On Sun, May 24, 2020 at 3:38 PM William Stein <[email protected]> wrote:
> On Sun, May 24, 2020 at 12:16 PM Preston Wake <[email protected]> > wrote: > > > > Hello, > > > > I've found that the modular symbols code in Sage may detect interesting > examples of torsion modular forms. For example: > > > > G=DirichletGroup(13); > > e=G.gen()^6; # e has order 2 > > M=ModularSymbols(e,2,-1); # M is 0 > > ep=e.change_ring(GF(3)); > > Mp=ModularSymbols(ep,2,-1); # Mp is one-dimensional > > > > I think this is coming from the famous counterexample to the naive > version of Serre's conjecture: there is a mod-3 modular form of weight 2 > and level 13 with quadratic character such that any lift to characteristic > zero has non-quadratic character. This is something peculiar to mod-p forms > for p=2 or 3. > > > > But it also sometimes gives confusing results: > > > > M=ModularSymbols(7,8,1).cuspidal_subspace(); # M has dimension 3 > > Mp=ModularSymbols(7,8,1,GF(5)).cuspidal_subspace(); # M has dimension 4 > > > > I don't think this should happen: mod-5 modular forms of weight 8 and > level Gamma0(7) should lift to characteristic 0 forms of the same type. I > assume that what is happening is that Mp is not really computing > H^1(X_0(7),\F_5)^+, which is I thing I think modular symbols should be. > > > > Math question: Where is this extra dimension coming from? Does it have > any interesting number theoretic meaning? > > > > Sage question: Is this a bug? Should Sage warn us that, with > finite-field coefficients, ModularSymbols might not be computing the thing > I think it is computing? > > > > ModularSymbols are indeed not computing what you think it is > computing. Is what I want already implemented somewhere in Sage? Can I compute the space of mod-p modular forms? (Especially in cases where this is different from modular forms mod p.) -Preston > This is not a bug. What it computes is precisely > defined, meaningful and in some cases may be much faster than > computing what you want. However, to use it as input to a > something else, you have to understand what it is really doing... > > It would be reasonable to add to the docs here > > > https://doc.sagemath.org/html/en/reference/modsym/sage/modular/modsym/modsym.html > > to say something like "ModularSymbols in characteristic p in Sage > might not compute what you think they compute. Do not make > assumptions about them without also consulting the Sage source code > and understanding what is actually implemented (which is approximately > using Manin symbols and the same relations that are used in > characteristic 0)." > > > Best wishes, > > Preston > > > > -- > > You received this message because you are subscribed to the Google > Groups "sage-nt" group. > > To unsubscribe from this group and stop receiving emails from it, send > an email to [email protected]. > > To view this discussion on the web visit > https://groups.google.com/d/msgid/sage-nt/18758bbb-b636-4db9-815f-ebe79354fbb4%40googlegroups.com > . > > > > -- > William (http://wstein.org) > > -- > You received this message because you are subscribed to the Google Groups > "sage-nt" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected]. > To view this discussion on the web visit > https://groups.google.com/d/msgid/sage-nt/CACLE5GCm9BO-%2B8YmV5HPovKb__WQaBsgbd1kQ3n%2BQYLBkQYCcg%40mail.gmail.com > . > -- You received this message because you are subscribed to the Google Groups "sage-nt" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/sage-nt/CAOjp0GGOEE%2BMeY43o_%2BVb3VGSe_aEHUyfGg-CSf_fW%2BH-VPsDg%40mail.gmail.com.
