sums of elliptic-curve morphisms

[yyyyx4]

Here we add a new EllipticCurveHom child class representing formal sums of elliptic-curve morphisms with the same domain and codomain. One of the main features is a conversion function from formal sums to a more explicit representation of the same morphism as an isogeny chain, which will come in handy for endomorphism-ring-related computations.

In the process, we also generalize point_of_order() from primes to prime powers.

There is certainly a lot of potential for optimizations; some ideas are mentioned in the code. Since I think it's fair to say that this version is already much better than what we currently have for sums of isogenies (i.e., nothing), I'd suggest merging this as is and dealing with bottlenecks later as they arise.

Cc: @JohnCremona @defeo @GiacomoPope @remyoudompheng

[JohnCremona]

I can review this, but not until Tuesday, sorry. Feel free to proceed without me!

[JohnCremona]

I have made a few comments, at least one of which requires a change, and will think further about the question of computing deg(phi+psi) more directly.

[JohnCremona]

I feel sure that there must be a way to compute the degree of the sum of two morphisms; if so you caould call it repeateldy here. But I'll need to think about that a bit more.

[yyyyx4]

I suppose what definitely works is the identity $\deg(\varphi+\psi) = \deg(\varphi)+\deg(\psi)+\mathrm{tr}(\varphi\widehat\psi)$, now that we have #35546. But there should be cases where computing multiple such trace pairings will certainly be slower than the current one-shot method, so this might require some thinking...

[JohnCremona]

My stupid mistake, apologies.

…

On Tue, 7 Nov 2023, 18:56 Lorenz Panny, ***@***.***> wrote: ***@***.**** commented on this pull request. ------------------------------ In src/sage/schemes/elliptic_curves/ell_point.py <#36637 (comment)>: > @@ -1359,9 +1359,10 @@ def set_order(self, value=None, *, multiple=None, check=True): raise ValueError('Value %s illegal for point order' % value) E = self.curve() q = E.base_ring().cardinality() - low, hi = Hasse_bounds(q) - if value > hi: - raise ValueError('Value %s illegal: outside max Hasse bound' % value) + if q < oo: + _, hi = Hasse_bounds(q) + if value > hi: Really? It looks correct to me; for example, Hasse_bounds(101) returns $(82,122)$. — Reply to this email directly, view it on GitHub <#36637 (comment)>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/AAXU32LPDLC7KEDDUE7266TYDJ76BAVCNFSM6AAAAAA64LT3T6VHI2DSMVQWIX3LMV43YUDVNRWFEZLROVSXG5CSMV3GSZLXHMYTOMJYGQ4TCNJQGE> . You are receiving this because you were mentioned.Message ID: ***@***.***>

[JohnCremona]

Yes, that's what I was thinking of. Without experimenting I cannot tell which would be fastest.

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On Tue, 7 Nov 2023, 18:55 Lorenz Panny, ***@***.***> wrote: ***@***.**** commented on this pull request. ------------------------------ In src/sage/schemes/elliptic_curves/hom_sum.py <#36637 (comment)>: > + :: + + sage: E = EllipticCurve(GF(443), [1,1]) + sage: pi = E.frobenius_endomorphism() + sage: m1 = E.scalar_multiplication(1) + sage: (pi - m1)._degree_bound() + 486 + sage: (pi - m1).degree() + 433 + """ + B = ZZ(0) + for phi in self._phis: + B += phi.degree() + 2*sqrt(B * phi.degree()) + return integer_floor(B) + + def _compute_degree(self): I suppose what definitely works is the identity $\deg(\varphi+\psi) = \deg(\varphi)+\deg(\psi)+\mathrm{tr}(\varphi\widehat\psi)$, now that we have #35546 <#35546>. But there should be cases where computing multiple such trace pairings will certainly be slower than the current one-shot method, so this might require some thinking... — Reply to this email directly, view it on GitHub <#36637 (comment)>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/AAXU32JE3LKJP4YTVV4UOIDYDJ72RAVCNFSM6AAAAAA64LT3T6VHI2DSMVQWIX3LMV43YUDVNRWFEZLROVSXG5CSMV3GSZLXHMYTOMJYGQ4TAMJVHA> . You are receiving this because you were mentioned.Message ID: ***@***.***>

[JohnCremona]

OK!

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On Tue, 7 Nov 2023, 19:00 Lorenz Panny, ***@***.***> wrote: ***@***.**** commented on this pull request. ------------------------------ In src/sage/schemes/elliptic_curves/hom_sum.py <#36637 (comment)>: > + M = self._degree_bound() + 1 + deg = Mod(0,1) + for l in Primes(): + if deg.modulus() >= M: + break + try: + P = point_of_order(self._domain, l) + except ValueError: + continue # supersingular and l == p + + Q = self.dual()._eval(self._eval(P)) + d = discrete_log(Q, P, ord=l, operation='+') + deg = deg.crt(Mod(d, l)) + + self._degree = ZZ(deg) + self.dual()._degree = self._degree In this class .dual() is marked @cached_method, so the same object will be returned on subsequent calls. — Reply to this email directly, view it on GitHub <#36637 (comment)>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/AAXU32KH7PYM3VZODILVCMDYDKALNAVCNFSM6AAAAAA64LT3T6VHI2DSMVQWIX3LMV43YUDVNRWFEZLROVSXG5CSMV3GSZLXHMYTOMJYGQ4TQOBSGU> . You are receiving this because you were mentioned.Message ID: ***@***.***>

[github-actions]

Documentation preview for this PR (built with commit fbe441e; changes) is ready! 🎉

[JohnCremona]

All the suggestions I made in my review and follow-ups have been implemented (or corrected), so this can have a positive review from me assuming that tests pass.

[yyyyx4]

Thanks a lot, @JohnCremona! The failures appear to be unrelated to these changes.