JB asks whether there is a conceptual proof of Jacobi’s formula:
Here (to me) the best proof is one that requires the least calculation, not necessarily the “easiest.” Here is my attempt. We use the following property of , which follows from its moduli theoretic definition: the only zero of is a simple zero at the cusp, moreover, the evaluation of on the Tate curve is normalized so that the leading coefficient is .
Let be prime. I claim that
Observe that both these expressions are modular forms of level one and weight times the weight of . One can prove this “by hand,” but also by noting that the RHS is equal to the norm of on down to . On the other hand, the RHS also has a zero of order at , from which the result immediately follows, since the ratio will be holomorphic of weight zero. If one defines Hecke operators on -expansions in the usual way, it also immediately follows that the logarithmic derivative is (as a -expansion) an eigenform for of weight two with eigenvalue for all primes . In fact, the same argument as above (with replaced by ) implies that this derivative is also an eigenform for with eigenvalue . This is almost enough to determine the -expansion uniquely: in particular, it implies that
for some integer , from which it follows that
To finish the argument, it suffices to check that , or that . One way to do this is to note (by uniqueness) that is a Hecke eigenform, and then use the equation which implies that ; the cases are then ruled out by the equations and , and is ruled out by the fact that is not a modular form. Curiously enough, this determines without ever using the fact that it has weight . Another (more traditional way) is to show that . Is there a way to do this final step by pure thought?