establish interface for instantiated classical modular polynomials #36190
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…_polynomials SageMath version 10.2.beta9, Release Date: 2023-10-30
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Documentation preview for this PR (built with commit 372ed15; changes) is ready! 🎉 |
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Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>
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Done, thank you for reviewing! |
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Thank you. |
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Thanks for implementing this! |
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…lar polynomials
Generic modular polynomials (i.e., as elements of $\mathbb Z[X,Y]$) grow
very big very quickly. However, most algorithmic applications only
require evaluations $\Phi_\ell(j,Y)$ of those polynomials. There are
algorithms to compute such evaluations faster than computing-then-
evaluating the full modular polynomial, for example due to
[Sutherland](https://arxiv.org/abs/1202.3985). PARI implements both
kinds of algorithms in `polmodular()`.
This patch adds a common interface for (1) accessing Kohel's database of
modular polynomials if available, (2) calling PARI's `polmodular()` to
compute generic modular polynomials, and (3) calling PARI's
`polmodular()` to compute instantiated modular polynomials. Results are
opportunistically cached and reused whenever this seems to make sense.
URL: sagemath#36190
Reported by: Lorenz Panny
Reviewer(s): Lorenz Panny, Travis Scrimshaw
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…olynomial`
There is a bug in the new `classical_modular_polynomial` function
created in sagemath#36190.
A simple example to reproduce the bug is the following
```
p = 2^216 * 3^137 - 1
F.<i> = GF(p^2, modulus=[1,0,1])
E = EllipticCurve(F, [0, 6, 0, 1, 0])
classical_modular_polynomial(2, E.j_invariant())
```
This will fail with a `TypeError`.
The bug is due to the interface with the pari function `polmodular`. In
particular, contrary to the [documentation](http://pari.math.u-
bordeaux.fr/dochtml/ref/Polynomials_and_power_series.html#polmodular),
the `polmodular` function will only evaluate $\Phi_\ell(j, Y)$ for
$j$-invariants that are defined over $\mathbb F_p$, and not over any
generic finite field.
If however `j.parent()` is a generic finite field, but `j` itself is
defined over the prime field, pari will evaluate that and the current
implementation fails to convert back to a sage polynomial.
The proposed fix is to cast the result of the pari call to `ZZ['Y']`,
since the result of `polmodular` currently returns a polynomial in the
correct $\mathbb Z/n\mathbb Z[Y]$.
#sd123
URL: sagemath#37094
Reported by: Riccardo Zanotto
Reviewer(s): Frédéric Chapoton, Peter Bruin, Riccardo Zanotto
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Generic modular polynomials (i.e., as elements of$\mathbb Z[X,Y]$ ) grow very big very quickly. However, most algorithmic applications only require evaluations $\Phi_\ell(j,Y)$ of those polynomials. There are algorithms to compute such evaluations faster than computing-then-evaluating the full modular polynomial, for example due to Sutherland. PARI implements both kinds of algorithms in
polmodular().This patch adds a common interface for (1) accessing Kohel's database of modular polynomials if available, (2) calling PARI's
polmodular()to compute generic modular polynomials, and (3) calling PARI'spolmodular()to compute instantiated modular polynomials. Results are opportunistically cached and reused whenever this seems to make sense.