The $p$-adic Regulator

Fix an elliptic curve $E$ defined over $\mathbb{Q}$ and a prime $p$ of good ordinary reduction for $E$. In this section we define the $p$-adic regulator $\Reg _p(E)$. See [MTT86], [MST06] and [SW07] and the references listed there for a more general discussion of $p$-adic heights, especially for bad or supersingular primes, and for elliptic curves over number fields. See also forthcoming work of David Harvey for highly optimized computation of $p$-adic regulators.

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

Example 2.21   For example, in SAGE we compute the logs of a couple of non-unit elements of $\mathbb{Q}_5$ as follows:

sage: K = Qp(5,8); K
5-adic Field with capped relative precision 8
sage: a = K(-5^2*17); a
3*5^2 + 5^3 + 4*5^4 + 4*5^5 + 4*5^6 + 4*5^7 + 4*5^8 + 4*5^9 + O(5^10)
sage: u = a.unit_part()
3 + 5 + 4*5^2 + 4*5^3 + 4*5^4 + 4*5^5 + 4*5^6 + 4*5^7 + O(5^8)
sage: b = K(1235/5); b
2 + 4*5 + 4*5^2 + 5^3 + O(5^8)
sage: log(a)
5 + 3*5^2 + 3*5^3 + 4*5^4 + 4*5^5 + 5^6 + O(5^8)
sage: log(a*b) - log(a) - log(b)
O(5^8)

Note that we can recover $b$:

sage: c = a^b; c
2*5^494 + 4*5^496 + 2*5^497 + 5^499 + 3*5^500 + 5^501 + O(5^502)
sage: log(c)/log(a)
2 + 4*5 + 4*5^2 + 5^3 + O(5^7)

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 $E$. For example, given any point $Q\in E(\mathbb{Q})$ one can construct such a $P$ by multiplying it by the least common multiple of the Tamagawa numbers of $E$.

Exercise 2.22   Show that 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 $d>0$. (Hint: Use that $\mathbb{Z}$ is a unique factorization domain.)

The function $d(P)$ assigns to $P$ this square root $d$ of the denominator of the $x$-coordinate $x(P)$.

Example 2.23   We compute a point on a curve, and observe that the denominator of the $x$ coordinate is a perfect square.

sage: E = EllipticCurve('446d1')
sage: P = 3*E.gen(0); P
(32/49 : -510/343 : 1)

Let

$$x(t) = \frac{1}{t^2} + \cdots \in \mathbb{Z}_p((t))$$ (2.5.5)

be the formal power series that expresses $x$ in terms of the local parameter $t=-x/y$ at infinity. Similarly, let $y(t) = -x(t)/t$ be the corresponding series for $y$. If we do the change of variables $t=-x/y$ and $w=-1/y$, so $x=t/w$ and $y=-1/w$, then the Weierstrass equation for $E$ becomes

$$s = {t}^{3} + {{a_{1} s} t} + {{a_{2} w} {t}^{2} } + {a_{3} {w}^{2} } + {{a_{4} {w}^{2} } t} + {a_{6} {w}^{3} } = F(w,t).$$

Repeatedly substituting this equation into itself recursively yields a power series expansion for $w=-1/y$ in terms of $t$, hence for both $x$ and $y$.

Remark 2.24   The formal group of $E$ is a power series

$$F(t_1, t_2) \in R = \mathbb{Z}[a_1,\ldots, a_6][[t_1, t_2]].$$

defined as follows. Since $x(t)$ and $y(t)$ satisfy the equation of $E$, the points $P_1 = (x(t_1),y(t_1))$ and $P_2 = (x(t_2),y(t_2))$ are in $E(R)$. As explained explicitly in [Sil92, §IV.1], their sum is

$$Q = P_1 + P_2 = (x(F), y(F)) \in E(R)$$

for some $F = F(t_1, t_2) \in R$.

Example 2.25   We compute the above change of variables in SAGE:

sage: var('a1 a2 a3 a4 a6')
sage: E = EllipticCurve([a1,a2,a3,a4,a6]); E
Elliptic Curve defined by
     y^2 + a1*x*y + a3*y = x^3 + a2*x^2 + a4*x + a6
over Symbolic Ring
sage: eqn = SR(E); eqn
(y^2 + a1*x*y + a3*y) == (x^3 + a2*x^2 + a4*x + a6)
sage: F = eqn.lhs() - eqn.rhs(); F
y^2 + a1*x*y + a3*y - x^3 - a2*x^2 - a4*x - a6
sage: G = w^3*F(x=t/s, y=-1/w); G.expand()
-t^3 - a2*w*t^2 - a4*w^2*t - a1*w*t - a6*w^3 - a3*w^2 + w

Example 2.26   We use SAGE to compute the formal power series $x(t)$ and $y(t)$ for the rank $1$ elliptic curve 37a.

sage: E = EllipticCurve('37a'); E
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: F = E.formal_group(); F
Formal Group associated to the Elliptic Curve defined by
y^2 + y = x^3 - x over Rational Field
sage: x = F.x(prec=8); x
t^-2 - t + t^2 - t^4 + 2*t^5 - t^6 - 2*t^7 + O(t^8)
sage: y = F.y(prec=8); y
-t^-3 + 1 - t + t^3 - 2*t^4 + t^5 + 2*t^6 - 6*t^7 + O(t^8)

Notice that the power series satisfy the equation of the curve.

sage: y^2 + y == x^3 - x
True

Recall that $\omega_E = \frac{dx}{2y + \underline{a}_1 x + \underline{a}_3}$ is the differential on a fixed choice of Weierstrass equation for $E$. Let

$$\omega(t) = \frac{dx}{2y + \underline{a}_1 x + \underline{a}_3} \in \mathbb{Q}((t)) dt$$

be the formal invariant holomorphic differential on $E$.

Example 2.27   Continuing the above example, we compute the formal differential on $E$:

sage: F.differential(prec=8)
1 + 2*t^3 - 2*t^4 + 6*t^6 - 12*t^7 + O(t^8)

We can also compute $\omega(t)$ directly from the definition:

sage: x.derivative()/(2*y+1)
1 + 2*t^3 - 2*t^4 + 6*t^6 - 12*t^7 + 6*t^8 + 20*t^9 + O(t^10)

The following theorem, which is proved in [MT91], uniquely determines a power series  $\sigma\in t\mathbb{Z}_p[[t]]$ and constant $c\in\mathbb{Z}_p$.

Theorem 2.28 (Mazur-Tate)   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

$$x(t) + c = -\frac{d}{\omega}\left( \frac{1}{\sigma} \frac{d\sigma}{\omega}\right),$$ (2.5.6)

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

The above theorem produces a (very inefficient) algorithm to compute $c$ and $\sigma(t)$. Just view $c$ as a formal indeterminate and compute $\sigma(t) \in \mathbb{Q}[c][[t]]$, then obtain constraints on $c$ using that the coefficients of $\sigma$ must be in $\mathbb{Z}_p$. These determine $c$ to some precision, which increases as we compute $\sigma(t)$ to higher precision. Until recently this was the only known way to compute $c$ and $\sigma(t)$ - fortunately the method of [MST06] is much faster in general.

Definition 2.29 (Canonical $p$-adic Height)   Let $E$ be an elliptic curve over $\mathbb{Q}$ with good ordinary reduction at the odd prime $p$. Let $\log_p$, $d$, and $\sigma(t)$ be as above and suppose $P\in E(\mathbb{Q})$ and that $nP$ is a nonzero multiple of $P$ such that $nP$ reduces to the identity component of the Néron model of $E$ at each prime of bad reduction. Then the $p$-adic canonical height of $P$ is

$$h_p(P) = \frac{1}{n^2}\cdot \frac{1}{p} \cdot \log_p\left(\frac{\sigma(P)}{d(P)}\right).$$

Definition 2.30 ($p$-adic Regulator)   The $p$-adic regulator of $E$ is the discriminant (well defined up to sign) of the bilinear $\mathbb{Q}_p$-valued pairing

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

Conjecture 2.31 (Schneider)   The $p$-adic regulator $\Reg _p(E)$ is nonzero.

Theorem 2.32 (Kato, Schneider, et al.)   Let $E$ be an elliptic curve over $\mathbb{Q}$ with good ordinary reduction at the odd prime $p$ and assume that the $p$-adic Galois representation $\rho_{E,p}$ is surjective. If

$$\ord _T(\mathcal{L}_p(E,T)) \leq \rank E(\mathbb{Q}),$$

then $\char93 {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E/\mathbb{Q})(p)$ is finite. Moreover, if $\Reg _p(E)$ is nonzero, then

$$\ord _p(\char93 {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}... ...c{\mathcal{L}_p^*(E,0)}{\prod c_\ell \cdot \Reg _p(E)}\right).$$