An order $\mathcal O$ in an algebraic number field is called monogenic if it can be generated by one element over $\mathbb Z$. Since $\mathbb{Z} [\alpha] = \mathbb{Z}[\pm \alpha + c]$, for any integer $c$, we call two algebraic integers $\alpha$ and $\alpha^\prime$ equivalent if $\alpha +\alpha^\prime$ or $\alpha - \alpha^\prime$ is a rational integer. By a monogenization of $\mathcal O$, we mean an equivalence class of monogenizers of $\mathcal O$. Győry has shown that there are finitely many monogenizations for a given order. An interesting (and open) problem is to count the number of monogenizations of a given monogenic order. First we will note, for a given order $\mathcal O$, that $$\mathcal{O}=\mathbb Z[\alpha] \operatorname{in} \alpha,$$ is indeed a Diophantine equation, namely an index form equation. Then we will modify some algorithmic approaches for finding solutions of in-dex form equations in quartic number fields to obtain new and improved upper bounds for the number of monogenizations of a quartic order.

I survey recent work on the classical Lebesgue-Nagell equation $x^2+D=y^n$, when the prime divisors of $D$ are restricted to a fixed finite set $S$. This is joint work with Samir Siksek and, in part, with Philippe Michaud-Jacobs. Our results rely upon a combination of various results based upon the modularity of Galois representations, with bounds for linear forms in logarithms.

The degeneracies of a 1-motive M are all the geometrical phenomena which imply the decrease in the dimension of the motivic Galois group of M. We study the degeneracies of M related to the unipotent radical of its motivic Galois group. Since the dimension of the motivic Galois group of M is a lower bound for the transcendence degree of the field generated by periods of M, our results are linked with the Generalized Grothendieck Period Conjecture applied to 1-motives.

Painlevé VI equations provide a generalization of torsion sections on an elliptic pencil. Using a Manin map, we obtain a finiteness result for some of their solutions. We will then present a survey of the various statements encompassed by Manin's theorem of the kernel.

I will recall some striking elementary results of Don Zagier arising from the theory of period polynomials and rational period functions, and discuss possible generalisations growing out of the recent PhD thesis of Isabella Negrini.

For any set of algebraic numbers in a fixed number field K satisfying standard metric conditions in the theory (close enough to zero), we prove that the values of Lerch functions (essentially polylogarithms with ``shifts'') are linearly independent over K (join with M. Kawashima and N. Hirata-Kohno)."

We present our recent work in collaboration with Michel Waldschmidt: Let $\mathcal F$ be an infinite family of binary forms with integer coefficients, satisfying natural conditions. What can we say about the cardinality of the set of values $m$, such that $\vert m \vert \leq B$ and such that $m$ is represented by some form $F\in \mathcal F$, with degree $\geq d$ ($d$ is fixed and $B$ tends to infinity)? We will give typical examples of such families $\mathcal F$.

Relying on methods to show a linear independence criterion for values of generalized hypergeometric functions in http://arxiv.org/abs/2203.00207 (joint work with Sinnou David and Makoto Kawashima), Kawashima and Anthony Poels (available at http://arxiv.org/abs/2202.10782) proved a new estimate for binomial functions, in the archimedean and in the p-adic cases, by Padé approximation together with parametric geometry of numbers. We apply the Kawashima-Poels estimate to refine the number of the solutions to unit equations. This work is a collaboration with Kawashima, Poels and Yukiko Washio.

The index of a number field K is defined as follows: $$ I(K) = \gcd\{I(\theta), \theta \textrm{ is a primitive integral element of K}\} . $$ The first known example of a number field with index larger than $1$ was given by Dedekind. Let $I_p(K) = v_p(I(K))$, that is, $I_p(K)$ is the largest exponent $e$ such that $p^e$ divides $I (K )$. Ore conjectured that $I_p (K ) $ is not determined by the decomposition type of $p$ in K ? a matter settled by Engstrom in 1930. Engstrom provided the example of two number fields $K_1$ and $K_2$, each of degree $8$, wherein $3$ has the same splitting type but $I_3(K_1) = 2$ and $I_3(K_2) = 3.$ Determining $I_p(K)$ is listed as one of the unsolved problems in a book of Narkiewicz. In this talk, we will report on some new results regarding $I(K)$ and provide an application.

We consider heuristic predictions for small non-zero algebraic central values of twists of the L-function of an elliptic curve $E/\mathbb{Q}$ by Dirichlet characters. We provide computational evidence for these predictions and some consequences for an analogue of the Brauer-Siegel Theorem in this context.

Since Bilu (1997) we know that the conjugates of an algebraic number $\alpha$ of small height tend to be equidistributed near the unit circle. With Roger Baker we give a general refinement, which covers for example the norm of $1-\alpha$ or the values of $z/|z|=e^{i\theta}$ at the conjugates. We give also applications to the multiplicative dependence of $x,1-x$ as well as the number of conjugates in say the upper half plane: this number differs from half the degree $d$ by at most $8000h^{1/3}d$ for the absolute logarithmic height $h$ (now assumed non-zero) of $\alpha$.

Much is known about the location of zeros of modular forms, but until recently very few was known about the zeros of quasimodular forms, their multiplicity, their location and the way to count them. In a joint work with Sanoli Gun, we started this study in the case of critical points of Eisenstein series (i.e. zeros of their derivatives). Berend Ringeling and Jan-Willem van Ittersum then extended our work to the critical points of all real modular forms, and even to some extent to the zeros of quasimodular forms of arbitrary depth.

Let $(F_n)_{n\geq 0}$ be the Fibonacci sequence given by $F_0=0,\; F_1 = 1$ and $$F_{n+2}=F_{n+1} + F_{n},\; \mbox{for\;all}\; n\geq 0.$$ In this talk, we discuss about the positive integer solutions $(m,n,a,k)$ of the Diophantine equation $$F_n\pm \frac{a(10^m-1)}{9}=k!,$$ with $1\leq a\leq 9.$ The proof of the main result requires lower bounds for nonzero linear forms in two logarithms of algebraic numbers both in the complex and $p$-adic cases and some computer calculations. What can be said about a similar equation involving Pell numbers? We will also give an answer to this question. The talk is based on a series of joint papers with Adédji and Luca.

Lower bounds for linear forms in logarithms are a powerful tool that have found application to many number theory problems. In fact, many problems can be reduced to linear forms in two or three logarithms. As a result, good lower bounds for such linear forms have resulted in the complete solution of several outstanding problems [1], [2]

In the 1990s, one of the celebrants (MW) asked me to look at establishing such good lower bounds for linear forms in three logarithms. This resulted in two unpublished manuscripts.

Since the mid-2000s, a manuscript of Mignotte's entitled A kit on linear forms in three logarithms has circulated. The results in it, along with an earlier version, have played an essential role in solving some important problems (see, for example, [2]).

Recently, Mignotte and I undertook the task of making his kit manuscript ready for publication and this work is now complete. In this talk, I report on this work.

As in the initial manuscripts, we use Laurent's interpolation determinant approach [3], [4], [5]. This work also includes several improvements to the initial manuscript, such as an improved multiplicity estimate for the analytic function associated with the interpolation determinant and Laurent's zero estimate (together with its complete proof).
As a demonstration of our improvements, and to provide a fully worked example that others can follow for application of this kit to their own problems, we rework the lower bounds for the linear form in [2] used to show that there is no solution of $y^{p}=F_{n}$ for $n>12$. We obtain an upper bound on $p$ that is nearly $50$ times smaller than the one obtained in [2]. Lastly, we will discuss ongoing work to (hopefully) obtain simpler closed-form lower bounds for linear forms in three logs.

[1]   Y. Bilu, G. Hanrot and P. M. Voutier, (with an appendix by M. Mignotte), Existence of Primitive Divisors of Lucas and Lehmer Numbers, Crelle's J. 539 (2001), 75--122.

[2]   Y. Bugeaud, M. Mignotte and S. Siksek, Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers, Ann. Math. 163 (2006), 969--1018.

[3]   M. Laurent, Hauteurs de matrices d'interpolation, Approximations diophantiennes et nombres transcendants, Luminy (1990), ed. P. Philippon, de Gruyter (1992), 215 - 238.

[4]   M. Laurent, \emph{Linear forms in two logarithms and interpolation determinants}, Acta Arith. {\bf 66} (1994), 181--199.

[5]   M. Laurent, Linear forms in two logarithms and interpolation determinants II Acta Arith. 133 (2008), 325--348.