## Algebra & Number Theory

### Specializations of elliptic surfaces, and divisibility in the Mordell–Weil group

Patrick Ingram

#### Abstract

Let $ℰ→C$ be an elliptic surface defined over a number field $k$, let $P:C→ℰ$ be a section, and let $ℓ$ be a rational prime. We bound the number of points of low algebraic degree in the $ℓ$-division hull of $P$ at the fibre $ℰt$. Specifically, for $t∈C(k̄)$ with $[k(t):k]≤B1$ such that $ℰt$ is nonsingular, we obtain a bound on the number of $Q∈ℰt(k̄)$ such that $[k(Q):k]≤B2$, and such that $ℓnQ=Pt$ for some $n≥1$. This bound depends on $ℰ$, $P$, $ℓ$, $B1$, and $B2$, but is independent of $t$.

#### Article information

Source
Algebra Number Theory, Volume 5, Number 4 (2011), 465-493.

Dates
Revised: 10 March 2010
Accepted: 21 August 2010
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.ant/1513882223

Digital Object Identifier
doi:10.2140/ant.2011.5.465

Mathematical Reviews number (MathSciNet)
MR2870098

Zentralblatt MATH identifier
1244.11061

Subjects
Secondary: 14J27: Elliptic surfaces 14G05: Rational points

#### Citation

Ingram, Patrick. Specializations of elliptic surfaces, and divisibility in the Mordell–Weil group. Algebra Number Theory 5 (2011), no. 4, 465--493. doi:10.2140/ant.2011.5.465. https://projecteuclid.org/euclid.ant/1513882223

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