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On finiteness conjectures for endomorphism algebras of abelian surfaces

Author
Bruin, N.; Flynn, E. Victor; Gonzalez, J.; Rotger, V.
Type of activity
Journal article
Journal
Mathematical proceedings of the Cambridge Philosophical Society
Date of publication
2006-11
Volume
141
Number
3
First page
383
Last page
408
DOI
https://doi.org/10.1017/S0305004106009613 Open in new window
Project funding
Curvas de Shimura, Superficies Abelianas y Thetanullwerte
Repository
https://arxiv.org/pdf/math/0312443.pdf Open in new window
URL
https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/on-finiteness-conjectures-for-endomorphism-algebras-of-abelian-surfaces/C040C06CCB79728394F7BF6D4FCCCCC7 Open in new window
Abstract
It is conjectured that there exist only finitely many isomorphism classes of endomorphism algebras of abelian varieties of bounded dimension over a number field of bounded degree. We explore this conjecture when restricted to quaternion endomorphism algebras of abelian surfaces of GL(2)-type over Q by giving a moduli interpretation which translates the question into the diophantine, arithmetic of Shimura curves embedded in Hilbert surfaces. We address the resulting problems on these curves by lo...
Keywords
Chabauty methods using elliptic curves, Heegner points, Hilbert surfaces, Shimura curves
Group of research
TN - Number Theory Research Group

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