Background

The following is adapted from the introduction to [MST06].

Let $p$ be an odd prime number, and $E$ an elliptic curve over a global field $K$ that has good ordinary reduction at $p$ . Let $L$ be any (infinite degree) Galois extension with a continuous injective homomorphism $\rho$ of its Galois group to $\mathbb{Q}_p$ . To the data $(E,K,\rho)$ , one associates^6.1 a canonical (bilinear, symmetric) ($p$ -adic) height pairing

$$( , )_{\rho} : E(K)\times E(K) \longrightarrow \mathbb{Q}_p.$$

Such pairings are of great interest for the arithmetic of $E$ over $K$ , and they arise specifically in $p$ -adic analogues of the Birch and Swinnerton-Dyer conjecture.^6.2

The goal of this project is to investigate some computational questions regarding $p$ -adic height pairings. The main stumbling block to computing them efficiently is in calculating, for each of the completions $K_v$ at the places $v$ of $K$ dividing $p$ , the value of the $p$ -adic modular form ${\bf E}_2$ associated to the elliptic curve with a chosen Weierstrass form of good reduction over $K_v$ .

The paper [MST06] contains an algorithm for computing these quantities (for $K=\mathbb{Q}$ ), i.e., for computing the value of ${\bf E}_2$ of an elliptic curve (that builds on the works of Katz and Kedlaya listed in our bibliography).

Problem 6.1.1   Implement the full algorithm in SAGE. (It is completely implemented in MAGMA already.)

The paper [MST06] also discusses the $p$ -adic convergence rate of canonical expansions of the $p$ -adic modular form ${\bf E}_2$ on the Hasse domain, where for $p\ge 5$ we view ${\bf E}_2$ as an infinite sum of classical modular forms divided by powers of the (classical) modular form ${\bf E}_{p-1}$ , while for $p\le 5$ we view it as a sum of classical modular forms divided by powers of ${\bf E}_4$ .

Problem 6.1.2   Compute $p$ -adic heights for elliptic curves in families, e.g., for curves over $\mathbb{Q}(t)$ . Interpret the result in terms of log convergence.

We were led to our fast method of computing $\mathcal{E}_2$ by our realization that the more naive methods, of computing it by integrality or by approximations to it as function on the Hasse domain, were not practical, because the convergence is ``logarithmic'' in the sense that the $n$ th convergent gives only an accuracy of $\log_p(n)$ .

The reason why this constant ${\bf E}_2$ enters the calculation is because it is needed for the computation of the $p$ -adic sigma function [MT91], which in turn is the critical element in the formulas for height pairings.

For example, let us consider the cyclotomic $p$ -adic height pairing in the special case where $K=\mathbb{Q}$ and $p\geq 5$ .

If $G_{\mathbb{Q}}$ is the Galois group of an algebraic closure of $\mathbb{Q}$ over $\mathbb{Q}$ , we have the natural surjective continuous homomorphism $\chi: G_{\mathbb{Q}} \to \mathbb{Z}_p^*$ pinned down by the standard formula $g(\zeta) = \zeta^{\chi(g)}$ where $g \in G_{\mathbb{Q}}$ and $\zeta$ is any $p$ -power root of unity. The $p$ -adic logarithm $\log_p:\mathbb{Q}_p^* \to (\mathbb{Q}_p,+)$ is the unique group homomorphism with $\log_p(p)=0$ that extends the homomorphism $\log_p:1+p\mathbb{Z}_p \to \mathbb{Q}_p$ defined by the usual power series of $\log(x)$ about $1$ . Explicitly, if $x\in\mathbb{Q}_p^*$ , then

$$\log_p(x) = \frac{1}{p-1}\cdot \log_p(u^{p-1}),$$

where $u = p^{-{\mathrm{ord}}_p(x)} \cdot x$ is the unit part of $x$ , and the usual series for $\log$ converges at $u^{p-1}$ .

The composition $(\frac{1}{p}\cdot \log_p)\circ \chi$ is a cyclotomic linear functional $G_{\mathbb{Q}} \to \mathbb{Q}_p$ which, in the body of our text, will be dealt with (thanks to class field theory) as the idele class functional that we denote $\rho_{\mathbb{Q}}^{\rm cycl}$ .

Let $\mathcal{E}$ denote the Néron model of $E$ over  $\mathbb{Z}$ . Let $P\in E(\mathbb{Q})$ be a non-torsion point that reduces to $0\in E(\mathbb{F}_p)$ and to the connected component of $\mathcal{E}_{\mathbb{F}_\ell}$ at all primes $\ell$ of bad reduction for $E$ . Because $\mathbb{Z}$ is a unique factorization domain, any nonzero point $P=(x(P),y(P)) \in E(\mathbb{Q})$ can be written uniquely in the form $(a/d^2, b/d^3)$ , where $a,b,d \in \mathbb{Z}$ , $\gcd(a,d)=\gcd(b,d)=1$ , and $d>0$ . The function $d(P)$ assigns to $P$ this square root $d$ of the denominator of $x(P)$ .

Here is the formula for the cyclotomic $p$ -adic height of $P$ , i.e., the value of

$$h_p(P) := -{\frac{1}{2}}(P,P)_p \in \mathbb{Q}_p$$

where $( , )_p$ is the height pairing attached to $G_{\mathbb{Q}} \to \mathbb{Q}_p$ , the cyclotomic linear functional described above:

$$h_p(P) = \frac{1}{p}\cdot \log_p\left(\frac{\sigma(P)}{d(P)}\right) \in \mathbb{Q}_p.$$ (6.1)

Here $\sigma = \sigma_p$ is the $p$ -adic sigma function of [MT91] associated to the pair $(E,\omega)$ . The $\sigma$ -function depends only on $(E,\omega)$ and not on a choice of Weierstrass equation, and behaves like a modular form of weight $-1$ , that is $\sigma_{E,c\omega} =c\cdot \sigma_{E,\omega}$ . It is ``quadratic'' the sense that for any $m\in\mathbb{Z}$ and point $Q$ in the formal group $E^f(\overline{\mathbb{Z}}_p)$ , we have

$$\sigma(mQ) = \sigma(Q)^{m^2} \cdot f_m(Q),$$ (6.2)

where $f_m$ is the $m$ th division polynomial of $E$ relative to $\omega$ (as in [MT91, App. 1]). The $\sigma$ -function is ``bilinear'' in that for any $P,Q \in E^f(\mathbb{Z}_p)$ , we have

$$\frac{\sigma(P-Q)\cdot \sigma(P+Q)}{\sigma^2(P)\cdot \sigma^2(Q)} = x(Q) - x(P).$$ (6.3)

See [MT91, Thm. 3.1] for proofs of the above properties of $\sigma$ .

The height function $h_p$ of (6.1.1) extends uniquely to a function on the full Mordell-Weil group $E(\mathbb{Q})$ that satisfies $h_p(nQ) = n^2 h_p(Q)$ for all integers $n$ and $Q \in E(\mathbb{Q})$ . For $P,Q \in E(\mathbb{Q})$ , setting

$$( P, Q )_p = h_p(P)+h_p(Q) -h_p(P+Q),$$

we obtain a pairing on $E(\mathbb{Q})$ . The $p$ -adic regulator of $E$ is the discriminant of the induced pairing on $E(\mathbb{Q})_{/{\mathrm{tor}}}$ (well defined up to sign), and we have the following standard conjecture about this height pairing.

Conjecture 6.1.3   The cyclotomic height pairing $( , )_p$ is nondegenerate; equivalently, the $p$ -adic regulator is nonzero.

Problem 6.1.4   Gather substantial experimental evidence for Conjecture 6.1.3.

Remark 6.1.5   Height pairings attached to other $p$ -adic linear functionals can be degenerate; in fact, given an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at $p$ , and $K$ a quadratic imaginary field over which the Mordell-Weil group $E(K)$ is of odd rank, the $p$ -adic anticyclotomic height pairing for $E$ over $K$ is always degenerate.

Problem 6.1.6   Implement in SAGE the algorithm from Challenge 6.2.7 (see below).

Problem 6.1.7   Use Challenge 6.1.6 to do a massive computation for Mazur and Rubin related to their longterm explorations of universal norms.

The $p$ -adic $\sigma$ function is the most mysterious quantity in (6.1.1). There are many ways to define $\sigma$ , e.g., [MT91] contains $11$ different characterizations of $\sigma$ ! We now describe a characterization that leads directly to a (slow!) algorithm to compute $\sigma(t)$ . Let

$$x(t) = \frac{1}{t^2} + \cdots \in \mathbb{Z}_p((t))$$ (6.4)

be the formal power series that expresses $x$ in terms of the local parameter $t=-x/y$ at infinity. The following theorem, which is proved in [MT91], uniquely determines $\sigma$ and $c$ .

Theorem 6.1.8   There is exactly one odd function $\sigma(t) = t + \cdots \in t\mathbb{Z}_p[[t]]$ and constant $c\in \mathbb{Z}_p$ that together satisfy the differential equation

$$x(t) + c = -\frac{d}{\omega}\left( \frac{1}{\sigma} \frac{d\sigma}{\omega}\right),$$ (6.5)

where $\omega$ is the invariant differential $dx/(2y+a_1x+a_3)$ associated with our chosen Weierstrass equation for $E$ .

Remark 6.1.9   The condition that $\sigma$ is odd and that the coefficient of $t$ is $1$ are essential.

In (6.1.1), by $\sigma(P)$ we mean $\sigma(-x/y)$ , where $P=(x,y)$ . We have thus given a complete definition of $h_p(Q)$ for any point $Q \in E(\mathbb{Q})$ and a prime $p\geq 5$ of good ordinary reduction for $E$ .