Related papers: On the $v$-adic values of G-functions I

In this third part in this series we continue from \cite{papaspadicpart1}, the study of relations among values of G-functions associated to a $1$-parameter family of principally polarized abelian surfaces. In particular, we establish…

This is the second paper in a series by the author, centered on the study of values of G-functions associated to a $1$-parameter family of abelian varieties $f:\CX\rightarrow S$ and a point $s_0\in S(K)$ over some number field $K$. Here we…

In this survey, we outline the role of G-functions in arithmetic geometry, notably their link with Picard-Fuchs differential equations and periods. We explain how polynomial relations between special values of G-functions arising from a…

We describe a general method for giving $p$-adic interpretations of $G$-functions arising from degenerating periods of smooth projective algebraic varieties. Using this, we are able to implement a strategy due to Andr\'e for bounding…

We propose an axiomatic approach towards studying unlikely intersections by introducing the framework of distinguished categories. This includes commutative algebraic groups and mixed Shimura varieties. It allows us to define all basic…

We show that the strategy of point counting in o-minimal structures can be applied to various problems on unlikely intersections that go beyond the conjectures of Manin-Mumford and Andr\'e-Oort. We verify the so-called Zilber-Pink…

We prove the Zilber-Pink conjecture for curves in $Y(1)^3$ that intersect a modular curve in the boundary. We also give an unconditional result for unlikely intersection points having few places of supersingular reduction where they are…

This is the first installment in a series of papers devoted to examining certain aspects of the asymptotic value distribution and distribution of zeros manifested by members of a broad class of linear combinations of L-functions in the…

Following our work in \cite{papas2022height}, we extend the height bounds established by Y. Andr\'e in his seminal research monograph \cite{andre1989g} for $1$-parameter families of abelian varieties defined over number fields. In our…

We investigate the relations between the rings ${\bf E}$, ${\bf G}$ and ${\bf D}$ of values taken at algebraic points by arithmetic Gevrey series of order either $-1$ ($E$-functions), $0$ (analytic continuations of $G$-functions) or $1$…

This paper has two objectives. First, we study lattices with skew-Hermitian forms over division algebras with positive involutions. For division algebras of Albert types I and II, we show that such a lattice contains an "orthogonal" basis…

We prove the Zilber--Pink conjecture for curves in $Y(1)^n$ whose Zariski closure in $(\mathbb{P}^1)^n$ passes through the point $(\infty, \ldots, \infty)$, going beyond the asymmetry condition of Habegger and Pila. Our proof is based on a…

In this paper we study definable families of functions from an ordered abelian group into various naturally arising definable quotients. We show that for an ordered abelian group $G$ and definable family of convex subgroups…

Concepts and techniques from the theory of G-structures of higher order are applied to the study of certain structures (volume forms, conformal structures, linear connections and projective structures) defined on a pseudo-Riemanniann…

In this article, we prove that the Zilber-Pink conjecture for abelian varieties over an arbitrary field of characteristic $0$ is implied by the same statement for abelian varieties over the algebraic numbers. More precisely, the conjecture…

In [90] the first-named author gave a working definition of a family of automorphic L-functions. Since then there have been a number of works [33], [107], [67] [47], [66] and especially [98] by the second and third-named authors which make…

The purpose of this paper is to study families of Artinian or one dimensional quotients of a polynomial ring $R$ with a special look to level algebras. Let $\GradAlg^H(R)$ be the scheme parametrizing graded quotients of $R$ with Hilbert…

This paper investigates the \textbf{graphical $r$-Stirling numbers of the first kind}, denoted by $\str{G}{k}$, which enumerate partitions of a vertex set $V(G)$ into $k$ disjoint cycles such that $r$ specified vertices occupy distinct…

We study relationships between certain algebraic properties of groups and rings definable in a first order structure or $*$-closed in a compact $G$-space. As a consequence, we obtain a few structural results about $\omega$-categorical rings…

We consider a family, depending on a parameter, of multiplicative extensions of an elliptic curve with complex multiplications. They form a 3-dimensional variety $G$ which admits a dense set of special curves, known as Ribet curves, which…