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≠0 in C[X] with A2–DB2=1 for D=X6+X+t. We also consider equations A2–DB2=c′X+c, where the situation is quite different.

Torsion points on families of simple abelian surfaces and Pell's equation over polynomial rings (with an appendix by E. V. Flynn)

Zannier, Umberto;
2015

Abstract

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≠0 in C[X] with A2–DB2=1 for D=X6+X+t. We also consider equations A2–DB2=c′X+c, where the situation is quite different.
Settore MAT/03 - Geometria
Torsion point, abelian surface scheme, Pell equation, Jacobian variety, Chabauty’s theorem
File in questo prodotto:
File Dimensione Formato  
JEMS-2015-017-009-10.pdf

accesso aperto

Tipologia: Published version
Licenza: Accesso gratuito (sola lettura)
Dimensione 267.84 kB
Formato Adobe PDF
267.84 kB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11384/69076
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 23
  • ???jsp.display-item.citation.isi??? 22
social impact