Anastassiades, Christos

To each regular algebraic, conjugate self-dual, cuspidal automorphic representation $\Pi$ of $\mathrm{GL}(N)$ over a CM number field $E$ (or, more generally, to a regular algebraic isobaric sum of conjugate self-dual, cuspidal representations), we can attach a continuous $\ell$-adic Galois representation $r(\Pi)$ of the absolute Galois group of $E$...

Carney, Alexander

In one of the fundamental results of Arakelov’s arithmetic intersection theory, Faltings and Hriljac (independently) proved the Hodge-index theorem for arithmetic surfaces by relating the intersection pairing to the negative of the Neron-Tate height pairing. More recently, Moriwaki and Yuan–Zhang generalized this to higher dimension. In this work, ...

Nguyen, Jennifer

Let F_q be a finite field and let D ⊆ F_q. Let m be a positive integer and let k be an integer such that 1 ≤ k ≤ |D|. For b = (b_1,...,b_m) ∈ (F_q)^m , let N_m(k,b) denote the number of subsets S ⊆ D with cardinality k such that for i = 1,...,m, the sum, over a ∈ S, of a^i = b_i. The Moments Subset Sum Problem is to determine if N_m(k,b) > 0. There...

McAdam, Taylor Jane

We study the asymptotic distribution of almost-prime entries in horospherical flows on the quotient of SL(n,R) by a lattice, where the lattice is either cocompact or SL(n,Z). In the cocompact case, we obtain a result that implies density for almost-primes in horospherical flows where the number of prime factors is independent of the basepoint, and ...

Riffaut, Antonin

À partir du théorème d’André en 1998, qui est la première contribution non triviale à la conjecture de André-Oort sur les sous-variétés spéciales des variétés de Shimura, la principale problématique de cette thèse est d’étudier les propriétés diophantiennes des modules singuliers, en caractérisant les points de multiplication complexe (x; y) satisf...

Emme, Jordan Prikhod'ko, Alexander

Let $s_2(x)$ denote the number of digits ``$1$'' in a binary expansion of any $x \in \mathbb{N}$. We study the mean distribution $\mu_a$ of the quantity $s_2(x+a)-s_2(x)$ for a fixed positive integer $a$.It is shown that solutions of the equation$$ s_2(x+a)-s_2(x)= d $$are uniquely identified by a finite set of prefixes in $\{0,1\}^*$, and that the...

Ben Letaïef, Khaled

In this work, we make some Python simulations of the modular properties of associated Stirling numbers in order to illustrate our previous results and to make new conjectures.

Ben Letaïef, Khaled

In previous papers [1] [2], we have examined two types of 'rotations' in associated Stirling numbers of first and second kind at any order r, respectively : 1. Geometrical rotations, which helped to compact all those Stirling numbers in an arithmeti-cal triangle's structure. 2. Modular rotations which determined their arithmetical properties using ...

Ben Letaïef, Khaled

In a previous article [1], we have found that associated Stirling numbers of first and second kind can be compacted, at any order and using a linear transformation, in a structure of arithmetical triangle. We show then a strong link between the congruence of such numbers and this common geometrical layout. It leads to non-trivial combinatorial and ...

Ben Letaïef, Khaled

Associated Stirling numbers of first and second kind are usually found in the literature in various forms of stairs depending on their order r. Yet, it is shown in this note that all of these numbers can be arranged, through a linear transformation, in the same arithmetical triangle structure as the 'Pascal's triangle'.