BACKGROUND

Here are 3 theorems of varying difficulty. Let M be the Z/2 subspace of Z/2[[x]] spanned by f^k, with the k>0 and odd, and f=x+x^9+x^25+x^49+... For g in M, let S(g) consist of the primes, p, for which the coefficient of x^p in g is 1. Note that each p in S(f^k) is congruent to k mod 8.

T1.----- If k=3 or 5, S(f^k) consists of the p that are k mod 8

T2.----- S(f^7) consists of the p that are 7 mod 16

T3.----- If k=19 or 21, then S(f^k) consists of the p that are k or k+8 mod 32.

To prove T1 when k=3, we write f^k as f*f^2 and use the fact that if p is 3 mod 8, then p is uniquely the sum of a square and twice a square. When k=5 we argue similarly using Fermat's two square theorem.

As I indicated in a comment on a recent MO question of Joel Bellaiche, "Primes and x^2+2y^2+4z^2" ,T2 follows from a result of Hasse on the class number of Q(root(-2p)), using Gauss' theorem that the number of representations of 2p as a sum of 3 squares is 12*(this class number). Hasse's proof is an application of the Gauss theory of genera and ambiguous forms.

T3 is thornier. Because f is the mod 2 reduction of (the Fourier expansion of) the normalized weight 12 cusp form for the full modular group, each g is the mod 2 reduction of a modular form of integral weight. A profound result of Deligne, relating Hecke eigenforms to Galois representations, then shows that S(g) is a "Frobenian set". Nicolas, Serre and Bellaiche, continuing in this vein, developed a theory of level 1 modular forms in characteristic 2 that led to more precise results. Their investigations motivated me to try to determine S(f^k) empirically for small k, and I was led to conjecture T3. Joel then applied his methods to give a proof. But this is very hard, and so I ask:

QUESTION

Does there exist an "elementary proof" of T3, using the theory of binary quadratic forms, along the lines of the Hasse-Gauss argument?