Hi all! Today’s post is going to be a little brief, but interesting.
Let be an elliptic curve over . Choose a prime , and denote by the number . The zeta-function of is defined to be:
Where is the trivial character modulo . The -function of is defined to be:
This in turn has a special expansion as a sum . Now, given an elliptic curve, one can also associate a formal group law as follows. Suppose is the generalized Weierstrass form of the elliptic curve. Make the following change of coordinates: let and let . The identity is at the origin if we look at the affine piece. The generalized Weierstrass equation is now:
We may now substitute for in this equation. Expanding this after repeated substitution gives a power series, denoted , which satisfies .
Recall that given a formal group law , we can write as where is the formal logarithm of . Suppose be a modular elliptic curve with cusp form ; then the coefficients of are the same as that of the -series . Consequently, integrating the formal invariant differential gives us . The main result of this post (which we won’t prove) is the following statement:
Theorem (Honda): Over , is isomorphic to the formal group law of .
(This is restated as Theorem 33.1.16 of Hazewinkel’s Formal groups and applications.) This is incredibly interesting, because you are recovering the formal group law of the elliptic curve from its -function! The formal connection between and is as follows. Define a function as follows:
Then . To see this, suppose with for some (what this means is that as , the fraction is bounded) is a modular form. Then:
Let ; then becomes:
But is just the -function of ! What about more general -functions, say of Dirichlet characters? Do you get a similar “formal logarithm”? What’s the explicit formal group law associated to a Dirichlet character in this manner? This is something I’m studying with a friend, Nina Anikeeva. I hope to write up what we’ve done in a future post.