Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Generic rank of Betti map and unlikely intersections   Let $\mathcal{A} \rightarrow S$ be an abelian scheme over an irreducible
variety over $\mathbb{C}$ of relative dimension $g$. For any simply-connected
subset $\Delta$ of $S^{\mathrm{an}}$ one can define the Betti map from
$\mathcal{A}_{\Delta}$ to $\mathbb{T}^{2g}$, the real torus of dimension $2g$,
by identifying each closed fiber of $\mathcal{A}_{\Delta} \rightarrow \Delta$
with $\mathbb{T}^{2g}$ via the Betti homology. Computing the generic rank of
the Betti map restricted to a subvariety $X$ of $\mathcal{A}$ is useful to
study Diophantine problems, e.g. proving the Geometric Bogomolov Conjecture
over characteristic $0$ and studying the relative Manin-Mumford conjecture. In
this paper we give a geometric criterion to detect this rank. As an application
we show that it is maximal after taking a large enough fibered power (if $X$
satisfies some conditions): it is an important step to prove the bound for the
number of rational points on curves [DGH20]. Another application is to answer a
question of Andr\'e-Corvaja-Zannier and improve a result of Voisin. We also
systematically study its link with the relative Manin-Mumford conjecture,
reducing the latter to a simpler conjecture. Our tools are functional
transcendence and unlikely intersections for mixed Shimura varieties.

14
The categories of this paper are: math-ph
36
The categories of this paper are: cmp-lg
7
The categories of this paper are: hep-th
34
The categories of this paper are: chem-ph
28
The categories of this paper are: alg-geom
22
The categories of this paper are: chao-dyn
24
The categories of this paper are: dg-ga
29
The categories of this paper are: funct-an
4
The categories of this paper are: cond-mat
0
The categories of this paper are: hep-ph
32
The categories of this paper are: atom-ph
19
The categories of this paper are: eess
33
The categories of this paper are: plasm-ph
31
The categories of this paper are: ao-sci
20
The categories of this paper are: acc-phys
15
The categories of this paper are: nucl-ex