Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The average number of integral points on elliptic curves is bounded   We prove that, when elliptic curves $E/\mathbb{Q}$ are ordered by height, the
average number of integral points $\#|E(\mathbb{Z})|$ is bounded, and in fact
is less than $66$ (and at most $\frac{8}{9}$ on the minimalist conjecture). By
"$E(\mathbb{Z})$" we mean the integral points on the corresponding quasiminimal
Weierstrass model $E_{A,B}: y^2 = x^3 + Ax + B$ with which one computes the
na\"{\i}ve height. The methods combine ideas from work of Silverman, Helfgott,
and Helfgott-Venkatesh with work of Bhargava-Shankar and a careful analysis of
local heights for "most" elliptic curves. The same methods work to bound
integral points on average over the families $y^2 = x^3 + B$, $y^2 = x^3 + Ax$,
and $y^2 = x^3 - D^2 x$.

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The categories of this paper are: quant-ph
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The categories of this paper are: astro-ph
32
The categories of this paper are: atom-ph
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The categories of this paper are: adap-org
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The categories of this paper are: q-bio
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The categories of this paper are: bayes-an
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The categories of this paper are: alg-geom
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The categories of this paper are: patt-sol
0
The categories of this paper are: hep-ph
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The categories of this paper are: eess
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The categories of this paper are: q-fin
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The categories of this paper are: plasm-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: math-ph
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The categories of this paper are: solv-int
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The categories of this paper are: acc-phys