Existence of certain identities involving characteristic 2 "thetas" Let l=2m+1 be prime. In my previous MO question, "What are the polynomial relations between these characteristic 2 thetas?", I defined a subring of Z/2[[x]] as follows:
The subring, S, is generated by [1],...,[m] where [i] is the sum of the x^(n^2), n running over all integers congruent to i mod l.
QUESTION...... Let F=x+x^9+x^25+x^49+...,G=F(x^l), and H=G(x^l). Are G and H in S?
The answer is yes when l=3,5 or 7. When l=7, if we set a=[1],b=[2] and c=[3], we have the curious identities H=(abc)^3*(abc+ba^3+cb^3+ac^3), and G=(abc)^2+a^7+b^7+c^7+H.
Remark 1... Kevin Buzzard explained to me that one can decide whether an explicitly given identity such as the ones we've displayed holds by using the theory of characteristic 2 modular forms and computer calculation. But how does one produce these putative identities?
Remark 2... For all l one can show in an elementary way that H is in the field of fractions of S. In fact if a=[i], b=[2i] and c=[4i], then H is the quotient of a^8(a^8+b^2) by b^4+c. Furthermore for l at most 13, H is in S. (One shows that the quotient lies in S, by combining the "quintic relations" of my MO question cited earlier with Groebner basis computer calculations.)
I'll sketch an argument giving the l=7 identities. Let C be the curve in affine 3-space defined by the ideal of quintic relations. C has 3 linear branches at the origin and 3
linear branches at each of the seven points (r,r^4,r^9) with r^7=1. Passing to projective 3-space we find that (the Zariski closure of) C has 14 simple points at infinity. The formula for H as a quotient shows that H has zeros of order 49 at the branches at the origin, simple zeros at the branches at the other singular points, and poles of order 12 at infinity. This
leads to the identity for H. To get the identity for G one notes that (GH)+(GH)^2+(G+H)^8=0--see my MO question, "What's known about the reduction...?" It follows from this that if G is in the field of fractions of S then G+H has zeros of order 7 at the branches at the origin, of order 3 at the branches at the other singular points, and poles of order 6 at infinity. This suggests that G+H=(abc)^2+a^7+b^7+c^7. To verify this we set J=(abc)^2+a^7+b^7+c^7+H, and use Groebner basis computer calculations to show that JH+(JH)^2+(J+H)^8=0; it then follows that J=G.
EDIT: I think I can now show that when l=11, G is NOT in the field of fractions of S,
even though H is in S. I'll make this an answer once I'm surer of it.
EDIT #2: My supposed counterexample when l=11 is incorrect; G like H is in S. I had
the wrong modular equation of degree 11 relating G and H. Once I found the correct equation,
in Cayley's article, I was able to argue as in the case l=7.
FINAL(?) EDIT: As I've shown in my answer, G and H are indeed always in S. And I've produced a simple conjectural explicit formula for G+H that holds for l<1500. Whether there is anything comparably simple for H isn't clear. At any rate here are formulas for H when l<24. I write C(a,b,c) for the sum of the [ra][rb][rc] where r runs from 1 through (l-1)/2;
more generally (a,b,c) can be replaced by any multi-set. P is the product of the [r] where
r runs from 1 through (l-1)/2. The identity when l=17 is striking.
l=3..........              [1]^9  +[1]^12
l=5..........              P^5  +P^6
l=7.......... (P^3)(P+C(1,1,1,2))
l=11.........(P^2)(C(1,1,3)+C(1,1,2,4))
l=13.........P*(P+[1][2][3][5]+[1][4][5][6]+[2][3][4][6]+C(1,1,2,2,2,5))
l=17.........P*([1][2][4][8]+[3][5][6][7])
l=19.........P*([2][3][5]+[4][6][9]+[1][7][8]+C(3,3,2,4))
l=23.........P*(C(1,2,3,3)+C(1,2,4,5)+C(1,4,4,6)+C(1,2,2,5,9))
 A: In the first version of this answer I gave a (necessarily incorrect) proof of the false statement that when the prime,l, is 11, then G is not in the field generated over Z/2 by the [j]. In the second version I found my error, and gave a computer-aided proof that for this l, G is in the ring generated over Z/2 by the [j].
In this completely rewritten answer I state the following conjecture and explain why it holds when l is congruent to 1 mod 4 or to 3 mod 8.
CONJECTURE: Let l be an odd prime. Then there is a C in the ring generated by the [j] such that C^2+C=G+H. In particular, G like H is in the field generated by the [j], and if H is in the ring generated by the [j], the same is true of G.
Proofsketch when l=1 mod 4 or l=3 mod 8.------When l=1 mod 4, take r with r^2=-1 mod l. Then [j][rj] only depends on the coset of {1,r,-1,-r} in (Z/l)* that contains j. Take C to be the sum of the [j][rj] where j runs over a set of representatives of the cosets. For example when l=13, C=[1][5]+[2][3]+[4][6]. It's an exercise in the arithmetic of Z[i] to show that C^2+C=G+H. When l=3 mod 8, take r with r^2=-2 mod l, and let C be the sum of the [rj][j][j] where j runs over representatives of the cosets of {1,-1} in the multiplicative group of Z/l. Now the result is proven using the arithmetic of Z[Root(-2)]. 
Remark: When l=7 mod 8, I may present evidence for the truth of the conjecture in a separate question. But now it seems that ternary rather than binary quadratic forms enter the picture.
EDIT(11/23/11)
I believe I can now prove the above conjecture. But since my proof uses the fact that G is
a polynomial (over the algebraic closure, K, of Z/2) in my theta series, it doesn't supersede my other (self-accepted) answer.
Here's the idea. Let q=x^l, and E be the elliptic curve Y^2+XY=X^3+(q+q^9+q^25+...) defined
over the field of fractions of Z/2[[q]]. The j-invariant of E is 1/(q+q^9+q^25+...) "=" (E_4)^3/(Delta). Using this fact one shows that E is the characteristic 2 Tate curve. The
study I've performed of the field, L, generated over K by the theta-series shows that L is the field generated over K by the x co-ordinates of the l-division points of E. (In the proof of this I use the fact that G and H are in L). But one can write these x co-ordinates
explicitly as power series, using a characteristic 2 analogue of the Weierstrass P-function (see Roquette's book). It turns out that there are (l-1)/2 of these division points for which, when their x co-ordinates are summed, one gets a power series C with C^2+C=G+H. So C is in L. Once this is known it's straightforward to see that C is a polynomial in the theta-series. But why the remarkable empirical formulas for C in terms of the theta-series hold when l is 7 mod 8 remains a mystery.
