using pari for elliptic and Eisenstein L-series

[fchapoton]

trying to move away from Dokchitser's auld scripts

• no longer allowing their use for elliptic curves
• same in L-function of number fields
• same in L-function of the modular form Delta
• trying to get rid of them for Eisenstein L-functions
• trying to get rid of them for Gross-Zagier L-functions

help welcome !

Note: the handling of general modular forms is best kept for another pull request.

📝 Checklist

• The title is concise and informative.
• The description explains in detail what this PR is about.
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[fchapoton]

@JohnCremona : I would appreciate help from Lmfdb and Pari experts on the case of Eisenstein series

[fchapoton]

Is this a failure or an improvement ?

File "src/sage/modular/modform/eis_series.py", line 422, in sage.modular.modform.eis_series.?
Failed example:
    L(2)
Expected:
    -5.0235535164599797471968418348135050804419155747868718371029
Got:
    -5.0235535164599799477225675531508296063579361580211036347215

ANSWER: this was a failure

[JohnCremona]

I'll look but not until Thursday

[fchapoton]

now there are some issues with the handling of bit precision..

and more precisely with the number of terms required, that are insufficient to check the functional equation

[fchapoton]

I am also annoyed because for Eisenstein series the only path I found to relative success was to pass the residue of the uncompleted L-function to pari, instead of the simpler residues of the completed L-function.

[fchapoton]

now remains only a failure

Expected:
    -5.0235535164599797471968418348135050804419155747868718371029
Got:
    -5.0235535164599800135315437325373372537800575000382643457910

which is apparently really a failure, as the expected value is

sage: f=zeta(2)*zeta(-17)
sage: f.n(200)
-5.0235535164599797471968418348135050804419155747868718371029

[fchapoton]

It seems to me that the handling of precision is still faulty, but I cannot investigate further.

[user202729]

the precision issues are mostly correct already? maybe just one # rel tol 1e-14 at L(2) + E.lseries_gross_zagier(A^2)(2) (if no explicit precision parameter are passed then one can expect roughly 53 bits)

[fchapoton]

@JohnCremona and @roed314 : would you please advertise to LMFDB people what I am trying to do here, namely get rid of the original Dokchitser scripts in favor of PARI ?

[github-actions]

Documentation preview for this PR (built with commit abe56d2; changes) is ready! 🎉
This preview will update shortly after each push to this PR.

[JohnCremona]

Documentation preview for this PR (built with commit af8a5e5; changes) is ready! 🎉 This preview will update shortly after each push to this PR.

I will leave this to @roed314 since he is, and I am no longer, a managing editor of LMFDB

[roed314]

I've advertised this PR on the LMFDB Zulip, and also sent @fchapoton an invitation to join (let me know if I got your address wrong: I took it from the Sage Zulip).