count_points breaks on even-degree hyperelliptic curves

[kedlaya]

The following is currently broken:

sage: x = polygen(GF(4099))
sage: H = HyperellipticCurve(x^6 + x + 1)
sage: H.count_points(1)
...
ValueError: Q must be a monic polynomial of odd degree >= 3

The problem is that ultimately this is trying to call hypellfrob, which is only implemented for odd degree polynomials. One can patch around this using either PARI's hyperellcharpoly or the naive point counting code.

In the longer term, some more intelligent switching needs to be implemented for this method, as well as the related zeta_function and zeta_series methods; these do not all optimize the same way. See #965, #16931, #18916, #20309 for related discussions.

[kedlaya]

There is a commit on #18916 that addresses this breakage.

[kedlaya]

Please close after #18916 is merged.

[kedlaya]

As reported by cremona, this is not completely resolved by #18916 after all. This call syntax should also be covered:

sage: F=GF(23) 
sage: x=polygen(F) 
sage: C=HyperellipticCurve(x^8+1) 
sage: C.cardinality()
... 
ValueError: Q must be a monic polynomial of odd degree >= 3 

[cremona]

Thanks for the update -- I had not noticed this ticket before.

[kedlaya]

This is ultimately a dispatching issue: the code in #18916 includes some updated decision-making about what algorithm to use, but the cardinality method doesn't currently route through this decision point. This should be easy to fix!

[kedlaya]

Apparently what I did for count_points was the trivial patch: in the even-degree case, just drop down to counting points. This is of course not the correct solution long-term, but it does fix the reported bug. Anyway, I copied that trivial fix over to cardinality.

---

New commits:

​dabfdd8

Avoid breakage of cardinality for even-degree curves

[jpflori]

Ok let's go with this at the moment.