The $p$-Adic $L$-function

Define the $p$-adic $L$-function as a function on characters

\begin{displaymath}
\chi \in \Hom (\mathbb{Z}_p^*,\mathbb{C}_p^*)
\end{displaymath}

as follows. Send a character $\chi$ to

\begin{displaymath}
L_p(E,\chi) = \int_{\mathbb{Z}_p^*} \chi   d \mu_E.
\end{displaymath}

We will later make the integral on the right more precise, as a limit of Riemann sums (see Section 1.6).

Remark 1.14   For any Dirichlet character $\chi:\mathbb{Z}/n\mathbb{Z}\to\mathbb{C}$, let $L(E,\chi,s)$ be the entire $L$-function defined by the Dirichlet series

\begin{displaymath}
\sum_{n=1}^{\infty} \frac{\chi(n) a_n }{n^s}.
\end{displaymath}

The standard interpolation property of $L_p$ is that for any primitive Dirichlet character $\chi$ of conductor $p^n$ (for any $n$), we have[*]
\begin{displaymath}
L_p(E,\chi) =
\begin{cases}
p^{n} \cdot g(\chi) \cdot L(E,\...
...ha^{-1})^2 L(E,1)/\Omega_E & \text{ if $\chi = 1$},
\end{cases}\end{displaymath} (1.5.1)

where $g(\chi)$ is the Gauss sum:

\begin{displaymath}
g(\chi) = \sum_{a \mod p^n} \chi(a) e^{\frac{2\pi i a}{p^n}}.
\end{displaymath}

Note, in particular, that $L(E,1)\neq 0$ if and only if $L_p(E,1)\neq 0$.

In order to obtain a Taylor series attached to $L_p$, we view $L_p$ as a $p$-adic analytic function on the open disk

\begin{displaymath}
D = \{u \in \mathbb{C}_p   :   \vert u-1\vert _p < 1 \},
\end{displaymath}

as follows. We have that $\gamma=1+p$ is a topological generator for $1+p\mathbb{Z}_p$. For any $u\in D$, let $\psi_u:1+p\mathbb{Z}_p \to \mathbb{C}_p^*$ be the character given by sending $\gamma$ to $u$ and extending by using the group law and continuity. Extend $\psi_u$ to a character $\chi_u:\mathbb{Z}_p^* \to \mathbb{C}_p^*$ by letting $\chi_u(x) = \psi_u(\langle x \rangle)$. Finally, overloading notation, let

\begin{displaymath}
L_p(E,u) = L_p(E,\chi_u).
\end{displaymath}

Theorem 1.15   The function $L_p(E,u)$ is a $p$-adic analytic function on $D$ with Taylor series about $u=1$ in the variable $T$

\begin{displaymath}
\mathcal{L}_p(E,T) \in \mathbb{Q}_p[[T]].
\end{displaymath}

that converges on $\{z \in \mathbb{C}_p  :  \vert z\vert _p < 1\}$. (Note that $L_p(E,u) = \mathcal{L}_p(E, u-1)$.)

It is $\mathcal{L}_p(E,T)$ that we will compute explicitly.

Conjecture 1.16 (Mazur, Tate, Teitelbaum)  

\begin{displaymath}
\ord _{T} \mathcal{L}_p(E,T) = \rank E(\mathbb{Q}).
\end{displaymath}

Proposition 1.17   Conjecture 1.16 is true if $\ord _{T}\mathcal{L}_p(E,T) \leq 1$.


\begin{proof}
% latex2html id marker 2059
[Sketch of Proof]
By Remark \ref{rm:in...
...=1$.
Kolyvagin's theorem then implies that
$\rank E(\mathbb{Q})=1$.
\end{proof}

Remark 1.18   Mazur, Tate, and Teitelbaum also define an analogue of $\mathcal{L}_p(E,T)$ for primes of bad multiplicative reduction and make a conjecture. A prime $p$ is supersingular for $E$ if $a_p \equiv 0 \pmod{p}$; it is a theorem of Elkies [Elk87] that for any elliptic curve $E$ there are infinitely many supersingular primes $p$. Perrin-Riou, Pollack, Greenberg and others have studied $\mathcal{L}_p(E,T)$ at good supersingular primes. More works needs to be done on finding a definition of $\mathcal{L}_p(E,T)$ when $p$ is a prime of bad additive reduction for $E$.

Remark 1.19   A theorem of Rohrlich implies that there is some character as in (1.5.1) such that $L(E,\chi,1)\neq 0$, so $\mathcal{L}_p(E,T)$ is not identically zero. Thus $\ord _T \mathcal{L}_p(T)<\infty$.

William 2007-05-25