The $p$-adic $\mathcal{L}$-series

Fix an elliptic curve $E$ defined over $\mathbb{Q}$. We say a prime $p$ is a prime of good ordinary reduction for $E$ if $p\nmid N_E$ and $a_p \not\equiv 0 \pmod{p}$. The Hasse bound, i.e., that $\vert a_p\vert< 2\sqrt{p}$ on implies that if $p\geq 5$ then ordinary at $p$ is the same as $a_p\neq 0$.

In this section, we define for each odd prime number $p$ of good ordinary reduction for $E$ a $p$-adic $L$-function $L_p(E,T)$. This is a $p$-adic analogue of the complex $L$-function $L(E,s)$ about which there are similar analogue of the BSD conjecture.