# 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,
SKD