Optimal bounds for the difference between the Néron–Tate height and the Weil height on elliptic curves over Q

I · MathInstitut f ¨ur Mathematik

Optimal bounds for

the difference between

the N ´eron–Tate height

and the Weil height

on elliptic curves over Q

Peter Bruin (work in progress)

Introduction

We consider elliptic curves E /Q , given by a Weierstrass

equation

y2+a1xy + a3y = x3+a2x2+a4x + a6.

There are two height functions E (Q) → R: the na¨ıve (Weil)

height h and the canonical (N ´eron–Tate) height ˆh .

N ´eron (1965) noted in his fundamental work on height functions that the function

h − ˆh : E (Q) → R

is bounded.

Explicit bounds on h − ˆh are important, e.g. for finding

generators for Mordell–Weil groups of elliptic curves over number fields.

Heights on P

Let x ∈P1(Q). We define the height of x as

h_P1(x ) = 1 [K :Q] X v place of K log max{|a|v, |b|v}

if x = (a : b) with a ,b in some number field K ⊂ Q . Here

| |_v is the normalised absolute value corresponding to the

place v .

The right-hand side is independent of the representation x = (a : b) because of the product formula, and independent of K because of the behaviour of valuations under field extensions. We therefore get a well-defined function

Heights on E

To every point P ∈ E (Q) there are associated two heights:

– the na¨ıve height of P ,

h(P) = h_P1(x (P)).

Advantage: easy to compute. – the canonical height of P ,

ˆ h(P) = lim n→∞ 1 n2h(nP) = lim_m→∞ 1 4mh(2 m_P).

Advantages: ˆh is a quadratic form, ˆh(P) ≥ 0 with

equality if and only if P is a torsion point, and it occurs in the conjecture of Birch and Swinnerton-Dyer.

Goal: compute sup(h − ˆh), inf(h − ˆh) to given precision. The result will depend on E and on the choice of coordinate

Earlier work

N ´eron (1965): remarked that h − ˆh is bounded.

Dem0janenko (1968), Zimmer (1976): first explicit bounds.

Tate (1979, unpublished): recipe for computing ˆh .

Buhler, Gross and Zagier (1985): for the curve 5077A1

(y2+y = x3− 7x + 6), which has rank 3, sharp bounds for

h − ˆh onQ-points are

−1.205 . . . ≤ h(P) − ˆh(P) ≤ 0,

where the lower bound is attained for P = (−1, 3) and the upper bound for P = O .

Silverman (1990): more precise bounds than those of

Dem0janenko and Zimmer, by decomposing into local

Earlier work

Siksek (1995): sharper bounds for h − ˆh for rational points

of elliptic curves over number fields, also using local contributions.

Cremona, Prickett and Siksek (2006): by analysing the various reduction types at finite places, they improve

Siksek’s algorithm to make it easier to implement (and make it rigorous at complex places).

Remark: the above methods all rely in some way or another on the properties of the multiplication-by-2 map on elliptic curves. This leads in general to non-optimal bounds.

Cremona, Prickett and Siksek (unpublished), Uchida (2008): consider multiplication-by-m map for general m .

I will now explain how to avoid multiplication maps, using an approach that leads to optimal bounds at complex places.

Very brief overview of Arakelov theory

Let K be a number field, ZK its ring of integers.

An arithmetic surface is a projective flat ZK-scheme X

which is regular and of Krull dimension 2. (Think of the minimal regular model of an elliptic curve.) For every infinite

place v of X , write Xv for the Riemann surface X ( ¯Kv).

An Arakelov divisor is a formal linear combination of integral

1-dimensional subschemes of X (with Z-coefficients) and of

symbols Xv (with R-coefficients), where v runs over the

infinite places of K . One can associate to every f ∈ K (X ) a principal Arakelov divisor. Let

b

Cl X = {Arakelov divisors}/{principal divisors}. There is an intersection pairing

Green functions

Let X → SpecZK be an arithmetic surface, and let P 6= Q

be two sections. Let Kfin, Kinf be the sets of finite and

infinite places of K , respectively. The intersection pairing

( . ) is a sum of local contributions:

(P . Q) = (P . Q)fin+

X

v ∈Kinf

(P . Q)v.

The local contribution to (P . Q) at a place v ∈ Kinf is

(P . Q)v = −[Kv :R] grXv(Pv,Qv).

Here Pv,Qv ∈ Xv come from P, Q by base extension and

gr_Xv: Xv× Xv \ diagonal → R

is the canonical Green function on Xv. With respect to a

local coordinate z on Xv, one has

The Green function of an elliptic curve

For τ, w ∈C with =τ > 0, we put, with q = exp(2πiτ ),

η(τ ) =q1/24Y n≥1 (1 − qn), θ1(w ; τ ) = −i X n∈Z (−1)nq(n+1/2)2/2exp(2πi(n + 1/2)w
, kηk(τ ) = (=τ )1/4|η(τ )|, kθ1k(w ; τ ) = (=τ )1/4exp  −π(=w ) 2 =τ  |θ1(w ; τ )|.

Key formula 1 (Faltings): for a lattice Λ = Zω₁⊕ Zω2⊂ C,

the Green function of the complex elliptic curve C/Λ is

gr_C/Λ(0, z) = logkθ1k(z/ω2; ω1/ω2)

kηk(ω1/ω2)

Na¨ıve heights and intersection theory

Let E be an elliptic curve over Q , and let P ∈ E (Q). We

take a number field K that is sufficiently large in the sense that E is defined over K and has split semi-stable reduction

over ZK, and P ∈ E (K ). For every infinite place v of K , we

identify ¯Kv with C and put

E_v =E ( ¯Kv) ∼= C/Λv,

where Λv ⊂ C is the period lattice of E( ¯Kv).

Simplifying assumption: the given equation for E defines

a minimal Weierstrass model over SpecZ_K.

In this situation, one can show using the definition of local intersection numbers at the finite places that

[K :Q]h(P) = 2(O . P)fin+ X

v ∈Kinf

log max{1, |x (P)|v},

Na¨ıve heights and intersection theory

Let ℘v be the Weierstrass ℘-function with respect to Λv,

and let Pv be the image of P in Ev =C/Λv. We note that

℘v(Pv) =x (P) + b2 12 in ¯Kv ∼ = C. This implies h(P) = 2(O . P) + X v ∈Kinf [Kv :R]φv(Pv),

where φv: Ev → R is the continuous function defined by

φv(z) = log max n 1,   ℘v(z) − b2 12    o +2 gr_Ev(0, z).

Canonical heights and intersection theory

Faltings–Hriljac formula relating heights to the intersection

pairing on the semi-stable arithmetic surface E → SpecZK:

(P − O − Ψ . P − O) = −[K :Q]ˆh(P),

where Ψ is a fibral Q-divisor such that P − O − Ψ has

intersection number 0 with every vertical divisor.

Let DE /K be the discriminant ideal of E over K :

D_{E /K} = Y

v ∈Kfin

pnv

v ,

where pv ⊂ ZK is the maximal ideal corresponding to v and

nv is the number of components of the reduction of E at v .

Szpiro’s formula for the self-intersection of a point:

(P . P) = − 1

Canonical heights and intersection theory

For every finite place v of K , let mv(P) ∈ {0, 1, . . . , nv − 1}

be the component of the reduction of E modulo v to which P reduces.

Using the Faltings–Hriljac formula and Szpiro’s formula, one can prove [K :Q]ˆh(P) = 2(O . P) + X v ∈Kfin B2  mv(P) nv  nvlog #k (v ),

where B2(t) = t2− t + 1/6 is the second Bernoulli function:

-1/12 0 1/6

Comparing the two heights

Recall our expressions for h and ˆh :

[K :Q]h(P) = 2(O . P) + X v ∈Kinf [Kv :R]φv(Pv), [K :Q]ˆh(P) = 2(O . P) + X v ∈Kfin B2  mv(P) nv  nvlog #k (v ).

Subtracting them, we get [K :Q](h(P) − ˆh(P)) = X v ∈Kinf [Kv :R]φv(Pv) − X v ∈Kfin B2  mv(P) nv  nvlog #k (v ).

Height bounds

Theorem: Let E be an elliptic curve over Q . Suppose E is

given by a Weierstrass equation over a number field K ⊂ Q

such that E has semi-stable reduction over K and such that the given equation defines a minimal Weierstrass model. Let

aE,bE ∈ R be defined by [K :Q]aE = X v ∈Kinf [Kv :R] inf z∈Ev φv(z) − 1 6log norm DE /K, [K :Q]bE = X v ∈Kinf [Kv :R] sup z∈Ev φv(z) + 1 12log norm DE /K.

Then the height difference h − ˆh satisfies

a_E ≤ h(P) − ˆh(P) ≤ b_E for all P ∈ E (Q),

Height bounds

Sketch of the proof:

– After enlarging K , we may assume P ∈ E (K ). – Use our earlier formula for the height difference.

– Use the fact that −1/12 ≤ B2(x ) ≤ 1/6.

– To prove optimality of the bounds: after enlarging K , we may assume that K is totally complex. Then by

approximation on P1(K ), we can choose an

x -coordinate at which all local height difference functions are arbitrarily close to their minima/maxima. We now want to turn this theorem into an algorithm by

showing that aE and bE are computable.

For this it remains to show that we can compute the infimum

Complex local heights and Green functions

Here ±t ∈C/Λv are the two points where x = 0, and c is

the unique real number making this function continuous.

This is a very useful expression for φv(z):

– Thanks to Faltings’s formula for gr_C/Λ in terms of θ1, it

is easy to compute φv(z) to any desired precision.

– The derivatives of φv (outside

℘_v(z) −₁₂b2 

=1) can be

bounded relatively easily, so the infimum and supremum

Example

Consider the curve 11A3 (= modular curve X1(11)):

y2+y = x3− x2.

Then our algorithm gives

−0.556 ≤ h(P) − ˆh(P) ≤ 0.798 . . . for all P ∈ E (Q).

The lower bound is very sharp for the points with x = −1,

which are defined over Q(√−7).

The upper bound is very sharp for the points with

x = 37/61, which are defined over Q(√7 · 11 · 17 · 61 · 73).

This x was found by approximation in P1_{(Q): the local}

height differences are maximal when x is close to −3 in the 11-adic topology and close to 0.606 in the real topology.

Silverman’s bounds are −4.219 . . . ≤ h − ˆh ≤ 4.695 . . .

(Cremona–Prickett–)Siksek’s upper bound for Q-points is

Plot of

φ(

z

)

for the curve 11A3

Example

Consider the curve 15A4:

y2+xy + y = x3+x2+35x − 28.

Then our algorithm gives

−1.982 . . . ≤ h(P) − ˆh(P) ≤ 3.768 . . . for all P ∈ E (Q).

Silverman’s bounds are −6.546 . . . ≤ h − ˆh ≤ 8.184 . . .

(Cremona–Prickett–)Siksek’s upper bound for Q-points is

Plot of

φ(

z

)

for the curve 15A4

Example

Consider the curve 5077A1 (rank 3, studied by Buhler, Gross and Zagier):

y2+y = x3− 7x + 6.

Then our algorithm gives

−1.205 . . . ≤ h(P) − ˆh(P) ≤ 2.133 . . . for all P ∈ E (Q).

Silverman’s bounds are −5.048 . . . ≤ h − ˆh ≤ 6.284 . . .

(Cremona–Prickett–)Siksek’s upper bound for Q-points is

Plot of

φ(

z

)

for the curve 5077A1