0

I, like the author of this post, am severely lacking the background to make the connection between

  1. reducing modular forms' $q$-expansions modulo $p$ at various cusps, and
  2. $q$-expansions of modular forms modulo $p$ in Katz's sense, e.g. in this paper.

It seems as though Matt E's answer in the post above essentially says these are equivalent, so long as the weight $k \geq 2$ and level $N$ is prime to $p$. David Loeffler's answer below that also affirms the equivalence between 1 and 2 in more explicit language. So these answer my question (and the author of the original post) but without explanation. So, I'm looking for a reference that explicitly mentions this equivalence, if one even exists.

Freddie
  • 1,799
  • The link between the two is that a weight $2$ meromorphic modular form with rational coefficients gives a meromorphic differential $2i\pi f(z)dz = f(q)dq=h dj$ with $h\in \Bbb{Q}(j,j_N)=\Bbb{Q}[j][Y]/(\varphi_N(j,Y))$. This function field and its differentials have an obvious analog over $\Bbb{F}_p$ and in a differential over $\Bbb{F}_p$ you can replace $j,j_N$ by their q-expansion in $\Bbb{Z}[[q^{1/N}]]$ to get a $f(q)dq$, the q-expansion of a modular form over $\Bbb{F}_p$. I don't see the point of the scheme theoretic stuff for this. – reuns Dec 16 '21 at 01:12
  • fwiw you might enjoy Benedict Gross's presentation of Serre's conjecture in the FLT seminar of yore... https://www.youtube.com/watch?v=C5WwRah3MPY - (as I recall he handwaves at Katz's mod forms). The 1st of G's lectures = setup for serre's conjecture, and 2nd = statement + "SC => 1 board proof of FLT" + sketch of Ribet's level lowering ( if I remember correctly!) – peter a g Dec 16 '21 at 04:41
  • @reuns Thanks. So there's an easier way to see the link with weight 2, but it gets complicated with higher weights, yes? I know the higher weights are "sections of the tensor product sheaf", to borrow some language from another reading. Perhaps this is why scheme theory is needed? – Freddie Dec 16 '21 at 17:39
  • 1
    Higher even weights aren't more complicated as meromorphic modular forms are of the form $f (dj)^k$ for some $f\in \Bbb{Q}(j,j_N)$, where $(dj)^k = (j'(q))^k (dq)^k$ is a $q$-expansion times the formal $dq\otimes \ldots \otimes dq$. This is an algebraic expression in $j,j_N$ so you need to define "having no poles" from it, at the cusps it is a matter of $q$-expansion, there is a bit of mess at the elliptic points, on the other points it should reduce to something like $f\in \Bbb{Q}[j,j_N,j^{-1},(j-1728)^{-1}]$ – reuns Dec 16 '21 at 17:50
  • The $p | N$ problem is because $j(pmz) = j(mz)^p \bmod p$ that is the modular polynomial $\varphi_N$ doesn't stay irreducible $\bmod p$, also $[p]$ is inseparable on elliptic curves over finite fields so the level $N$ moduli problem gets different. – reuns Dec 16 '21 at 18:35

0 Answers0