Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Self-intersection of the relative dualizing sheaf on modular curves
  $X_1(N)$   Let $N$ be an odd and squarefree positive integer divisible by at least two
relative prime integers bigger or equal than 4. Our main theorem is an
asymptotic formula solely in terms of $N$ for the stable arithmetic
self-intersection number of the relative dualizing sheaf for modular curves
$X_1(N)/ \mathbb{Q}$. From our main theorem we obtain an asymptotic formula for
the stable Faltings height of the Jacobian $J_1(N) / \mathbb{Q}$ of $X_1(N)/
\mathbb{Q}$, and, for sufficiently large N, an effective version of Bogomolov's
conjecture for $X_1(N) / \mathbb{Q}$.

5
The categories of this paper are: gr-qc
35
The categories of this paper are: mtrl-th
4
The categories of this paper are: cond-mat
1
The categories of this paper are: cs
28
The categories of this paper are: alg-geom
31
The categories of this paper are: ao-sci
6
The categories of this paper are: astro-ph
13
The categories of this paper are: quant-ph
33
The categories of this paper are: plasm-ph
0
The categories of this paper are: hep-ph
20
The categories of this paper are: acc-phys
32
The categories of this paper are: atom-ph
26
The categories of this paper are: bayes-an
37
The categories of this paper are: supr-con
8
The categories of this paper are: hep-ex
34
The categories of this paper are: chem-ph