z) + log(zi )F(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 sequences of factorial ratios) and it generalizes the criterion in the case of one variable presented in [E. Delaygue, Critère pour l’intégralité des coefficients de Taylor des applications miroir, J. Reine Angew. Math. 662 (2012) 205–252]. One of the techniques used to prove this criterion is a generalization of a version of a theorem of Dwork on formal congruences between formal series, proved by Krattenthaler and Rivoal in [C. Krattenthaler, T. Rivoal, Multivariate p-adic formal congruences and integrality of Taylor coefficients of mirror maps, in: L. Di Vizio, T. Rivoal (Eds.), Theories Galoisiennes et Arithmétiques des Équations Différentielles, in: Séminaire et Congrès, vol. 27, Soc. Math. France, Paris, 2011, pp. 279–307].

We demonstrate the integrality of the Taylor coefficients of roots of formal power series q(z) = z exp(G(z)/F(z)), where F(z) and G(z) + log(z)F(z) are particular solutions of certain hypergeometric differential equations. This allows 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: « Critère pour l’intégralité des coefficients de Taylor des applications miroir »[J. Reine Angew. Math.].

Nous donnons une condition nécessaire et suffisante pour que les coefficients de Taylor de séries de la forme q(z) = z exp(G(z)/F(z)) soient entiers, où F(z) et G(z) + log(z)F(z) sont des solutions particulières de certaines équations diérentielles hypergéométriques généralisées. Ce critère est basé sur les propriétés analytiques de l'application de Landau (classiquement associée aux suites de quotients de factorielles) et il généralise les résultats de Krattenthaler-Rivoal dans On the integrality of the Taylor coefficients of mirror maps, Duke Math. Journal. Pour démontrer ce critère, nous généralisons entre autres un théorème de Dwork concernant les congruences formelles entre séries formelles dans On p-adic dierential equations IV : generalized hypergeometric functions as p-adic functions in one variable, Annales scientiques de l'ENS.

Mirror maps are power series which occur in Mirror Symmetry as the inverse for composition of q(z) = exp(f(z)/g(z)), called local q-coordinates, where f and g are particular solutions of the Picard–Fuchs differential equations associated with certain one-parameter families of Calabi–Yau varieties. In several cases, it has been observed that such power series have integral Taylor coefficients at the origin. In the case of hypergeometric equations, we discuss p-adic tools and techniques that enable one to prove a criterion for the integrality of the coefficients of mirror maps. This is a joint work with T. Rivoal and J. Roques. This note is an extended abstract of the talk given by the author in January 2017 at the conference “Hypergeometric motives and Calabi–Yau differential equations” in Creswick, Australia.