Points on elliptic curves
with several integral multiples:
algebra, geometry,
and some applications
Noam D. Elkies
Harvard University
1
0. Overview
We explicitly parametrize elliptic curves E with
a points P such that nP is integral⋆ for each
n ∈ S. The case S = {1, 2, 3, 4} is particularly
nice, leading to an open set in P2. Applications
include:
∼ 16);
• Explicit formulas for X1(N ) (N up to =
• Parametrizations for yet larger S, by blowups of P2;
• Elliptic surfaces with non-torsion sections of
low height;
• Curves E/Q and nontorsion P ∈ E(Q) with
low height and/or many integral multiples.
(⋆) This is intentionally vague, since it applies
in arithmetic as well as geometric settings.
2
1. P , 2P , 3P , 4P integral: Algebra
We parametrize pairs (E, P ) where E is an elliptic curve and P is an integral point whose
first few multiples are integral as well. We use
repeatedly a consequence of the group law: if
P, Q are integral then P + Q is integral iff the
secant P Q (or the tangent at P = Q) has integral slope.
(i) P integral
We write E in extended Weierstrass form
y 2 + a1xy + a3y = x3 + a2x2 + a4x + a6
(often specified by just the list
(a1, a2, a3, a4, a6)
of coefficients), with coordinates x, y chosen so
that P is at the origin (0, 0) — this is possible
because P is integral. Thus a6 = 0 .
3
(ii) P , 2P integral
Now 2P is integral iff the tangent to E at P
has integer slope. This slope is a4/a3.
NB The denominator a3 vanishes iff
2P = 0, so we have a (redundant) parametrization of the modular curve X1(2).
If a4/a3 is integral, we translate y by (a4/a3)x
to make the slope zero. We then have an
equivalent equation for E with a4 = 0 (and
a6 = 0 as well). From here on, we’ll retain
this normalization, so our curves will be of the
form
E : y 2 + a1xy + a3y = x3 + a2x2
and we’ll specify them by just listing (a1, a2, a3).
4
(iii) P , 2P , 3P integral
Now P = (0, 0) and 2P = (a2, a1a2 − a3). The
secant from P to 2P has slope (a3/a2) − a1.
NB The denominator a2 vanishes iff
3P = 0, so we have parametrized the
modular curve X1(3): (a1, 0, a3, 0, 0).
So the slope, and equivalently 3P , is integral
iff a3 = b1a2 for some integral b1. Then 2P =
(−a2, (a1 − b1)a2), and
3P = (b1(b1 − a1), b1(a1b1 − a2 − b2
1 )).
5
(iv) P , 2P , 3P , 4P integral
The slope of the secant between from P to 3P
is then (a2/(b1 − a1)) − b1.
NB The denominator b1 − a1 vanishes
iff 4P = 0, so we have parametrized the
modular curve X1(4): (a1, a2, a1a2, 0, 0).
So the slope, and equivalently 4P , is integral
iff a2 = c1(b1 − a1) for some integral c1. Now
that all our parameters have weight 1, we drop
the subscripts and change variables by writing
a (née a1) as b − d, obtaining finally:
(a1, a2, a3) = (b − d, cd, bcd) .
The first four multiples are then P = (0, 0),
2P = (−cd, −cd2), 3P = (bd, −bd(b + c)),
4P = (c(b + c), c2(b + c + d)).
6
2. P , 2P , 3P , 4P integral: Geometry
Changing (b, c, d) to (λb, λc, λd) yields an isomorphic curve (and if b, c, d have a common
factor then our equation for E was not minimal). So we have a birational parametrization of the (E, P ) moduli space by an open set
in P2.
The discriminant is b2c3d4 times the cubic
δ = b3 + (c − 3d)b2 + (3d − 20c)bd − d(4c + d)2,
whose zero-locus has a cusp at (b : c : d) =
(−32 : 27 : 4), and is isomorphic with P1 by
(b : c : d) = (µ(µ + 2)2 : 1 : µ2(µ + 1)).
Picture of the discriminant locus:
7
d/c
µ = −2/3
µ = −1
µ=0
b/c
µ = −2
[Multiplicity 1 on the cubic, 2 at b = 0 (vertical
axis), 3 at c = 0 (line at infinity), and 4 at
d = 0 (horizontal axis); the cubic meets c = 0
triply at the µ = ∞ point (1 : 0 : 1).]
8
Now a curve C of degree d in this P2 amounts
to an elliptic surface S→C of discriminant degree 12d, together with a section P : C→S.
At least if C does not contain (0 : 1 : 0), this
section as well as 2P, 3P, 4P are integral: they
do not meet the zero-section.
Moreover, a generic intersection of C with the
line b = 0, c = 0, or d = 0 gives a multiplicative
fiber of type I2, I3, or I4 respectively, with P
going through a component adjacent to the
identity component. This is what we denote
by a [1/2], [1/3], or [1/4] fiber in the ANTS7 paper. The cubic gives I1 [0/1]. Multiple
intersections combine; e.g. if C goes through
the point c = d = 0 it has a [2/7] fiber there,
assuming c and d have simple zeros; if b is a
simple and c a double zero, we get [3/8] (since
3 = 1 + 2 and 8 = 1 · 2 + 2 · 3).
9
If nP is integral then h(nP ) = 2d, so ĥ(P ) =
n−2ĥ(nP ) ≤ 2d/n2. Conversely, nontorsion P
of very low canonical height tend to have nP
integral for several small n. For d = 1, 2, 3
and the smallest positive ĥ(P ), these n include
1, 2, 3, 4 so we easily recover the surfaces from
our P2 picture.
d = 1: Here the minimal ĥ(P ) is 1/30 [OguisoShioda 1991], P meeting non-identity components [1/5], [1/3], [1/2]. This yields lines
through the point (b : c : d) = (−1 : 1 : 0),
at which d = δ = 0. The general such line
(s′ − s : −s′ : qs) yields the Oguisa-Shioda surfaces. (Lines through b = c = 0 yield ĥ(P ) =
1/20, the next-smallest value, with components [2/5], [1/4].)
10
d = 2: Here the minimum is 11/420 [Nishiyama
1996], with components
[1/7], [2/5], [1/4], [1/3], [1/2].
So, conics through b = c = 0 (to combine
[1/2], [1/3]→[2/5]) that meet δ = 0 to order 3
at b + c = d = 0 (to promote one [1/4] to
[1/7]). Again a 1-dimensional linear system.
[K3 surfaces with NS(S) of rank 19 and disc. 22,
parametrized by X0(11)/w, “etc.”]
d = 3: Now the minimal ĥ(P ) is 23/840, at
least for C rational (NDE 2002), with components
[1/8], [3/7], [1/5], [1/4], [1/3], [1/3], [1/2].
So, cubics tangent to b = 0 at b+c = d = 0 (for
the [3/7]) with a double point at b + c = d = 0,
one of whose branches meets δ = 0 to order 4
there. Yet again a 1-dimensional linear system
of rational curves C.
11
In each case, we prove minimality by trying all
possible fiber configurations.
[This can be done also for d > 3, but combinatorial explosion sets in — not of the total
number of configurations (which is manageable at least through d = 7) but of apparent
record configurations that are overdetermined
must be excluded case by case.]
Curiously for each of d = 1, 2, 3 the minimal
ĥ(P ) is also characterized by the maximal M
s.t. nP is integral for each n ∈ {1, 2, . . . , M }
(namely M = 6, 8, 9). [This is no longer true
fr d = 4, and probably not for any d > 3.]
So, how to extend the {1, 2, 3, 4} analysis to
M > 4?
12
d/c
b/c
6P = 0
5P = 0
13
3. Beyond {1, 2, 3, 4}
We continue as before. The modular curves
X1(5) and X1(6) are the lines b + c = 0 and
b + c + d = 0 respectively; and 5P or 6P is
integral iff (b+c)|cd or (b+c+d)|d2 respectively.
We make (b + c)|cd by taking
b + c = rs, c = rs′, d = r′s
with gcd(r, r′) = gcd(s, s′) = gcd(r, s) = 1.
Geometrically, we blow up the pts. (b : c : d) =
(−1 : 1 : 0) and (0 : 0 : 1) where b + c = 0
meets c = 0 and d = 0; we may then blow
down the line connecting them (on which P is
5-torsion) to obtain P1 × P1.
We deal with 6P similarly, obtaining a known
but less familiar rational surface called F1.
With more work, we can keep going this way
for a while — though not forever: no way to
blow down X1(N ) for N = 11 or N > 12.
One good stopping place is:
14
b = (u + 1)2 (u + A + 1) (Au2 − Au − 2u − A − 1),
c = −(u + 1) (u + A + 1)
(Au3 −Au2−2u2−Au−2u−A−1),
d = −u (Au − A − 1) (Au2 + u2 + u + A + 1).
Here each of the lines X1(n) for n ≤ 8 has been
blown down, so nP is integral for n ≤ 8; also
X1(9), X1(10), and X1(12) are simply A = −1,
A = −1/2, and A = 0 respectively; and for
N = 11, 13, 14, 15, the curve X1(N ) is given by
(u2 − 2u − 1)A2 = (4u + 1)A + u
(u3 − u2 + u + 1)A2 + (u3 − u2 + 2u + 1)A = u2,
(u2 − 1)A2 + (2u2 − u − 1)A + u2 = 0,
(3u2 − 1)A2 + (3u2 − u − 1)A + u2 = 0.
15
Each of those is quadratic in A and yields a
hyperelliptic equation for X1(N ). (For N =
16, 17, 18, . . . we still get reasonably simple formulas but with some singularities.)
Other applications: specializing A, u to Q yields
E/Q with nontorsion points of small height,
e.g. the record in the Cremona tables extended
to conductor 12000, located by Wm. Stein
[cond.= 3990], appears at (A, u) = (1/2, 3/7);
suitable specializations yield infinite (“Pell”)
families of (E/Q, P ) with nP is integral for
n ≤ 12, and one case where 13P and 14P (and
also 18P ) are integral as well, namely (A, u) =
(−45/41, −12/5) [conductor 1029210]; etc.
16
4. Further directions
Ongoing work: deal with d = 4 and beyond;
curves with nontrivial torsion and niP + Ti integral; curves with points P, Q and miP + niQ
integral, and/or det ĥ(P, Q) small; etc.
Example (Sonal Jain): If T is 2-torsion then
P , P + T , 2P , 2P + T all integral iff E is
y 2 = x3 + (a2 − 2bc)x2 + (b2 − a2)c2x
for some (a : b : c) ∈ P2, with T = (0, 0) and
P = ((a + b)c, (a + b)ac). The minimal ĥ(P ) for
d = 1 is 1/12, for lines in the (a : b : c) plane
through the point (0 : 1 : 1) where the components a = 0, 4c(b − c) = a2 of the discriminant
locus meet.
Discriminant picture:
17
b/c
a/c
[∆ vanishes on the conic 4c(b − c) = a2 with
multiplicity 1, on a = 0 (vertical axis) and b =
±a (diagonals) with multiplicity 2, and on c = 0
(line at infinity) with multiplicity 4; the conic
meets the lines a = 0, c = 0 at (0 : 1 : 0); the
involution a ↔ −a takes P = ((a+b)c, (a+b)ac)
to P + T .]
18
c=0
a=b
a=0
a = -b
19