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Mathematics > Algebraic Geometry

arXiv:1812.06495 (math)
[Submitted on 16 Dec 2018]

Title:Period relations for Riemann surfaces with many automorphisms

Authors:Luca Candelori, Jack Fogliasso, Christopher Marks, Skip Moses
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Abstract:By employing the theory of vector-valued automorphic forms for non-unitarizable representations, we provide a new bound for the number of linear relations with algebraic coefficients between the periods of an algebraic Riemann surface with many automorphisms. The previous best-known general bound for this number was the genus of the Riemann surface, a result due to Wolfart. Our new bound significantly improves on this estimate, and it can be computed explicitly from the canonical representation of the Riemann surface. As observed by Shiga and Wolfart, this bound may then be used to estimate the dimension of the endomorphism algebra of the Jacobian of the Riemann surface. We demonstrate with a few examples how this improved bound allows one, in some instances, to actually compute the dimension of this endomorphism algebra, and to determine whether the Jacobian has complex multiplication.
Subjects: Algebraic Geometry (math.AG); Number Theory (math.NT)
Cite as: arXiv:1812.06495 [math.AG]
  (or arXiv:1812.06495v1 [math.AG] for this version)
  https://doi.org/10.48550/arXiv.1812.06495

Submission history

From: Luca Candelori [view email]
[v1] Sun, 16 Dec 2018 16:12:04 UTC (17 KB)
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