Let $E$ be an elliptic curve over $\mathbb{Q}$ with Weierstrass equation: $$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$ with coefficients $a_i\in\mathbb{Z}$ . For $p$ a prime in $\mathbb{Z}$ , define $N_p$ as the number of points in the reduction of the curve modulo $p$ , this is, the number of points in: $$\{O\}\cup\{(x,y)\in{\mathbb{F}_p}^2\colon y^2+a_1xy+a_3y-x^3-a_2x^2-a_4x-a_6\equiv 0\ mod\ p\}$$ where $O$ is the point at infinity. Also, let $a_p=p+1-N_p$ . We define the local part at $p$ of the L-series to be:

Note: The product converges and gives an analytic function for all $Re(s)>3/2$ . This follows from the fact that $\mid a_p \mid \leq 2\sqrt{p}$ . However, far more is true:

The number $w$ above is usually called the root number of $E$ , and it has an important conjectural meaning (see Birch and Swinnerton-Dyer conjecture).

This result was known for elliptic curves having complex multiplication (Deuring, Weil) until the general result was finally proven.

Bibliography

1

James Milne, Elliptic Curves, online course notes.

2

Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.

3

Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1994.

4

Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions. Princeton University Press, Princeton, New Jersey, 1971.