Formal group laws and L-functions

Hi all! Today’s post is going to be a little brief, but interesting.

Let E be an elliptic curve over \mathbf{Q}. Choose a prime p, and denote by a_{p^n} the number p^n+1-\# E(\mathbf{F}_{p^n}). The zeta-function of E is defined to be:

\displaystyle Z_p(E,X)=\exp\left(\sum_{n\geq 1}\frac{a_{p^n}}{n}X^n\right)=(1-a_pX+\mathbf{1}_E(p)pX^2)^{-1}

Where \mathbf{1}_E(p) is the trivial character modulo N. The L-function of E is defined to be:

\displaystyle L(E,s)=\prod_pZ_p(E,p^{-s})=\prod_p(1-a_pX+\mathbf{1}_E(p)pX^2)^{-1}

This in turn has a special expansion as a sum \sum_nb_nn^{-s}. Now, given an elliptic curve, one can also associate a formal group law as follows. Suppose y^2+a_1xy+a_2y=x^3+a_2x^4+a_4x+a_6 is the generalized Weierstrass form of the elliptic curve. Make the following change of coordinates: let z=-x/y and let w=1/y. The identity is at the origin if we look at the affine piece. The generalized Weierstrass equation is now:

\displaystyle w=z^3+(a_1z+a_2z^2)w+(a_3a_4z)w^2+a_6w^3=\eta(z,w)

We may now substitute \eta(z,w) for w in this equation. Expanding this after repeated substitution gives a power series, denoted w(z), which satisfies \eta(z,w(z))=w(z).

Recall that given a formal group law F(X,Y), we can write F as \ell^{-1}(\ell(x)+\ell(y)) where \ell is the formal logarithm of F. Suppose E be a modular elliptic curve with cusp form \alpha(q)=\sum_nb_nq^{n}; then the coefficients of \alpha(q) are the same as that of the L-series L(E,s)=\sum_nb_nn^{-s}. Consequently, integrating the formal invariant differential \omega=\alpha(q)\mathrm{d}q/q gives us f(q)=\sum_nb_nq^n/n. The main result of this post (which we won’t prove) is the following statement:

Theorem (Honda): Over \mathbf{Z}, f^{-1}(f(X)+f(Y))=G(X,Y) is isomorphic to the formal group law F(X,Y) of E.

(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 L -function! The formal connection between f(q) and L(E,s) is as follows. Define a function g(s) as follows:

\displaystyle g(s)=\int^{i\infty}_0f(z)z^{s-1}dz

Then g(s)=(-2\pi i)^{-s}\Gamma(s)L(E,s). To see this, suppose f(z) = \sum^\infty_{n=1}b_nq^n with |b_n|=O(n^c) for some c\in\mathbf{R} (what this means is that as n\to\infty, the fraction a_n/n^c is bounded) is a modular form. Then:

\displaystyle g(s)= \sum^\infty_{n=1}b_n\int^{i\infty}_0z^{s-1}e^{2\pi i nz}dz

Let u=-2\pi i nz; then g(s) becomes:

\displaystyle g(s) = \sum^\infty_{n=1}b_n\left(\dfrac{-1}{2\pi i n}\right)^s\int^\infty_0u^{s-1}e^udu = (-2\pi i)^{-s}\Gamma(s)\sum^\infty_{n=1}b_nn^{-s}

But \sum^\infty_{n=1}b_nn^{-s} is just the L-function of E! What about more general L-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.

Have fun,


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