- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:11
Journal of the European Mathematical Society
J. Eur. Math. Soc.
JEMS
1435-9855
1435-9863
General
10.4171/JEMS
http://www.ems-ph.org/doi/10.4171/JEMS
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
17
2015
9
Torsion points on families of simple abelian surfaces and Pell's equation over polynomial rings (with an appendix by E. V. Flynn)
David
Masser
Universität Basel, BASEL, SWITZERLAND
Umberto
Zannier
Scuola Normale Superiore, PISA, ITALY
Torsion point, abelian surface scheme, Pell equation, Jacobian variety, Chabauty’s theorem
In recent papers we proved a special case of a variant of Pink's Conjecture for a variety inside a semiabelian scheme: namely for any curve inside anything isogenous to a product of two elliptic schemes. Here we go beyond the elliptic situation by settling the crucial case of any simple abelian surface scheme defined over the field of algebraic numbers, thus confirming an earlier conjecture of Shou-Wu Zhang. This is of particular relevance in the topic, also in view of very recent counterexamples by Bertrand. Furthermore there are applications to the study of Pell equations over polynomial rings; for example we deduce that there are at most finitely many complex $t$ for which there exist $A,B \neq 0$ in ${\mathbb C}[X]$ with $A^2 – DB^2 = 1$ for $D = X^6 + X + t$. We also consider equations $A^2 – DB^2 = c'X + c$, where the situation is quite different.
Number theory
Algebraic geometry
2379
2416
10.4171/JEMS/560
http://www.ems-ph.org/doi/10.4171/JEMS/560