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L-Functions and Modular Forms Database
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List reflex fields of CM fields #4902

Open AndrewVSutherland opened 2 years ago

AndrewVSutherland commented 2 years ago

For number fields that are CM fields, it would be nice to list the possible CM types (choice of embedding for each complex place) and the corresponding reflex fields (even just a list of the reflex fields would be useful, they should all be in the database).

rbommel commented 2 years ago

Note to anyone who wants to think about this in the future (maybe myself): some fields to start trying to compute this for could be https://beta.lmfdb.org/NumberField/14.0.121781095161992523.1 or https://beta.lmfdb.org/NumberField/36.0.2167466163148151967744828835955985745949336362631879785820717056.1.

rbommel commented 2 years ago

Today, we discussed only computing this for CM fields of degree at most 12, and to only compute equations for the reflex field is that field itself has degree at most 36. We also discussed computing whether or not the reflex field is CM, but after a bit more consideration, I think they are always CM.

I did a computation on lovelace, and everything finished within 50 cpu hours or so, so this is definitely feasible.

AndrewVSutherland commented 2 years ago

The reflex field of a CM type is always a CM field, see Proposition 1.18 of Milne's notes.

The reflex field of a Shimura variety can be totally real or CM, but here we are talking about reflex fields of CM types (associated to CM abelian varieties).

JohnCremona commented 1 year ago

I am working with Marco Streng and Pinar Kilicer at COUNT to display the CM types for each reflex field, not just the multiplicities. For this we are adding 2 columns to nf_fields to hold the real and imaginary parts of the embeddings of the field generator (only for the CM fields of degree up to 12 at first), and one column to nf_fields_reflex to hold a list of CM-types, each encoded as a string like "+-+-". This will require more computation by @rbommel.

jwj61 commented 1 year ago

Is this done and can be closed?

JohnCremona commented 1 year ago

Is this done and can be closed?

No. Work on it started at Luminy in Feb/March, and a branch exists with a table with a few test rows, but the work still needs to be done to do the remaining computations.

jwj61 commented 1 year ago

The table has almost 3 million entries.

JohnCremona commented 1 year ago

I'm clearly out of date on this one!

rbommel commented 1 year ago

Whether this is still the same issue or not, we are working on an upgrade for this, including complex embeddings of the reflex fields, Galois orbits, and reflex reflex fields (which are 'fields of definition' for the CM types). For more info, I refer to the discussion on the COUNT Zulip.