IMB > Recherche > Séminaires

# Séminaire Théorie des Nombres

Responsables : Renaud Coulangeon, Baptiste Morin et Nicola Mazzari

• Le 29 mars 2019 à 14:00
• Salle de Conférences
Jürg Kramer (Humboldt Universität Berlin)
On formal Fourier-Jacobi expansions
It is a classical fact that Siegel modular forms possess so-called Fourier-Jacobi expansions. The question then arises, given such an expansion, when does it originate from a Siegel modular form. In the complex setting, J. Bruinier and M. Raum gave a necessary and sufficient criterion when Fourier-Jacobi expansions give rise to Siegel modular forms. In our talk we would like to revisit this problem however using the arithmetic compactifications of the moduli space of principally polarized abelian varieties established by G. Faltings and C.-L. Chai. In particular, this will allow us to generalize the result of J. Bruinier and M. Raum to the arithmetic setting.
• Le 5 avril 2019 à 14:00
• Salle de Conférences
Florian Luca (University of the Witwatersrand/University of Ostrava)
$Y$-coordinates of Pell equations in binary recurrences
Let $d>1$ be an integer which is not a square and $(X_n,Y_n)$ be the $n$th solution of the Pell equation $X^2-dY^2=\pm 1$. Given an interesting set of positive integers $U$, we ask how many positive integer solutions $n$ can the equation $Y_n\in U$ have. We show that under mild assumptions on $U$ (for example, when $1\in U$ and $U$ contains infinitely many even integers), then the equation $Y_n\in U$ has two solutions $n$ for infinitely many $d$. We show that this is best possible whenever $U$ is the set of values of a binary recurrent sequence $\{u_m\}_{m\ge 1}$ with real roots and $d$ is large enough (with respect to $U$). We also show that for the particular case when $u_m=2^m-1$, the equation $Y_n=2^m-1$ has at most two positive integer solutions $(n,m)$ for all $d$. The proofs use linear forms in logarithms. This is joint work with Bernadette Faye.
• Le 12 avril 2019 à 14:00
• Salle de Conférences
César Martinez Metzmeier (Universität Regensburg)
TNC

• Le 19 avril 2019 à 14:00
• Salle de Conférences
Diego Izquierdo (MPIM)

• Le 3 mai 2019 à 14:00
• Salle de Conférences
Joao Pedro Dos Santos (IMJ)

• Le 24 mai 2019 à 14:00
• Salle de Conférences