In my answer to my earlier question, "Existence of certain identities involving characteristic 2 thetas", I established some curious identities when the thetas have prime "level" congruent to 1 mod 4 or to 3 mod 8. This question concerns the case when the level is 7 mod 8.

I reprise notation from earlier questions. l is an odd prime and [j] is the sum of the x^(n^2), where n runs over the integers congruent to j mod l; we view the "theta series" [j] as elements of Z/2[[x]]. F is the power series x+x^9+x^25+x^49+x^81..., G=F(x^l) and H=G(x^l). My identities involve G,H and the various [j].

There is evidently a unique C in Z/2[[x]], having constant term 0, with C^2+C=G+H. I showed that when l is 1 mod 4 or 3 mod 8 (or when l=7), then C can be written explicitly as a 
polynomial in the [j]. Here is what the computer suggests when l=7 mod 8 and is < 50. First some notation. If (r,s,t) is a triple of integers, we define C(r,s,t) to be the sum of the power series [rj][sj][tj] where j runs from 1 to (l-1)/2. Define C(r,s,t,u) similarly. (When l is 3 mod 8, I showed that C is C(1,1,t) where t^2 is congruent to -2 mod l).

(1) When l=7, I can show that C=C(1,1,1,2)+C(1,2,3)

(2) When l=23 I think that C=C(3,3,1,2)+C(1,3,6)

(3) When l=31 I think that C=C(3,3,2,3)+C(2,3,7) (In my original post I wrote C(2,5,8), but C(2,3,7)=C(2,5,8))

(4) When l=47 I think that C=C(3,3,2,5)+C(2,3,9)

(Note that the sum of the squares of 3,3,2 and 5 is 47, etc.)

QUESTION 1: Can one establish the truth of (2),(3) and (4)? Kevin Buzzard explained to me that it's enough to show that the power series expansions agree up to a certain exponent, but I'm not sure what that exponent is, and I doubt that I have the computer power.

QUESTION 2: Are there identities like those above for l>50? And if so, what are these
identities explicitly?

EDIT: Let V be the space spanned by the C(r1,r2,r3,r4) with r1=r2 and l dividing the sum of the squares of r1,r2,r3 and r4, together with the C(s1,s2,s3) with l dividing the sum of the squares of s1,s2 and s3. When l=7 mod 16 I can use Jacobi's 4-square theorem to show that C is in V. It's then possible to prove identities like those of (2) above by exploiting the
geometry of of Spec R where R is the subring of Z/2[[x]] generated by the theta series [j].

-----One can show that an element of V has at most l(l-1)(l+1)/6 poles, counted with 
multiplicity, on the obvious projective completion of this curve. So if it has a zero of large enough order at the origin, it vanishes. I applied this technique for various l congruent to 7 mod 16; the results boggled my mind. It's only necessary to use 2 terms in 
the power series expansion of each theta series. When l=23, I got (2) above. 


When l=71, I found that C=C(3,3,2,7)+C(5,6,9)


When l=103, I got 5 different expressions for C! Explicitly:

a----C(3,3,6,7)+C(2,9,11)

b----C(7,7,1,2)+C(5,9,10)

c----C(5,5,2,7)+C(1,3,14)

d----C(3,3,2,9)+C(6,7,11)

e----C(1,1,1,10)+C(1,6,13)

It seems possible to me that in general, for l=7 mod 8, one gets h/4 formulae of this
sort where h is the class-number of Q(Root(-2l)). I've discussed the case l=31 in the comment to ARupinski. When l=47, I can show that C(3,3,2,5)+C(2,3,9)=C(1,1,3,6)+C(3,6,7).
So if (4) above holds, there's a second formula for C in this case, just as in the case l=31. But I can't prove that C is in V when l=15 mod 16.