I · Math
Institut für Mathematik
Optimal bounds for
the difference between
the Néron–Tate height
and the Weil height
on elliptic curves over Q
Peter Bruin
(work in progress)
Selmer Groups, Descent and the Distribution of Ranks – Warwick, 28 September 2012
Introduction
We consider elliptic curves E/Q , given by a Weierstrass
equation
y 2 + a1 xy + a3 y = x 3 + a2 x 2 + a4 x + a6 .
There are two height functions E(Q) → R: the naı̈ve (Weil)
height h and the canonical (Néron–Tate) height ĥ .
Néron (1965) noted in his fundamental work on height
functions that the function
h − ĥ : E(Q) → R
is bounded.
Explicit bounds on h − ĥ are important, e.g. for finding
generators for Mordell–Weil groups of elliptic curves over
number fields.
Heights on P1
Let x ∈ P1 (Q). We define the height of x as
hP1 (x) =
1
[K : Q]
X
log max{|a|v , |b|v }
v place of K
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
hP1 : P1 (Q) → R.
Heights on E
To every point P ∈ E(Q) there are associated two heights:
– the naı̈ve height of P ,
h(P) = hP1 (x(P)).
Advantage: easy to compute.
– the canonical height of P ,
ĥ(P) = lim
n→∞
1
1
h(nP) = lim m h(2m P).
m→∞ 4
n2
Advantages: ĥ is a quadratic form, ĥ(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 − ĥ), inf(h − ĥ) to given precision.
The result will depend on E and on the choice of coordinate
function x ∈ Γ(E, OE (2O)), but not on y .
Earlier work
Néron (1965): remarked that h − ĥ is bounded.
Dem 0 janenko (1968), Zimmer (1976): first explicit bounds.
Tate (1979, unpublished): recipe for computing ĥ .
Buhler, Gross and Zagier (1985): for the curve 5077A1
(y 2 + y = x 3 − 7x + 6), which has rank 3, sharp bounds for
h − ĥ on Q-points are
−1.205 . . . ≤ h(P) − ĥ(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
Dem 0 janenko and Zimmer, by decomposing into local
heights (following an idea of Lang).
Earlier work
Siksek (1995): sharper bounds for h − ĥ 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 (K̄v ).
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 X = {Arakelov divisors}/{principal divisors}.
Cl
There is an intersection pairing
b X × Cl
b X → R.
( . ) : Cl
Green functions
Let X → Spec ZK be an arithmetic surface, and let P 6= Q
be two sections. Let K fin , K inf be the sets of finite and
infinite places of K , respectively. The intersection pairing
( . ) is a sum of local contributions:
X
(P . Q) = (P . Q)fin +
(P . Q)v .
v ∈K inf
The local contribution to (P . Q) at a place v ∈ K inf is
(P . Q)v = −[Kv : R] grXv (Pv , Qv ).
Here Pv , Qv ∈ Xv come from P, Q by base extension and
grXv : Xv × Xv \ diagonal → R
is the canonical Green function on Xv . With respect to a
local coordinate z on Xv , one has
grXv (P, Q) = log |z(P) − z(Q)| + O(1).
The Green function of an elliptic curve
For τ, w ∈ C with =τ > 0, we put, with q = exp(2πiτ ),
Y
η(τ ) = q 1/24
(1 − q n ),
n≥1
θ1 (w; τ ) = −i
X
2 /2
(−1)n q (n+1/2)
exp(2πi(n + 1/2)w ,
n∈Z
kηk(τ ) = (=τ )1/4 |η(τ )|,
(=w)2
1/4
kθ1 k(w; τ ) = (=τ ) exp −π
|θ1 (w; τ )|.
=τ
Key formula 1 (Faltings): for a lattice Λ = Zω1 ⊕ Zω2 ⊂ C,
the Green function of the complex elliptic curve C/Λ is
grC/Λ (0, z) = log
kθ1 k(z/ω2 ; ω1 /ω2 )
.
kηk(ω1 /ω2 )
Naı̈ve 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 K̄v with C and put
∼ C/Λv ,
Ev = E(K̄v ) =
where Λv ⊂ C is the period lattice of E(K̄v ).
Simplifying assumption: the given equation for E defines
a minimal Weierstrass model over Spec ZK .
In this situation, one can show using the definition of local
intersection numbers at the finite places that
X
[K : Q]h(P) = 2(O . P)fin +
log max{1, |x(P)|v },
v ∈K inf
where O , P are now viewed as sections of E → Spec ZK .
Naı̈ve 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
b2
12
in K̄v ∼
= C.
X
[Kv : R]φv (Pv ),
℘v (Pv ) = x(P) +
This implies
h(P) = 2(O . P) +
v ∈K inf
where φv : Ev → R is the continuous function defined by
n φv (z) = log max 1, ℘v (z) −
o
b2 12 + 2 grEv (0, z).
Canonical heights and intersection theory
Faltings–Hriljac formula relating heights to the intersection
pairing on the semi-stable arithmetic surface E → Spec ZK :
(P − O − Ψ . P − O) = −[K : Q]ĥ(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 :
Y
DE/K =
pnv v ,
v ∈K fin
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
log norm DE/K .
12
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
X
mv (P)
[K : Q]ĥ(P) = 2(O . P) +
B2
nv log #k(v ),
nv
fin
v ∈K
where B2 (t) = t 2 − t + 1/6 is the second Bernoulli function:
1/6
0
-1/12
0
1/2
1
Comparing the two heights
Recall our expressions for h and ĥ :
X
[K : Q]h(P) = 2(O . P) +
[Kv : R]φv (Pv ),
v ∈K inf
X
[K : Q]ĥ(P) = 2(O . P) +
B2
v ∈K fin
mv (P)
nv
nv log #k(v ).
Subtracting them, we get
[K : Q](h(P) − ĥ(P)) =
X
[Kv : R]φv (Pv )
v ∈K inf
−
X
v ∈K fin
B2
mv (P)
nv
nv log #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 ∈K inf
[K : Q]bE =
X
v ∈K inf
[Kv : R] inf φv (z) −
1
log norm DE/K ,
6
[Kv : R] sup φv (z) +
1
log norm DE/K .
12
z∈Ev
z∈Ev
Then the height difference h − ĥ satisfies
aE ≤ h(P) − ĥ(P) ≤ bE
and these bounds are optimal.
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
and supremum of the functions φv to given accuracy.
Complex local heights and Green functions
Key formula 2:
(
φv (z) =
2 grEv (0, z)
grEv (t, z) + grEv (−t, z) + c
if
if
℘ (z) −
v
℘v (z) −
b2 12 b2 12
≤ 1,
> 1.
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 grC/Λ in terms of θ1 , it
is easy to compute φv (z) to any desired precision.
b2 – The derivatives of φv (outside ℘v (z) − 12
= 1) can be
bounded relatively easily, so the infimum and supremum
of φv can be computed to any desired precision.
Example
Consider the curve 11A3 (= modular curve X1 (11)):
y 2 + y = x 3 − x 2.
Then our algorithm gives
−0.556 ≤ h(P) − ĥ(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 − ĥ ≤ 4.695 . . .
(Cremona–Prickett–)Siksek’s upper bound for Q-points is
0.299 . . .
0
1
2
3
4
5
z
6
7
8
0
-0.2
-0.4
-0.6
-0.8
-1
-1.2
-1.4
-1.6
9 10
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
φ(z)
Plot of φ(z ) for the curve 11A3
Example
Consider the curve 15A4:
y 2 + xy + y = x 3 + x 2 + 35x − 28.
Then our algorithm gives
−1.982 . . . ≤ h(P) − ĥ(P) ≤ 3.768 . . .
for all P ∈ E(Q).
Silverman’s bounds are −6.546 . . . ≤ h − ĥ ≤ 8.184 . . .
(Cremona–Prickett–)Siksek’s upper bound for Q-points is
3.915 . . ..
0
0.5
1
1.5
z
2
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
2.5
2.6
2.4
2.2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
φ(z)
Plot of φ(z ) for the curve 15A4
Example
Consider the curve 5077A1 (rank 3, studied by Buhler,
Gross and Zagier):
y 2 + y = x 3 − 7x + 6.
Then our algorithm gives
−1.205 . . . ≤ h(P) − ĥ(P) ≤ 2.133 . . .
for all P ∈ E(Q).
Silverman’s bounds are −5.048 . . . ≤ h − ĥ ≤ 6.284 . . .
(Cremona–Prickett–)Siksek’s upper bound for Q-points is
exactly 0.
0
0.5
1
1.5
z
2
0
-0.2
-0.4
-0.6
-0.8
-1
-1.2
-1.4
-1.6
2.5
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
φ(z)
Plot of φ(z ) for the curve 5077A1