Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Strong minimality and the j-function   We show that the order three algebraic differential equation over ${\mathbb
Q}$ satisfied by the analytic $j$-function defines a non-$\aleph_0$-categorical
strongly minimal set with trivial forking geometry relative to the theory of
differentially closed fields of characteristic zero answering a long-standing
open problem about the existence of such sets. The theorem follows from Pila's
modular Ax-Lindemann-Weierstrass with derivatives theorem using Seidenberg's
embedding theorem and a theorem of Nishioka on the differential equations
satisfied by automorphic functions. As a by product of this analysis, we obtain
a more general version of the modular Ax-Lindemann-Weierstrass theorem, which,
in particular, applies to automorphic functions for arbitrary arithmetic
subgroups of $SL_2 ({\mathbb Z})$. We then apply the results to prove effective
finiteness results for intersections of subvarieties of products of modular
curves with isogeny classes. For example, we show that if $\psi:{\mathbb P}^1
\to {\mathbb P}^1$ is any non-identity automorphism of the projective line and
$t \in {\mathbb A}^1({\mathbb C}) \smallsetminus {\mathbb A}^1({\mathbb
Q}^\text{alg})$, then the set of $s \in {\mathbb A}^1({\mathbb C})$ for which
the elliptic curve with $j$-invariant $s$ is isogenous to the elliptic curve
with $j$-invariant $t$ and the elliptic curve with $j$-invariant $\psi(s)$ is
isogenous to the elliptic curve with $j$-invariant $\psi(t)$ has size at most
$36^7$. In general, we prove that if $V$ is a Kolchin-closed subset of
${\mathbb A}^n$, then the Zariski closure of the intersection of $V$ with the
isogeny class of a tuple of transcendental elements is a finite union of weakly
special subvarieties. We bound the sum of the degrees of the irreducible
components of this union by a function of the degree and order of $V$.

35
The categories of this paper are: mtrl-th
26
The categories of this paper are: bayes-an
15
The categories of this paper are: nucl-ex
4
The categories of this paper are: cond-mat
30
The categories of this paper are: q-alg
5
The categories of this paper are: gr-qc
22
The categories of this paper are: chao-dyn
25
The categories of this paper are: patt-sol
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The categories of this paper are: alg-geom
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The categories of this paper are: supr-con
7
The categories of this paper are: hep-th
20
The categories of this paper are: acc-phys
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The categories of this paper are: solv-int
10
The categories of this paper are: q-bio
3
The categories of this paper are: physics
8
The categories of this paper are: hep-ex