Computing $\mathcal{L}_p(E,T)$

Fix notation as in Section 1.5. In particular, $E$ is an elliptic curve over $\mathbb{Q}$, $p$ is an odd prime of good ordinary reduction for $E$, and $\alpha$ is the root of $x^2-a_px+p$ with $\vert\alpha\vert _p = 1$.

For each integer $n\geq 1$, define a polynomial

$$P_{n}(T) = \sum_{a=1}^{p-1} \left( \sum_{j=0}^{p^{n-1}-1} ... ...\mathbb{Z}_p\right) \cdot (1+T)^j \right) \in \mathbb{Q}_p[T].$$

Recall that $\tau(a)\in\mathbb{Z}_p^*$ is the Teichmuller lift of $a$.

Proposition 1.20   We have that the $p$-adic limit of these polynomials is the $p$-adic $L$-series:

$$\lim_{n\to\infty} P_n(T) = \mathcal{L}_p(E, T).$$

This convergence is coefficient-by-coefficient, in the sense that if $P_{n}(T) = \sum_j a_{n,j} T^j$ and $\mathcal{L}_p(E,T) = \sum_j a_j T^j$, then

$$\lim_{n \to \infty} a_{n,j} = a_j.$$

We now give a proof of this convergence and in doing so obtain an upper bound for $\vert a_j - a_{n,j}\vert$.

For any choice $\zeta_r$ of $p^r$-th root of unity in $\mathbb{C}_p$, let $\chi_r$ be the $\mathbb{C}_p$-valued character of $\mathbb{Z}_p^\times$ of order $p^r$ which factors through $1+p\mathbb{Z}_p$ and sends $1+p$ to $\zeta_r$. Note that the conductor of $\chi_r$ is $p^{r+1}$.

Lemma 1.21   Let $\zeta_r$ be a $p^r$-th root of unity with $1 \leq r \leq n-1$, and let $\chi_r$ be the corresponding character of order $p^{r+1}$, as above. Then

$$P_{n}(\zeta_r-1) = \int_{\mathbb{Z}_p^\times} \chi_r d\mu_{E}$$

In particular, note that the right hand side does not depend on $n$.

For each positive integer $n$, let $w_n(T) = (1+T)^{p^n}-1$.

Corollary 1.22   We have that

$$w_{n-1}(T) \text{ divides } P_{n+1}(T) - P_{n}(T).$$

Lemma 1.23   Let $f(T) = \sum_j b_j T^j$ and $g(T)=\sum_j c_j T^j$ be in $\O[T]$ with $\O$ a finite extension of $\mathbb{Z}_p$. If $f(T)$ divides $g(T)$, then

$$\ord _p(c_j) \geq \min_{0 \leq i \leq j} \ord _p(b_i).$$

As above, let $a_{n,j}$ be the $j$th coefficient of the polynomial $P_n(T)$. Let

$$c_n = \max(0, - \min_j \ord _p(a_{n,j}))$$

so that $p^{c_n} P_{n}(T) \in \mathbb{Z}_p[T]$, i.e., $c_n$ is the smallest power of $p$ that clears the denominator. Note that $c_n$ is an integer since $a_{n,j} \in \mathbb{Q}$. Probably if $E[p]$ is irreducible then $c_n=0$ - see Question 1.13. Also, for any $j>0$, let

$$e_{n,j} = \min_{1 \leq i \leq j} \ord _p \binom{p^n}{i}.$$

be the min of the valuations of the coefficients of $w_{n}(T)$, as in Lemma 1.23.

Proposition 1.24   For all $n\geq 0$, we have $a_{n+1,0} = a_{n,0},$ and for $j>0$,

$$\ord _p(a_{n+1,j} - a_{n,j}) \geq e_{n-1,j} - \max(c_{n}, c_{n+1}).$$

For $j$ fixed, $e_{n-1,j} - \max(c_{n+1}, c_n)$ goes to infinity as $n$ grows since the $c_k$ are uniformly bounded (they are bounded by the power of $p$ that divides the order of the cuspidal subgroup of $E$). Thus, $\{ a_{n,j} \}$ is a Cauchy and Proposition 1.24 implies that that

$$\ord _p(a_j - a_{n,j}) \geq e_{n-1,j} - \max(c_{n+1},c_n).$$

Remark 1.25   Recall that presently there is not a single example where we can provably show that $\ord _{s=1} L(E,s) \geq 4$. Amazingly $\ord _{T} \mathcal{L}_p(E,T)$ is ``computable in practice'' because Kato has proved, using his Euler system in $K_2$, that $\rank E(\mathbb{Q}) \leq \ord _{T} \mathcal{L}_p(E,T)$ by proving a divisibility predicted by Iwasawa Theory. Thus computing elements of $E(\mathbb{Q})$ gives a provable lower bound, and approximating $\mathcal{L}_p(E,T)$ using Riemann sums gives a provable upper bound - in practice these meet.