The $p$-adic BSD Conjectural Formula

Let $E$ be an elliptic curve over $\mathbb{Q}$ and let $p$ be a prime of good ordinary reduction for $E$.

In Chapter 1 (see Theorem 1.15) we defined a $p$-adic $L$-series

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

Conjecture 1.16 asserted that $\ord _{T} \mathcal{L}_p(E,T) = \rank E(\mathbb{Q}).$ Just as is the cases for $L(E,s)$, there is a conjectural formula for the leading coefficient of the power series $\mathcal{L}_p(E,T)$. This formula is due to Mazur, Tate, and Teitelbaum [MTT86].

First, suppose $\ord _T \mathcal{L}_p(E,T) = 0$, i.e., $\mathcal{L}_p(E,0) \neq 0$. Recall that the interpolation property (1.5.1) for $\mathcal{L}_p(E,T)$ implies that

$$\mathcal{L}_p(E,0) = \varepsilon _p \cdot L(E,1)/\Omega_E,$$

where

$$\varepsilon _p = (1-\alpha^{-1})^2,$$ (2.5.1)

and $\alpha\in \mathbb{Z}_p$ is the unit root of $x^2 - a_p x + p=0$. Thus the usual BSD conjecture predicts that if the rank is $1$, then

$$\mathcal{L}_p(E,0) = \varepsilon _p \cdot \frac{\prod_\ell ... ...E/\mathbb{Q}) \cdot \Reg (E)}{\char93 E(\mathbb{Q})_{\tor }^2}$$ (2.5.2)

Notice in (2.5.2) that since $E(\mathbb{Q})$ has rank $0$, we have $\Reg (E) = 1$, so there is no issue with the left hand side being a $p$-adic number and the right hand side not making sense. It would be natural to try to generalize (2.5.2) to higher order of vanishing as follows. Let $\mathcal{L}_p^*(E,0)$ denote the leading coefficient of the power series $\mathcal{L}_p(E,T)$. Then

$$\mathcal{L}_p^*(E,0) \lq\lq =\text{''} \varepsilon _p \cdot \fra... ...E)}{\char93 E(\mathbb{Q})_{\tor }^2}\qquad\text{(nonsense!!).}$$ (2.5.3)

Unfortunately (2.5.2) is total nonsense when the rank is bigger than $0$. The problem is that $\Reg (E)\in\mathbb{R}$ is a real number, whereas $\varepsilon _p$ and $\mathcal{L}_p^*(E,0)$ are both $p$-adic numbers.

The key new idea needed to make a conjecture is to replace the real-number regulator $\Reg (E)$ with a $p$-adic regulator $\Reg _p(E) \in \mathbb{Q}_p$. This new regulator is defined in a way analogous to the classical regulator, but where many classical complex analytic objects are replaced by $p$-adic analogues. Moreover, the $p$-adic regulator was, until recently (see [MST06]), much more difficult to compute than the classical real regulator. We will define the $p$-adic number $\Reg _p(E) \in \mathbb{Q}_p$ in the next section.

Conjecture 2.19 (Mazur, Tate, and Teitelbaum)  

Let $E$ be an elliptic curve over $\mathbb{Q}$ and let $p$ be a prime of good ordinary reduction for $E$. Then the rank of $E$ equals $\ord _T(\mathcal{L}_p(E,T))$ and

$$\mathcal{L}_p^*(E,0) = \varepsilon _p \cdot \frac{\prod_\el... ...mathbb{Q}) \cdot \Reg _p(E)}{\char93 E(\mathbb{Q})_{\tor }^2},$$ (2.5.4)

where $\varepsilon _p$ is as in (2.5.1), and the $p$-adic regulator $\Reg _p(E) \in \mathbb{Q}_p$ will be defined below.

Remark 2.20   There are analogous conjectures in many other cases, e.g., good supersingular, bad multiplicative, etc. See [SW07] for more details.