# Function fields of characteristic p modular curves, and mod p reductions of the classical modular equation

Let l and p be distinct primes, l>2. There are "characteristic p modular curves" X_0(l) and X(l), defined over an algebraic closure, K, of Z/p, solving moduli problems for elliptic curves with some additional level-l structure. Each of these curves has the same genus as the corresponding characteristic 0 object; in particular the genus of X(l) is (l-3)(l-5)(l+2)/24.

There is also an irreducible symmetric f in Z[x,y] with f(j(lz),j(z))=0, where j is the elliptic modular function. This is the "classical modular equation". Let f* be the mod p reduction of f. I'm looking for a proof that certain well-known relations between f and the function fields of the characteristic zero X_0(l) and X(l) persist when f is replaced by f*, and X_0 and X are replaced by their characteristic p counterparts. I'd like an argument showing:

1) f* is irreducible in K[x,y]

2) The Galois group of f* over K(y) identifies with PSL_2(Z/l).

3) The function field (over K) of the curve defined by f* identifies with the function field of the characteristic p X_0.

4) If L is the splitting field (over K(y)) of f*, then L identifies with the function field (over K) of the characteristic p X.

Remarks:

a) I would guess that 1)---4) somehow follow from the existence of moduli schemes defined over Z[1/l]. But can someone provide a reference and details?

b) A weaker form of 4) whose statement doesn't involve the theory of modular forms in characteristic p, is that the genus of L/K equals the genus of the classical X(l). As an old dog who has trouble with new tricks, I'd be happiest with a classical proof of this result.

c) I'm mostly interested in the case p=2, where I can prove 1) and 2). This is all related to an MO question of mine about the genus of a curve coming from the theory of characteristic 2 theta functions.

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I think most of these things can be found in a paper of Igusa from the 50's (I think). I don't have the reference handy but should be easy to locate with mathscinet. Here is some  to prettify your question. – Felipe Voloch Oct 14 '11 at 9:46
Thanks, Felipe! I found the papers in Amer. J. of Math., v.81, (1959). They're not easy for me to read, but I get the drift, and they seem to be exactly what I was looking for. – paul Monsky Oct 16 '11 at 6:23