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. 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.
