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Level Raising and Lowering Modulo $ p^n$

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Ribet's theorems about level raising and level lowering have been central in a huge amount of modern work on modular forms. For example, they play a famous role in the proof of Fermat's last theorem. You should read about these theorems somewhere. One introduction is [RS01].

Diamond (see [Dia95], etc.) and Diamond-Taylor (in their ``Nonoptimal levels'' paper), and Russ Mann in his Ph.D. thesis, have all also done important work related to level lowering and raising.

Unfortunately, it seems that nobody has proved or even formulated a conjectural analogue of these results for congruences modulo $ p^n$ between eigenforms. There is work about higher congruences in that comes up when studying $ p$ -adic modular forms (see, e.g., [Col03]).

Some7.1 have expressed doubt that there can even be a good level raising or lowering theorem modulo $ p^n$ .

Remark 7.0.1 (From Richard Taylor.)   I share Jetchev's pessimism. You can presumably come up with a criterion for a congruence to a new mod $ p^n$ eigenform (ie a homomorphism from the new part of the Hecke alg to $ \mathbb{Z}/p^n\mathbb{Z}$ ). But as soon as $ n>1$ this does not imply that there is a characteristic 0 newform which reduces to this modulo $ p^n$ . Rather the mod $ p^n$ eigenform could result from several newforms congruent to the original form modulo $ p$ .



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William Stein 2006-10-20