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Mod-l-galois-representations

Where is what?

code/mfmodell

Contains Sage + Magma code to compute mod ell reductions of modular forms

  • code/mfmodell/modell.py: run on legendre.mit.edu in lmfdb directory see comments in code for detailed instructions

  • code/mfmodell/bigspaces.m: requires mf.m from CMFs/magma. To run this, add a symlink to it in CMFs/magma/ and run magma there.

data/mfmodell

Contains data output from modell.py: data/mfmodell/ for ell=0,2,3,5,7 contains data in characteristic ell

ell=2

  • data/mfmodell/2/1-1000.txt contains 12739 mod 2 forms from levels 1-1000 (1960 excluding repeats) format: one per line, fields separated by :, listed in code/mfmodell/modell.py's nf_to_string(). NB if a newform has several different mod-ell reductions this is not recorded in the labelling in this file

  • data/mfmodell/2/x1-1000.txt contains 1368 undecomposed spaces of dim>50, not processed for mod 2 forms

  • data/mfmodell/2/mod2bb1000.txt contains BlackBox output for 12739 mod 2 reductions:

    • one line each with fields separated by ":".
    • First field is the modular form label with additional "-1", "-2", etc to distinguish multiple reductions of the same modular form.
    • Second field is "reducible" or "S3" or "C3".
    • Third field (for irreducibles) is a cubic defining the splitting field.

    e.g.

    5.4.a.a-1:reducible 11.2.a.a-1:S3:x^3-x^2-x-1 49.4.a.c-1:C3:x^3-7*x-7

  • data/mfmodell/2/mod2dim2reps.yxy is a dump from the database modlgal table of rows with ell=2 and dim=2.

  • data/mfmodell/2/label_map_2.txt has one line for each database entry for ell=2 showing which modular forms match it (a_p for good p<100 and splitting fields match).

  • data/mfmodell/2/reverse_label_map_2.txt: inverse of previous.

ell=3

  • data/mfmodell/3/1-1000.txt contains 14293 mod 3 forms from levels 1-1000 (8008 excluding repeats)

  • data/mfmodell/3/x1-1000.txt contains 2254 undecomposed spaces of dim>50, not processed for mod 3 forms

  • data/mfmodell/3/mod3bb1000.txt contains BlackBox output for 24569 mod 3 reductions:

    • one line each with fields separated by ":".
    • First field is the modular form label with additional "-1", "-2", etc to distinguish multiple reductions of the same modular form.
    • 2nd field is "reducible" or "GL(2,3)" or "Nn" or "Ns" (image).
    • 3rd field (for irreducibles) is an octic defining the splitting field.
    • 4th field (for irreducibles) is "S4" or "V4" or "D4" (projective image)
    • 5th field (for irreducibles) is a quartic defining the projective splitting field.

    5.4.a.a-1:GL(2,3):x^8-4x^7+7x^6-7x^5+4x^4-x^3-4x^2+4x-1:S4:x^4-x^3+4x-1 7.3.b.a-1:Ns:(x^4-x^3-3x^2-x+1)(x^4-x^3+2x+1):V4:(x^2-21)(x^2+3) 7.4.a.a-1:GL(2,3):x^8-x^7-3x^6+2x^5+4x^4+3x^3-5x^2-7x-3:S4:x^4-x^3-3x^2-7*x+1 8.3.d.a-1:reducible

  • data/mfmodell/3/mod3dim2reps.yxy is a dump from the database modlgal table of rows with ell=3 and dim=2.

  • data/mfmodell/3/label_map_3.txt has one line for each database entry for ell=3 showing which modular forms match it (a_p for good p<100 and splitting fields and projective splitting fields match).

  • data/mfmodell/3/reverse_label_map_3.txt: inverse of previous.

ell=5

  • data/mfmodell/5/1-1000.txt contains 24851 mod 5 forms from levels 1-1000 (21177 excluding repeats)
  • data/mfmodell/5/x1-1000.txt contains 3145 undecomposed spaces of dim>50, not processed for mod 5 forms

ell=7

  • data/mfmodell/7/1-1000.txt contains 32761 mod 7 forms from levels 1-1000 (30425 excluding repeats)
  • data/mfmodell/7/x1-1000.txt contains 4213 undecomposed spaces of dim>50, not processed for mod 7 forms

ell=0

  • Contains magma output for many spaces of dimention 21-50 used in computing the above, but I stopped adding these to the repo as they were getting too big (>5G in all).