On Tue, Jul 5, 2016 at 9:05 PM, Rob Harron <[email protected]> wrote: > Heh, that's got me worried now! I'll be more careful. Is your thesis a > reference for what is being computed?
http://wstein.org/books/modform/ > I suppose I could also read the code. That's always the canonical real reference. > What I think I'm computing is specialization mod p of modular symbols over > Zp and mod p specializations of Hecke algebras over Zp, and, by extension, > specializations of Hida's ordinary p-adic Hecke algebras localized at > maximal ideals. Does that sound reasonable? It depends on p. If p = 2 or 3 it's easy to find examples where you're NOT computing those things (and other p can be problems in general when the character is nontrivial): Try this: for N in [11..100]: print N, dimension_cusp_forms(N), ModularSymbols(Gamma0(N),base_ring=GF(2),sign=1).cuspidal_subspace().dimension() When the dimensions match up, modular symbols in Sage with a char p base ring are just the reduction of char 0 modular symbols. When they don't match up, obviously they are different. In that case, bad things can happen -- e.g., I think the Hecke operators might even fail to commute on that extra torsion in some cases. There's definitely a paper to be written about this topic... I first seriously ran into these subtleties when writing http://wstein.org/papers/artin/ William > > Rob > > On Tuesday, July 5, 2016 at 8:34:29 PM UTC-6, William stein wrote: >> >> On Tue, Jul 5, 2016 at 3:25 PM, Rob Harron <[email protected]> wrote: >> > Kiran, thanks for getting on this so quickly! >> > >> > Maarten and Vincent, it would be a shame to not be able to do mod p >> > computations; they are pretty essential when doing anything p-adic. It >> > is >> > true that I've found several problems when working mod p, but also with >> > Qp. >> > Do note that sage only allows base rings that are fields, so that one >> > can't >> > literally work over ZZ, and also note that magma does allow finite >> > fields, I >> > wonder if William wrote that part. >> >> I wrote the code in both Magma and Sage. It's the same algorithm with >> the same shortcomings. There's no difference in what each program >> allows in this regard, except the bug involving the P1list >> implementation in this thread. >> >> You can work with the modulo symbols presentation modulo p, but you >> had better clearly understand what is really being computed if you >> want to draw any conclusions from the results you get. >> >> -- William >> >> -- >> 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 post to this group, send email to [email protected]. > Visit this group at https://groups.google.com/group/sage-nt. > For more options, visit https://groups.google.com/d/optout. -- 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 post to this group, send an email to [email protected]. Visit this group at https://groups.google.com/group/sage-nt. For more options, visit https://groups.google.com/d/optout.
