>
>
> *Regarding the documentation*: when I look at
> https://doc.sagemath.org/html/en/reference/modsym/sage/modular/modsym/modsym.html,
> it's not easy for me to see what the mathematical definition of "modular
> symbol" being used is. Given that different authors use the term "modular
> symbols" to mean different things, it might be useful to add to the
> documentation either the definition or a link to a place where the
> definition can be read. It seems like ModularSymbols(G,k,base_ring=R) is
> computing the object called [image: \mathbb{M}_k(G;R)] in Definition 1.23
> here: https://wstein.org/books/modform/modform/modular_symbols.html.
>
I realized that this is wrong: ModularSymbols(G,k,base_ring=R) is not
computing [image: \mathbb{M}_k(G;R)]. According to the definition [image:
\mathbb{M}_k(G;R)=\mathbb{M}_k(G)\otimes_\mathbb{Z} R] and [image:
\mathbb{M}_k(G)] is a free [image: \mathbb{Z}]-module of finite rank,
so [image:
{\rm dim}_{\mathbb{F}_p}\mathbb{M}_k(G;\mathbb{F}_p) = {\rm dim}_\mathbb{Q}
\mathbb{M}_k(G;\mathbb{Q})] for any prime [image: p]. This is contradicted
by my example.
Does anyone know what ModularSymbols(G,k,base_ring=R) *is *computing? Is
maybe it is [image: (\mathbb{M}_k)_G \otimes_\mathbb{Z} R], where
subscript-G means coinvariants?
-Preston
>
> Best,
> Preston
>
> On Tue, May 26, 2020 at 3:34 AM John Cremona <[email protected]>
> wrote:
>
>> Computing mod p modular forms is quite tricky, and has not in general
>> been implemented in Sage (or anywhere else). As well as the valid
>> points William has already made, consider the following: suppose one
>> wants to find all mod-2 modular forms which are ductions of
>> characteristic zero modular forms. Take a char. 0 newforms with
>> coefficients in Q(sqrt(5)). the coefficients are algebraic integers,
>> hence in Z[(1+sqrt(5))/2]. That ring has no prime ideals with
>> quotient GF(2) (since 2 is inert in that field) so you think that
>> there is no mod-2 reduction of the newform. BUT in fact, in this case
>> (which can happen) the coefficients lie in the non-maximal order
>> Z[sqrt(5)], and that ring does have a prime of index 2, so there is a
>> GF(2) modular form obtained by reduction (replacing sqrt(5) as a
>> coefficient by 1 mod 2).
>>
>> In 2018 I was part of a small group starting to do this sort of thing
>> systematically (the others are Samuele Anni, Andrew Sutherland, David
>> Roberts and Peter Bruin). We made a start only. One intention is to
>> make a database of mod-ell modular forms in the LMFDB. One reason for
>> the delay in making progress on this was that at the time we started,
>> the LMFDB database of classical modular forms was rather small and not
>> so well organised. That situation is now vastly improved (see
>> https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/ and the preprint
>> https://arxiv.org/abs/2002.04717 for many details of the large-scale
>> computations behind that).
>>
>> John
>>
>> On Sun, 24 May 2020 at 23:32, Preston Wake <[email protected]>
>> wrote:
>> >
>> >
>> >
>> > 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
>> >> >
>> >> > --
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>> .
>> >>
>> >>
>> >>
>> >> --
>> >> William (http://wstein.org)
>> >>
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