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L-Functions and Modular Forms Database
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Twisted base change for HMFs and BMFs #2132

Open jvoight opened 7 years ago

jvoight commented 7 years ago

This notion is more general than base change on the nose and is important for all sorts of reasons.

sanni85 commented 6 years ago

Is this the \sigma-\tau aligned notion? i.e. the Galois action on the base number field (which I suppose is Galois) and the choice of embedding of the Hecke eigenvalue field leave the form stable : a{\tau(\mathfrak{p}}=\sigma(a{\mathfrak{p})

Do you want to have a search on this? or add info on the pages?

JohnCremona commented 6 years ago

I have computed this for "my" BMFs, i.e. the ones with rational eigenvalues. See for example http://beta.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/67600.6/d/ http://beta.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/16384.1/d/ http://beta.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/57967.9/a/ but to find some of these examples I needed to do a low-level search. I store a field 'bc' which is 0 for not-a-base-change-even-up-to-twist, >0 for base-change, <0 for twist-of-base-change, and the absolute value tells you the coefficient field of the underlying form.

jvoight commented 6 years ago

Samuele: no, twisted base change means just that, a twist of a base change.

The "sigma-tau-aligned" notion we discussed in Oldenburg is something different: sometimes such forms will arise from twisted base change, sometimes not.

sanni85 commented 6 years ago

Got it, but I think both notions are important enough to deserve more visibility, i.e. searches and displays.