In a recent work in collaboration with Makoto Kawashima with established new (effective) irrationality measures for values of functions which belong to a certain class of hypergeometric functions (including shifted logarithmic functions and binomial functions). In this talk, we will try to explain the ideas behind our proofs and constructions.

We first give conjecturally complete parametric evaluations of Gauss's $_2F_1$ hypergeometric functions in terms of finite sums of gamma quotients. We then generalize the connection due to Fricke with the Eisenstein series $E_4$ to give functional evaluations of $_2F_1$ corresponding to hyperbolic triangle groups. Finally, we show how a long search has enabled us to find almost $1000$ evaluations of $_2F_1(a,b,c,z)$ with all parameters rational and the result algebraic, and mention that they are probably $90\%$ complete for arithmetic triangle groups.

Almost nothing is known on the continued fraction expansion of an algebraic real number of degree at least three. The situation is different over the field of power series $\mathbb{F}_p((x^{-1}))$, where $p$ is a prime number. For instance, there are algebraic power series of degree at least three whose sequence of partial quotients have bounded degree. And there are as well algebraic power series of degree at least three which are very well approximable by rational fractions: the analogue of Liouville's theorem is best possible in $\mathbb{F}_p((x^{-1}))$. Recently, in a joint work with Han (built on a previous work by Han and Hu), we proved that, for any distinct nonconstant polynomials $a, b$ in $\mathbb{F}_2 [x]$, the power series $$ [a; b, b, a, b, a, a, b, \ldots ] = a + \frac{1}{b + \frac{1}{b + \cdots}} , $$ whose sequence of partial quotients is given by the Thue–Morse sequence, is algebraic of degree $4$ over $\mathbb{F}_2 (x)$. We discuss this and related results. Furthermore, we give a complete description of the continued fraction expansion of the algebraic power series $(1 + x^{-1})^{j/d}$ in $\mathbb{F}_p((x^{-1}))$, where $j, d$ are coprime integers with $d \ge 3$, $1 \le j < d/2$, and $\gcd(p, jd) = 1$. (Joint work with Han).

We explain in detail how to build upon Apéry's method in the framework of arithmetic holonomy bounds to prove the $\mathbb Q$-linear independence of $1, \log(1+1/m), \log(1+1/n)$, and $\log(1+1/m)\log(1+1/n)$ for the range $0 < |1-m/n| < \epsilon_0$ (a positive absolute constant), and how to upgrade the qualitative proof to a quantitative linear independence measure. We further discuss the relations to the classical theory of $G$-functions. This is a report on a joint work with Frank Calegari and Yunqing Tang.

Despite several significant advances over the last 30 years, guaranteeing the correctly rounded evaluation of elementary functions, such as the cosine, the exponential or the cube root for example, remains a difficult problem. It can be formulated as a Diophantine approximation problem, called the table maker's dilemma, which consists in determining points with integer coordinates close to a curve. In a recent work, we propose two algorithmic approaches to tackle this problem. They are closely linked to a celebrated work by Bombieri and Pila and to the so-called Coppersmith method, which has become classic in cryptography. We will present the initial context, one of our approaches and the results of practical experiments. This is joint work with Guillaume Hanrot (ENS Lyon).

Building on symbolic integration and numerical evaluation of D-finite functions, we can compute numerically periods of rational integrals with high precision. I will show the main lines of the method and develop three applications: (1) computing the volume of semialgebraic sets; (2) uncovering algebraic curves lying on complex surfaces; (3) computing the singular homology of projective hypersurfaces. This is based on joint work with Marc Mezzarobba, Eric Pichon-Pharabod, Mohab Safey El Din, Emre Sertöz, and Pierre Vanhove.

We consider meromorphic solutions of difference equations and prove that very few among them satisfy an algebraic differential equation. The basic tool is the difference Galois theory of functional equations.

We decompose the generating function $\sum_{n=0}^\infty\binom{2n}nP_n(y)^2z^n$ of the squares of Legendre polynomials as a product of periods of hyperelliptic curves. These periods satisfy second order differential equations which is highly unusual since four is the expected order for genus 2. These second order equations are arithmetic and yet their monodromy group is dense in $\rm{SL}_2(\mathbb{R})$. This implies that they cannot be solved in terms of hypergeometric functions, which is novel for an arithmetic equation that occurred naturally. This is joint work with Duco van Straten and Wadim Zudilin.

I will explain how differential operators coming from algebraic geometry produce interesting $p$-adic numbers. In a recent work with Frits Beukers we give examples of families of Calabi-Yau hypersurfaces in $n$ dimensions, for which one observes $p$-adic zeta values $\zeta_p(k)$ for $1 < k < n$. Appearance of $p$-adic zeta values for differential operators of Calabi-Yau type was conjectured by Candelas, de la Ossa and van Straten.

The class of $E$-functions has been introduced by Siegel in 1929; it contains the exponential and Bessel functions. Given a finite family of algebraically independent $E$-functions, we consider the set $S$ of algebraic points at which their values are algebraically dependent. The Siegel-Shidlovskii theorem, proved in 1955 and refined by several authors, implies that $S$ is finite. The aim of this talk is to give an algorithm that allows one to determine $S$. It is a joint work with Tanguy Rivoal.

$E$-functions were introduced by Siegel in 1929 to generalize Diophantine properties of the exponential function. After discussing the recent effective results on the algebraic values of $E$-functions at algebraic points, I will present a generalization of the linear formulation of the Lindemann-Weierstrass theorem for $E$-functions. As a consequence, I will show that the transcendental values at algebraic points of an entire hypergeometric function are linearly independent over the field of algebraic numbers. If time permits, I will give applications to the determination of linear relations between transcendental values of $E$-functions at algebraic points.

In this talk, I will present recent results obtained with Stéphane Fischler concerning the values taken at algebraic points by families of $G$-functions. These results are dual in some sense to a classical result of Chudnovsky. Our results hold not only for points inside the disk of convergence of a given $G$-function (the usual situation), but also for points in a suitable star-shaped domain at the origin to which the $G$-function can be extended.

The celebrated Skolem-Mahler-Lech Theorem states that the set of zeros of a linear recurrence sequence is the union of a finite set and finitely many arithmetic progressions. The corresponding computational question, the Skolem Problem, asks to determine whether a given linear recurrence sequence has a zero term. Although the Skolem-Mahler-Lech Theorem is almost 90 years old, decidability of the Skolem Problem remains open. The main contribution of this paper is an algorithm to solve the Skolem Problem for simple linear recurrence sequences (those with simple characteristic roots). Whenever the algorithm terminates, it produces a stand-alone certificate that its output is correct -- a set of zeros together with a collection of witnesses that no further zeros exist. We give a proof that the algorithm always terminates assuming two classical number-theoretic conjectures: the Skolem Conjecture (also known as the Exponential Local-Global Principle) and the $p$-adic Schanuel Conjecture. Preliminary experiments with an implementation of this algorithm within the tool Skolem point to the practical applicability of this method. Joint work with Yuri Bilu, Florian Luca, Joris Nieuwveld, Joël Ouaknine, David Purser.

There are only aleph-zero rational numbers, while there are 2 to the power aleph-zero real numbers. Hence the probability that a randomly chosen real number would be rational is 0. Yet proving rigorously that any specific, natural, real constant, is irrational is usually very hard, witness that there are still no proofs of the irrationality of the Euler-Mascheroni constant, the Catalan constant, or $\zeta(5)$. Inspired by Frits Beukers' elegant rendition of Apéry's seminal proofs of the irrationality of $\zeta(2)$ and $\zeta(3)$, and heavily using algorithmic proof theory, we systematically searched for other similar integrals, that lead to irrationality proofs. We found quite a few candidates for such proofs, including the square-root of $\pi$ times $\Gamma(7/3)/\Gamma(-1/6)$ and $\Gamma(19/6)/\Gamma(8/3)$ divided by the square-root of $\pi$. Joint work with Robert Dougherty-Bliss and Doron Zeilberger.

Regulator integrals are a certain kind of periods which arise in Beilinson's conjectures on special values of $L$-functions. In the case of modular forms, the regulator integrals introduced by Beilinson involve products of two Eisenstein series. We will describe a new regulator integral based on the Goncharov regulator map. Using triple modular values, we can compute this integral as the $L$-value of a weight $2$ modular form at $s=3$. As an application, we prove a conjecture of Boyd and Rodriguez Villegas on the Mahler measure of $(1+x)(1+y)+z$. This is (in part) joint work with Wadim Zudilin.

I will explain how ideas from algebraic geometry can inform the construction of rational approximations to zeta values, with an emphasis on the role of Poincaré duality. I will also report on joint work with Wadim Zudilin on rational approximations to $\zeta(5)$.