A brief note on computing p-adic L-series of elliptic curves

Robert Pollack

January 29, 2001

1 Definition of p-adic L-series

Let E/Q be an elliptic curve. Fix p a prime number. Denote by [\frac{r}{s}]^{+} the positive modular symbol of E associated to \frac{r}{s} (defined up to choice of sign). Let α be a root of x^{2} − a_{p}x + p with ord_{p}(α) < 1. Here a_{p} := p + 1 − E(F_{p}). Such an α is unique in the ordinary case and is actually in Z_{p}^{×}. In the supersingular case, there are two choices for alpha both conjugate in a quadratic extension of Q_{p}. Define a distribution on Z_{p}^{×} by

μ_{E,α}^{+}(a + p^{n}Z_{p}) = \frac{1}{α^{n}} [\frac{a}{p^{n}} ]^{+} − \frac{1}{α^{n+1}} [\frac{a}{p^{n−1}} ]^{+} .

Then the p-adic L-series of E is defined as a function on the C_{p}-valued characters of Z_{p}^{×} by integration with respect to μ_{E,α}^{+}. Here,

L_{p}(E, α, χ) = ∫_{Z_{p}^{×}} χ dμ_{E,α}^{+}

where χ : Z_{p}^{×} → C_{p} is a character.

2 Powers series expression for the p-adic L-series

Pick γ a topological generator of 1+pZ_{p}. Then characters of 1+pZ_{p} are defined uniquely by their value on γ. For u ∈ C_{p} with |u − 1|_{p} < 1, define a character χ_{u} on Z_{p}^{×} by first taking the natural projection of Z_{p}^{×} onto 1 + pZ_{p} and then mapping γ onto u.

The p-adic L-series L_{p}(E, α, χ_{u}) is then analytic in the variable u and we will denote its expansion about u = 1 by L_{E,p,α}(T) ∈ Q_{p}(α)[[T]]. This power series is convergent on the open unit disc of C_{p}. We have that

L_{E,p,α}(u − 1) = L_{p}(E, α, χ_{u}).

3 Explicit polynomial approximations of L_{E,p,α}(T)

We can approximate ∫_{Z_{p}^{×}} χ dμ_{E,α}^{+} via Riemann sums. The details of this calculation will not be done here. However, the end result is a sequence L_{n}(T) of polynomials in Q_{p}(α)[T] that converge to the p-adic L-series. Here,

L_{n}(T) = ∑_{j=0}^{p^{n−1}−1} (∑_{a=1}^{p−1} μ_{E,α}^{+} ({a}γ^{j} + p^{n}Z_{p})) · (1 + T)^{j}

where {a} is the Teichmuller lifting of a.

4 Computer calculations

The above formula for L_{n}(T) allows one to readily approximate them on a computer. One can only approximate them as they have Q_{p}(α) coefficients.

In the ordinary case, α is in Z_{p}^{×} and L_{n}(T) should have Z_{p} coefficients. This is known when E[p] is irreducible, but it is conjectured to happen generally. Analyzing the rate of convergence of the L_{n}(T), one can see that it suffices to compute everything mod p^{n} (taking care with possible p’s in the denominators of the modular symbols). Computing {a}, α and (1 + T)^{j} mod p^{n} is very easy.

All that is left to compute is [\frac{a}{p^{n}} ]^{+} and [\frac{a}{p^{n−1}} ]^{+} for a prime to p between 1 and p^{n}. This is the most time consuming part of the calculation.

In the supersingular case, α is in a quadratic extension of Q_{p} and μ_{E,α}^{+} has p’s in its denominators on order about p^{\frac{n}{2}} . In this case L_{n}(T) will not have Z_{p} coefficients. Nonetheless, no essential information is lost if one computes {a} and (1+T)^{j} only mod p^{n}. The end result should be that L_{n}(T) = G_{n}(T)+H_{n}(T)·α with G_{n}, H_{n} ∈ Q_{p}[T] and scaling by p^{\frac{n+2}{2}} will put both of them in Z_{p}[[T]].