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Congruences for newforms and the index of the Hecke algebra

Authors: Scott Ahlgren and Jeremy Rouse
Journal: Proc. Amer. Math. Soc. 139 (2011), 1247-1261
MSC (2010): Primary 11F33; Secondary 11F30
Published electronically: October 1, 2010
MathSciNet review: 2748418
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Abstract: We study congruences between newforms in the spaces $ S_4(\Gamma_0(p), \overline{\mathbb{Z}}_p)$ for primes $ p$. Under a suitable hypothesis (which is true for all $ p<5000$ with the exception of $ 139$ and $ 389$) we provide a complete description of the congruences between these forms, which leads to a formula (conjectured by Calegari and Stein) for the index of the Hecke algebra $ \mathbb{T}_{\mathbb{Z}_p}$ in its normalization. Since the hypothesis is amenable to computation, we are able to verify the conjectured formula for $ p<5000$. In 2004 Calegari and Stein gave a number of conjectures which provide an outline for the proof of this formula, and the results here clarify the dependencies between the various conjectures. Finally, we discuss similar results for the spaces $ S_6(\Gamma_0(p), \overline{\mathbb{Z}}_p)$.

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Additional Information

Scott Ahlgren
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801

Jeremy Rouse
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
Address at time of publication: Department of Mathematics, Wake Forest University, Winston-Salem, North Carolina 27109

Received by editor(s): April 20, 2010
Published electronically: October 1, 2010
Additional Notes: The second author was supported by NSF grant DMS-0901090
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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