Related papers: The large Galois orbits conjecture under multiplic…

We establish Large Galois orbits conjectures for points of unlikely intersections of curves in $Y(1)^n$, upon assumptions on the intersection of such curves with the boundary $X(1)^n\backslash Y(1)^n$, in the Zilber-Pink setting. As a…

The Zilber-Pink conjecture predicts that an algebraic curve in $\mathcal{A}_2$ has only finitely many intersections with the special curves, unless it is contained in a proper special subvariety. Under a large Galois orbits hypothesis, we…

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 prove quantitative versions of Borel and Harish-Chandra's theorems on reduction theory for arithmetic groups. Firstly, we obtain polynomial bounds on the lengths of reduced integral vectors in any rational representation of a reductive…

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…

Given a smooth proper family $g : X \to S$ of surfaces over a number field $K \subset \mathbb{C}$, with $S$ an irreducible curve and $\eta \in S$ its generic point, we consider the general problem of constraining the locus $\textrm{NL}(S)$…

We show that the Lang-Trotter conjecture for pairs of elliptic curves implies new cases of the Zilber-Pink conjecture for curves in $\mathcal{A}_3$. Unlike previous results for curves in $\mathcal{A}_g$, our result does not rely on any…

We prove the Zilber-Pink conjecture to the intersection of an irreducible Hodge generic algebraic subvariety $ V \subset \mathcal{A}_g$ with special subvarieties of all simple PEL types other than $\mathbb{Z}$, under the assumption of the…

We obtain results on the so-called Andre-Pink-Zannier conjecture which is a special case of a the Zilber-Pink conjecture on unlikely intersections in Shimura varieties. Our methods rely on an ergodic theorem of Richard-Zamojski and we are…

We give a proof of the Andr\'e-Oort conjecture for $\mathcal{A}_g$ - the moduli space of principally polarized abelian varieties. In particular, we show that a recently proven `averaged' version of the Colmez conjecture yields lower bounds…

We introduce and study the notion of a generalised Hecke orbit in a Shimura variety. We define a height function on such an orbit and study its properties. We obtain a lower bounds for the size of Galois orbits of points in a generalised…

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…

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…

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…

Consider a smooth irreducible Hodge generic curve $S$ defined over $\bar{\Q}$ in the Torelli locus $T_g\subset \mathcal{A}_g$. We establish Zilber-Pink-type statements for such curves depending on their intersection with the boundary of the…

We explore a strong categorical correspondence between isomorphism classes of sheaves of arbitrary rank on a given algebraic curve and twisted pairs on another algebraic curve, mostly from a linear-algebraic standpoint. In a particular…

We introduce a ``hybrid'' conjecture which is a common generalisation of the Andr\'e-Oort conjecture and the Andr\'e-Pink-Zannier conjecture and we prove that it is a consequence of the Zilber-Pink conjecture. We also show that our hybrid…

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…

We prove Breuil's lattice conjecture for higher Hodge-Tate weights in the case of $\mathrm{GL}_2(K)$ where $K$ is an unramified extension of $\mathbb{Q}_p$. More precisely, under some genericity conditions, we show that the lattice inside a…

We formulate a tropical analogue of Grothendieck's section conjecture: that for every stable graph G of genus g>2, and every field k, the generic curve with reduction type G over k satisfies the section conjecture. We prove many cases of…