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Factoring newparts of Jacobians of certain modular curves

Authors: M. Ram Murty and Kaneenika Sinha
Journal: Proc. Amer. Math. Soc. 138 (2010), 3481-3494
MSC (2010): Primary 11F11, 11F25, 11F30, 11G10, 11G18
Published electronically: May 5, 2010
MathSciNet review: 2661548
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Abstract: We prove a conjecture of Yamauchi which states that the level $ N$ for which the new part of $ J_0(N)$ is $ \mathbb{Q}$-isogenous to a product of elliptic curves is bounded. We also state and partially prove a higher-dimensional analogue of Yamauchi's conjecture. In order to prove the above results, we derive a formula for the trace of Hecke operators acting on spaces $ S^{new}(N,k)$ of newforms of weight $ k$ and level $ N.$ We use this trace formula to study the equidistribution of eigenvalues of Hecke operators on these spaces. For any $ d\geq 1,$ we estimate the number of normalized newforms of fixed weight and level, whose Fourier coefficients generate a number field of degree less than or equal to $ d.$

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Additional Information

M. Ram Murty
Affiliation: Department of Mathematics and Statistics, Queen’s University, Jeffery Hall, Kingston, Ontario, Canada K7L 3N6

Kaneenika Sinha
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

Keywords: Traces of Hecke operators, Fourier coefficients of modular cusp forms, modular curves, Jacobian varieties
Received by editor(s): August 4, 2009
Received by editor(s) in revised form: January 7, 2010
Published electronically: May 5, 2010
Additional Notes: The first author was partially supported by an NSERC grant.
The second author was partially supported by a PIMS fellowship.
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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