Computation of the Néron-Tate Height on Elliptic Curves€¦ · THE NERON-TATE HEIGHT ON ELLIPTIC CURVES 355 and k{6K - v(A)) A 4av 4 dv(2P) - 4dv(P) - M2(P)) 4 -|a(, if2P&, where

MATHEMATICS OF COMPUTATIONVOLUME 48, NUMBER 177JANUARY 19K7. PAGES 351-370

Computation of the Néron-Tate Height

on Elliptic Curves

By Heinz M. Tschöpe and Horst G. Zimmer

For Daniel Shanks on the occasion of his 10 th birthday

Abstract. Using Néron's reduction theory and a method of Täte, we develop a procedure for

calculating the local and global Néron-Tate height on an elliptic curve over the rationals. The

procedure is illustrated by means of two examples of Silverman and is then applied to

calculate the global Néron-Tate height of a series of rank-one curves of Bremner-Cassels and

of a series of rank-two curves of Selmer. In the latter case, the regulator is also computed, and

a conjecture of S. Lang is investigated numerically.

In dealing with the arithmetic of elliptic curves E over a global field K, the task arises of

computing the Néron-Tate height on the group E(K) of rational points of E over K. Solving

this task in an efficient manner is important, for instance, in view of calculations concerning

the Birch and Swinnerton-Dyer conjecture (see [2]) or of the conjectures of Serge Lang [6],

The purpose of this note is to suggest a procedure for performing the necessary calculations.

1. Multiplication Formulas. Let the elliptic curve E over any field K be defined by

a generalized Weierstrass equation

(E) y2 A axxy A a3y = x3 + a2x2 A a4x A a6 (a, e K).

As usual, we introduce the quantities (see [10], [11])

A2 = a2 A 4a2, b4 = axa3 A 2a4, b6 = aj A 4a6,

Ag = axa6 - axa3a4 A 4a2a6 A a2a\ - a\,

c4 = b\ - 24b4, c6 = -A3 + 36A2A4 - 216A6,

and the discriminant

A = -A22A8 - 8A43 - 27A2 + 9A2A4A6 * 0,

as well as the absolute invariant

j = cl/ts,

belonging to E over K.

The fact that E is nonsingular implies the nonvanishing of the partial derivatives

of the polynomial

F(x, y) — y2 A axxy A a3y - jc3 - a2x2 - a4x - a6

Received May 19, 1986; revised June 30, 1986.

1980 Mathematics Subject Classification. Primary 14-04, 14H45, 14K07, 14K.15; Secondary 10B10,

14G20,14G25,14H25.

©1987 American Mathematical Society

0025-5718/87 $1.00 + $.25 per page

351

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

352 HEINZ M.TSCHÖPE AND HORST G. ZIMMER

at every rational point P e E(K):

(Íf(,),fo>))*<o.o).The addition law in the additive Abelian group E(K) of rational points on E over K

is given by the following formulas:

For P = (xP, yP), Q = (xQ, yQ) e E(K), denote the sum by P A Q =

(xP+Q,yP+Q).Then,

i L. \Ayr~yQ\2, lyp-yç]xP+Q= -(xP + xQ)+ ——-- \ +ax — -a2,

y xP xqj yxP Xqj

yP - yQyp+Q = v - v (xp ~ xp + q) - aixp+Q - a3 - yP if P * Q

Xp Xq

and

x2P = -2xP A t2P A axtp - a2, y2P = tP(xP - x2P) - axx2P - a3- yP

(2) im 3x\A2a2xPAa<-axyP ,f

2yP + axxP A a3

Generalizing classical formulas (see [3], [4], [15]), we obtain

Proposition 1. For a rational point P e E(K) and an r e N, the r-fold rational

point has coordinates

where <I>r, ^r, and 2Qr are polynomials in x and y with coefficients in

Z[ax, a2, a3, a4, a6] given by the following recursion formulas:

•$! = x, 02 = xA - b4x2 - 2b6x - A8,

Sl1=y, % = 0, % = 1,% = 2yAa1xAa3,

% = 3x4 + b2x3 + 3b4x2 A 3b6x A A8,

*4 = *2[2;c6 + A2x5 + 5A4*4 + lOA«,*3 + 10A8x2 +(A2A8 - A4A6)x + A4A8 - A62]

and for r > 2,

<Pr = xyvr — Tr_1M/r+1,

2%ttr = *?_x%+2 - %^2+i - %%[ax<i>r + a,*,2],

\J> = vl/3* — \Ir \J>3*2r+l *r Tr+2 V-1V+1'

%%r=%[*r2-l%+2-%-2*r2+l\-

Moreover, í>r, as a polynomial in x, has degree r2 and leading coefficient 1, whereas

^ (resp. <í,21<fír), as a polynomial in x, has degree (r2 — l)/2 (resp. (r2 — 4)/2)

and leading coefficient r (resp. r/2) provided that r is odd (resp. even). If we assign

the weight 2, 3 or i to x, y or a¡, then each term of $r has weight 2r2 and each term of


THE NÉRON-TATE HEIGHT ON ELLIPTIC CURVES 353

^fr (resp. <^2lyi,r) has weight r2 - 1 (resp. r2 - 4). The coefficients of <L>r, tyr, as

polynomials in x, y, belong already to Z[A2, A4, A6, A8].

From Proposition 1, one derives the following

Corollary. For r g N, we put ^_r = -%. Then, for r, n g N, we have

(4) *?n(P) = %2r\P)^2(nP)

and, more generally,

(4') *2»(p) = n*2m2("~')(™>iP).v=l

Furthermore,

(5) X -X %+n(P)%-ÁP)() ,P "" ViPWiP) '

r+1 %(P A Q)%(P - Q),

*?(P)*?(Q)

f,s , ,V+1 ^r\r T A¿)Âr ~ S>) , ,r(6) Xrp-XrQ=(-l) -.T,2iPW,2/^-(Xp-tQ)

and finally, for r G N0,

(7) 4»2(2T) = $2,+ l(P)*2-,8(P), ^22(2rP) = ^2 + i(P)^2rs(P).

These formulas will be needed in the sequel.

2. Reduction Theory. Now let the elliptic curve E be defined by (E) over a

complete field K with respect to a discrete normalized additive valuation v, and

suppose that the corresponding residue field K of K is perfect. We assume the

equation defining E over K to be minimal with respect to the valuation v (see [11]).

Reducing E modulo v yields a cubic curve

(E) y2 + äxxy + ä3y = x3 + ä2x2 + ä4x + ä6 (à, G K)

over K with discriminant À. If Ä # 0, i.e., u(A) = 0, then Ë is an elliptic curve over

K, and E has good reduction at v. If, however, À = 0, i.e., u(A) > 0, then £ is a

rational curve over K, and E has bad reduction at v. In the latter case, E is said to

have multiplicative reduction or additive reduction modulo v, according as v(c4) = 0

or u(c4) > 0, respectively.

Denote by E0(K) the set of points in E(K) whose image under the reduction

map modulo v,

P: E(K)-*É(K),

is a nonsingular point on É over K. Then, EQ(K) is a subgroup of finite index in

E(K). Further, the set

EX(K) = [P = (*„ v>) g E(K)\v(xP) 4 -2, t;( v>) < -3}



is a subgroup of E0(K), sind the restriction p0 to E0(K) of the reduction map p

induces an injective homomorphism of the factor group

~Po: E0(K)/EX(K) ^ Ë0(K)

to the nonsingular part Ë0(K) of Ë(K).

We shall use the following result (see [11]).

Proposition 2. The above groups satisfy

E(K) = E0(K) if E has good reduction at v,

#(E(K)/E0(K)) divides v(j) if E has multiplicative reduction at v,

and

#(E(K)/E0(K)) 4 4 if E has additive reduction at v.

3. Definition of Height Functions. Now let K be a global field, that is, an

algebraic number field or a function field of finite transcendence degree over its field

of constants k. Then K possesses a complete set MK of nonequivalent additive

valuations v satisfying the sum formula

(S) Ya \vv(c) = 0 forO^ce K

with some positive multiplicities Xv g R (cf. [7], [13]).

For an elliptic curve E over K, given by the Weierstrass equation (E), we

introduce the quantities

(8) ix, = min{t;(A2),^(A4),it;(A6),if;(A8)}.

Let P = (xP, yP) g E(K) be any rational point and 6 = (oo,oo) designate the

point at infinity. Then we define the local Weil height on E(K) with respect to v by

setting

... . . /-imii.((.„ip(i,)) ili*»,

(9) *-(P>=(-K ur-0.

Then the global Weil height on E(K) is simply the sum, with multiplicities, over the

local Weil heights

d(p)= £ Kd„(P)veMK

(see [13]).

In order to define the global Néron-Tate height on E(K), we proceed in the same

way as with the global Weil height. However, before introducing the local Néron-Tate

height on E(K), we need some estimates.

Proposition 3. The local Weil height onE(K) satisfies the following estimates:

{-(6u(, - D(A)) + 5a, < d,(P AQ) + d„(P - Q)

(10) -2dv(P)-2dv(Q)-v(xp-xQ)

4-2av if P,Q,P ± Q + e,


THE NERON-TATE HEIGHT ON ELLIPTIC CURVES 355

and

k{6K - v(A)) A 4av 4 dv(2P) - 4dv(P) - M*2(P))

4 -|a(, if2P*&,

where the constant av can be chosen to be 0 or -log 2 according as the valuation v of K

is discrete or archimedean, respectively (see [13]).

These estimates are obtained as generalizations of those given in [13], [14]. At the

same time, they sharpen those cited.

Remark 1. It is interesting to note that the authors of [2] suggested that a

sharpening of the estimates in [13], [14] should be possible. Proposition 3 appears to

be a step in this direction.

Employing (10), the inequalities (11) can be further generalized.

Corollary. For any m G N, there are (recursively computable) nonnegative

constants c, m, c2 m G R depending on E, K, and v such that, given an arbitrary point

P G E(K) with mP # 0, we have

(IV) cUm 4 dv(mP) - m2dv(P) - Jd(*¿(P)) *S c2,m.

We are now in a position to define the local Néron-Tate height on E(K) with

respect to v. Let m, n g N and m > 2. Then, for a rational point P G E(K) such

that m"P =é <S for each n G N, we define the local Néron-Tate height of P with

respect to v by the limit formula

<*> v.<,)-B./4íî-líí3aáía\+±.(4).nôo \ m l m I í¿

Proposition 4. For an elliptic curve E defined by a Weierstrass equation (E) over a

global field K and any valuation v of K, the function 8V m, defined by (12) on the

rational point group E(K), exists, is independent of the choice of m g N, so that

8V m = Sv, and fulfills the relations

(13) 8V(P AQ)A 8V(P - Q) - 28„(P) - 28V(Q) - v(xP - xQ) A MA) = 0

for any two points P = (xP, yP), Q = (xQ, yQ) G E(K) such that P,Q,P ±Q* 0,

and

(14) 8„(rP) - r$P) - ±v{*?(P)) A ̂ -«(A) = 0

for any P = (xP, yP) G E(K) and r G N such that rP * 0.

Proof. The proof is an adaptation of the corresponding proof of the existence

theorem in [14]. Indeed, one exploits (10), (11) from Proposition 3 and (11') from the

corollary to establish the existence of 8V m. Then formulas (6) and (4) from the

corollary to Proposition 1 are utilized to prove that 8l: m fulfills the asserted relations



(13) and (14). Finally, the independence of 8V m on m is a consequence of the

following

Corollary 1. The function 8V m on E(K) is related to the local Weil height on

E(K) through the estimate

(15)

where

dv(P)A±v(A) <cm,

m

1 /I I I il— ■mnx{\clm\,\c2m$.

In fact, 8V m = 8V is uniquely determined by the properties (14) and (15) and hence is

independent of the choice of m.

We can now define the global Néron-Tate height on E(K) as the sum, with

multiplicities, over the local Néron-Tate heights as follows (see [14]):

, , Í I KMP) ap + o,(16) 8(P) = vgmk

lo ifP = 0.

By the sum formula (S) we then obtain on the basis of (13) and (14):

Corollary 2. The global Néron-Tate height onE(K) fulfills the relations

(13') 8(P A Q) A 8(P - Q) - 28(P) - 20(g) = 0

and, for r G N,

(14') 8(rP)-r28(P) = 0.

Remark 2. Corollary 2 shows that the global Néron-Tate height 8 is a quadratic

form on E(K), whereas Proposition 4 implies that the local Néron-Tate height 8V is

"almost" a quadratic form on E(K).

4. Computation of the Néron-Tate Height. Again, let the elliptic curve E be given

by (E) over a global field K. Fix a nonarchimedean (discrete) valuation v of K.

Suppose that P = (xP, yP) G E(K) is a rational point satisfying v(xP) < nir

By Proposition 1, on choosing an m g N such that m > 2 and v(m) = 0, we have

Xm"P=*2»(p)-

Now v(xp) < ju.t, together with v(a¡) ^ fiv entails

«(<MP)) - rn2"v(xP), D(**.(P)) = (m2" - l)v(xP).

Hence

f (xm»P) = v(xp).



Thus we obtain from the limit formula (12) and the definition (9) of dv the asserted

relation

«„(P) = - HxP) A -¿i;(A) = d.(P) + ¿p(A).

Proposition 5. Suppose that a rational point P = (xP, yP) g E(K) satisfies the

inequality v(xP) < u„ for a nonarchimedean (discrete) valuation v of the global field

K. Then the local Néron-Tate height of P essentially coincides with the local Weil height

of P with respect to v; more precisely,

8v(P) = du(P) + ~v(A).

From Proposition 5 we get the following theorem, which is crucial for the

calculation of the Néron-Tate height on E(K).

Theorem 1. Let E be an elliptic curve defined by a Weierstrass equation (E) over an

algebraic number field K. Choose a discrete normalized additive valuation v of K and

suppose that the equation (E) is minimal with respect to v.* Then, for each nontorsion

point P G E0(K), the local Néron-Tate height of P is essentially equal to the local Weil

height of P with respect to v; more precisely,

8v(P) = dv(P)A±v(à).

Proof. The theorem can be found in [9]. For the convenience of the reader,

however, we give a proof.

By Proposition 5, we may confine ourselves to the case in which v(xP)> ¡iv. The

subcase in which v(xP)> u„ > 0 would lead to a contradiction to the choice of

P G EQ(K). Hence it remains to consider the subcase in which

v(xP)>nv = 0.

The reduction map of Section 2,

p0: E0(K)/EX(K) -» É0(K),

is an injective homomorphism. Since K is a number field, the residue field K of K

with respect to v is finite and hence so is the group E0(K). Therefore, for any

P g E0(K), there exists a number r g N such that rP g Ex(K). Choose r g N

minimal with this property. Then we have

v{xrP) <ßu = 0-

From this, since v(xP)> u„ = 0 and v(a¡) > u„ = 0, we conclude that

v($r(P)) > 0 and v{%(P)) > 0.

We claim

(17) v(xrP) = -v{^(P)).

*The required minimal model of E is found by Tate's algorithm [11].


358 HEINZ M.TSCHÖPE AND HORST G ZIMMER

By Proposition 5, Formula (14) of Proposition 4, and the definition (9) of dv, this

claim yields the asserted identity

«.(*) = i\{8^rp) - riîp)) + ̂ »(a)

= dv(P)A±v(A)

since v(xP) > ¡iv = 0.

To prove (17) it suffices to show that

(18) u(*,(P)) = 0.

This is accomplished by verifying (18), first for the lower r g N and then for general

r G N.

Let r = 2.

If v(3xp A 2a2xP A a4 - axyP) > 0 we would get a contradiction to the assump-

tion that P g E0(K). Hence it is enough to consider v(3xp A 2a2xP A a4 - axyP)

= 0. But then the asserted relation (18) follows directly from the formula (2) for x2P

and Proposition 1.

Let r = 3.

By the minimal choice of r, we have

v(%(P)) = 0 sind v(%(P))>0.

Now the decomposition formula (which can be verified without trouble)

%(P) = %(P)[%(P)(6x2 + b2xP A b4) - %\P)}

yields v(ty4(P)) = 0, and hence the relation from Proposition 1,

<t>3(P) = xP%2(P)-%(P)%(P),

leads to the identity v(<t>3(P)) = 0, as asserted in (18).

Finally, let r > 4.

Again, by the choice of r, we have

v(%(P)) = v(%(P)) = ••• = v(%_1(P)) = 0 and

i>(*,(P))>0.

Then Formula (5) from the corollary to Proposition 1 yields

v(x2P-xP) = 0 and v(x(r_X)P - xP) > 0,

so that another consequence of Formula (5), viz.,

^r+l(P) = ~[(X(r-l)P ~ Xp) +(xP — x2P)] —; r— ,

leads to i;(^r+1(/>)) = 0. Now the identity from Proposition 1,

*r(P) = xP*?(P)-%_x(P)%+x(P),

reveals that v(<±>r(P)) = 0, as asserted in (18). This proves Theorem 1.



Remark 3. Theorem 1 makes it possible to calculate the local Néron-Tate height

8V(P) with respect to all discrete valuations v of the number field K for all

nontorsion points P G E(K).

This is true because Proposition 2 tells us that a suitable multiple rP of P belongs

to E0(K). Then we apply Theorem 1 to calculate 8v(rP) and use Formula (14) from

Proposition 4 to get the desired value of 8V(P) itself.**

Remark 4. Torsion points P G E(K) are of no interest in this connection since

their global Néron-Tate height is 8(P) = 0.

It remains to show how to compute the local Néron-Tate height S„ for archi-

medean valuations v of the number field K. From (4') in the corollary to Proposi-

tion 1, we get the formula

1 o(^.(P)) " 1 v{*Z,(m-lP))

which proves to be useful in the sequel.

Now, since we are interested here only in the case of K = Q, the field of rational

numbers, we confine ourselves to considering its completion Kx = QK = R with

respect to the ordinary absolute value v = vx = -log | |. Then Tate's method is best

suited for calculating 8V (see [12]).

Theorem 2. Let E be an elliptic curve defined by a Weierstrass equation (E) over

the field R of real numbers and denote by vx = -log| | the ordinary additive

archimedean valuation of R. Take an open subgroup T of P(R) such that all

P = (xP, yP) g r satisfy xP * 0.*"* For P g T such that 2"P * 6 for all n g N,

define the entities Tn, Wn, and Zn by putting

'T1 _ _n

' 1n+l - 7Xp z.n

where

Wn = 4T„ A b2T„2 A 2A4P„3 + A6P„4, Z„ = 1 - A4r„2 - 2A6P„3 - A8P„4.

Let

fi(P)- £ î^. \(P) = \l0g\Xp\ + \ß(P).n = 0 *•

Then the local Néron-Tate height of P with respect to vx is

«JP) = X(P)-^log|A|.

Proof See [12]. However, the assertion of Theorem 2 also follows from

Proposition 6. In the situation of Theorem 2 we have for n g N0,

1 *22(2"P) $2(2"P)i„ =-, W„ =-■-, Z„ =-:-.

Po = — ' Tn + i = ^T fom& N0,

\.2np i2np A2„p

** Added in proof. Joe Silverman, whom we wish to thank for some valuable hints, told us that he has

carried out similar height computations (unpublished) avoiding, however, the use of Proposition 2 by

employing Tate's local formulas (see [14]).

***Hence T is either £(R) or the identity component of £(H) according as £(R) is connected or

disconnected.



Proof. The proof is carried out easily by means of the formulas (7) in the corollary

to Proposition 1.

Remark 5. The simplest way of finding a subgroup T of E(R) of the type desired

in Theorem 2 is by applying a birational transformation to (E) to obtain a model

(E') such that A¿ < 0. Then T = £"(R) itself will do.

In the special case of K = Q we are interested in, the set MQ consists in the

p-stdic valuations v corresponding to the primes p of Q and the additive valuation

vx = - log | | corresponding to the unique archimedean absolute value | | on Q. Of

course, the multiplicities in the sum formula (S) are all \L, = 1.

5. Examples. We are now in a position to calculate the Néron-Tate height 5 on the

group E(K) of rational points on an elliptic curve E over the rational number field

K = Q. To this end, we use the defining formula (16) for 8 with multiplicities

X v = 1 to reduce the computation of 8 to that of the local Néron-Tate heights 8V on

E(K). For discrete valuations v of Q, the height 8V is calculated by means of

Theorem 1 in accordance with Remark 3, and for the archimedean absolute value

vx = -log| |, the calculation of 8V is performed on the basis of Theorem 2.

(i) Examples of Silverman. We illustrate our procedure by verifying the height

calculations of Silverman [9].

(A) E: y2 A 21xy A 494y = x3 A 26x2,

P = (0,0)g£(Q),

A = -213 • 133 • 192.

Silverman obtains

5(P) = 0.010,492,....

We have

(a) 8„ (P) = 0.038,612,393,...,

(b) P <£ £0(Q) for p = 2,13 and 19; and

5„(P) = 0 for all primes p * 2, 13 or 19.

Now

13P e £0(Q) for p = 2,

3Pg£0(Q) for/> = 13,

2Pg£0(Q) for/; = 19.

One computes

%(P) = 2 • 13 • 19, %(P) = 23 • 133 • 192, *13(P) = -280- 1356 • 1942

and

x2P = -2-13, x3P = -2-19, xX3P= -24 ■ 5 ■ 13 • 19.

This leads to

802(13P) = -gln2, «Di3(3P) = ilnl3, 8VJ2P) = | In 19.

Hence, by (14) of Proposition 4,

S02(P) = ^ln2, 8VJP)= -^lnl3, 8jP) = - -^ In 19.



By (16) this adds up to

8(P) = 0.010,492,061,....

(B) E: y2 + Uxy A my = x3 4- Sx2,

P = (0,0)g£(Q),

A = -211 • 52 ■ 19.

Silverman gets

8(P) = 0.010,284,...

and we obtain similarly to (A)

8(P) = 0.010,284,005,....

(ii) The Bremner-Cassels Curves. Our procedure turns out to be particularly useful

for calculating the global Néron-Tate height on the elliptic curves

E : y2 = x3 + px

for primes p of Q such that p = 5 (mod 8), as they were considered by Bremner and

Cassels [1]. The authors exhibit points P g E(Q) of infinite order on 43 curves of

this type, where

r2 r • tP = (xP,yP) with Xp = —-, yP = —— forr,í,íGZ

s s3

such that

g.c.d.(r,s) = 1; r,/# 0 (mod/?); and

r = / == 1 (mod2), ,r = 0(mod2).

One easily checks that P g £0(Q) for all primes p of Q and all points P G P(Q)

displayed in [1]. (Notice that 2yP and 3x2P + p are relatively prime.) This leads to

Proposition 7. For the points P g ^(Q) of infinite order on the Bremner-Cassels

curves in [1], the Néron-Tate height is

8(P) = 8JP)A ¿In|A|+lnM.

(iii) Modular Elliptic Curves. In [16, pp. 75-113], N. M. Stephens and J. Daven-

port list 68 modular elliptic curves E of rank 1 with a rational point P g E(Q) of

infinite order. We computed the Néron-Tate heights of these points P.* Comparison

of the Néron-Tate height of the generator of the 63rd curve in their table with the

Néron-Tate height of the point in Silverman's second example (see (i) (B) above)

shows that the two values agree. It turns out, as one easily checks, that the

corresponding two curves are birationally isomorphic (see Table 1).

+ We have compared the height values in our Table 1 with those in a corresponding (unpublished) table

of Silverman containing up to six digits behind the period. They agree (except for the sixth digits of the

curves 58A, 61A, 135A, 153A, 189C and for the fifth and sixth digit of the curve 185D).



Table 1

1 . ) al = 0 a2 = 0 a3 = 1 a4 =-1 aó

P = ( 0 ; 0 )global height: .0255557041237A

2 . ) al = 0 a2 = 1 a3 = 1 a4 = 0 a6P ■ ( 0 S 0 )global height: .03140825354443A

3 .) al = 1 a2 =-1 a3 = 1 a4 = 0 aóP = ( 0 ; 0 )

global height: .04649074231953A

4 .) al = 0 a2 =-1 a3 = 1 a4 =-2 aó

57E P = ( 2 ; 1 )global height: .018787296303

5 .) al = 1 a2 =-1 a3 = 0 a4 =-1 aó

P » ( 0 J 1 )global height: .0212101539258A

6 .) al = 1 a2 = 0 a3 = 0 a4 =-2 aóP = ( 1 ; 0 )global height: .039593865681

61A

7 . ) al = 1 a2 = 0 a3 = 0 a4 =-1 a6P = (-1 ¡ 1 )global height: .187757

65A

8 . ) al = 0 a2 = 0 a3 = 1 a4 = 2 aó

77FP = ( 2 ; 3 )global height: .049013989032

9 .) al = 1 a2 = 1 a3 = 1 a4 = -2 aóP = ( 0 ; 0 )

79A global height: .048832105054

10 .) al = 1 a2 = 0 a3 = 1 a4 =-2 aó

82A p = < ° ! ° »°¿il global height: .112353462459

11 . ) al = 1 a2 = 1 a3 = 1 a4 = 1 aó

83A P = ( 0 ; 0 )global height: . 033Ó4Ó1.47057

12 .) al = 0 a2 = 0 a3 = 0 a4 =-4 aó

88A P = í 2 i 2 )global height: .020132182168

13 .) al = 1 a2 = 1 a3 = 1 a4 =-1 aó

8qc P = ( 0 ! 0 )oy^ global height: .056052440615

14 . ) al = 0 a2 = 0 a3 = 1 a4 = 1 aó

91A P = < 0 ! 0 )^ global height: .071190075334

15 .) al = 0 a2 = 1 a3 = 1 a4 =-7 aó

q1R P = (-1 ; 3 )3 global height: .529622543205

16 .) al = 0 a2 = 0 a3 = 0 a4 =-1 aó

P ■ < 1 5 1 )9¿C" global height: .024904198649

99A

17 .) al = 1 a2 =-1 a3 = 1 a4 =-2 aóP = ( 0 ; 0 )global height: .151285092281


THE NÉRON-TATE HEIGHT ON ELLIPTIC CURVES

Table 1 (continued)

18 .) al = 0 a2 = 1 a3 = 1 a4 =-1 aó =-1

101A P " <_1 ' ° '' global height: .032351726475

19 .) al = 1 a2 = 1 a3 = 0 a4 =-2 aó = 0

102E P ■ ("I i 2 'u"* global height: .07162694647

20 .) al = 1 a2 = 1 a3 = 0 a4 =-7 aó = 5

106A P = < 2 ¡ 1 )global height: .034456340202

21 . ) al = 0 a2 = 1 a3 = 0 a4 = 0 a6 = 4P = ( 0 i 2 )

1 ltIV global height: .119959949363

22 .) al = 1 a2 =-1 a3 = 1 a4 = 4 aó = 6P = ( 0 i 2 )

117A global height: .56516781309

23 .) al = 1 a2 = 1 a3 = 0 a4 = 1 aó = 1

118A p = < ° ' 1 '1 IOA global height: .043953097838

24 .) al = 0 a2 =-1 a3 = 1 a4 =-7 aó = 1

121D P - ' « > 3 >*■ 'u global height: .04439257808

25 . ) al = 1 a2 = 0 a3 = 1 a4 = 2 aó = 0122A P = ' l ' 1 '

global height: .060421607704

26 .) al = 0 a2 = 1 a3 = 1 a4 =-10 a6 =123A P = ' 1 ' l 'u:->A global height: .420260708766

27 .) al = 0 a2 = 1 a3 = 0 a4 =-2 a6 = 1

124B P - < 1 I 1 )global height: .2602Ó5346941

28 . ) al = 0 a2 = 1 a3 = 0 a4 = 1 aó

128C P = « 0 ¡ 1 )global height: .216105582237

29 .) al = 0 a2 =-1 a3 = 1 a4 =-19 aó =

12QE P = ( 1 ¡ 4 )l¿yfi' global height: .04997957634

30 . ) al = 1 a2 = 0 a3 = 1 a4 =-33 aó =P = ( 2 i 2 )

"OH global height: .585232076797

31 .) al = 0 a2 =-1 a3 = 1 a4 = 1 aó = 0,,,. P = < o ; o )1-51A global height: .108047599334

136A

2 . ) al = 0 a2 = 1 a3 = 0 a4 =-4 aó

P = (-2 ! 2 )global height: .115753990413

33 . ) al = 1 a2 = 1 a3 = 0 a4 =-1 aó =

1W P = ( 0 ! 1 ),-5ÖI!, global height: .08368409567

34 .) al = 0 a2 = 1 a3 = 1 a4 =-12 aó

-wir. P = (-3 ! 4 )1411i global height: .017243387509



Table 1 (continued)

35 . ) a

1411 P

30 . ) a

142E I

37 . ) a

142F r,

38 . ) a

145A I

39 . ) a

148AÜ

40 . )

152A

41 . ) a

153AP

42 . )

153C

43 . ) a

155C I

44 . )

156E

45 . ) a

158D Pg

46 . ) a

158E pg

47 . )

162K

43 . )

163A

49 . )

'■Ak

50 . )

171A

51 . )

172A

= 0 a2 =■= ( 0 ; 0 )

obal height:

= 1 a2 == (-1 ¡ 1 )

obal height:

= 1 a2 =-= ( 1 ; 1 )

obal height:

= 1 a2 =-= ( 0 ¡ 1 )

obal height:

= 0 a2 =■= (-1 ¡ 2 )

obal height:

= 0 a2 =

= (-1 ! 2 )obal height:

= 0 a2 == ( 5 ; 13 )

obal height:

= 0 a2 == ( 0 ; 1 )

obal height:

= 0 a2 == ( 1 ; 0 )

obal height:

= 0 a2 =■= ( 1 ; 1 )

obal height:

= 1 a2 == ( 0 i 1 )

obal height:

= 1 a2 == (-1 ¡ 4 )

obal height:

= 1 a2 == ( 2 ; 0 )

obal height:

= 0 a2 =

= ( 1 ! 0 )obal height:

= 1 a2 =


= 0 a2 == ( 2 ¡ 4 )

obal hei ght:

= 0 a2 == ( 2 Î 1 )

obal he i ght:

1 a3 = 1

.099247618232

1 a3 = 0

.090456855492

1 a3 = 1

.010894571319

1 a3 = 1

.292228814932

1 a3 = 0

.043120589701

1 a3 = 0

.032707434794

0 a3 = 1

.056444869251

0 a3 = 1

.034740140542

1 a3 = 1

.092071901309

1 a3 = 0

.073707206024

1 a3 = 0

.03958438143

1 a3 = 1

.019495140155

1 a3 = 0

.152967441934

0 a3 = 1

.094954616249

1 a3 = 0

.044978395458

0 a3 = 1

.112983434413

1 a3 = 0

.380069831503

a4 =-1

a4 =-1

a4 =-12

a4 =-3

a4 =-5

a4 =-1

a4 = 6

a4 =-3

a4 =-1

a4 =-5

a4 =-3

a4 =-9

a4 =-6

a4 =-2

a4 =-6

a4 = ó

a4 =-13

aó = 0

a6 =-1

aó = 15

aó = 2

aó = 1

aó = 3

aó = 27

aó = 2

aó = 1

aó = 6

aó = 1

aó = 9

aó = 8

aó = 1

aó = 4

aó = 0

aó = 15


THE NÉRON-TATE HEIGHT ON ELLIPTIC CURVES

Table 1 (continued)

52 . )

175A

53 . )

175C

54 . )

176A

55 . )

184B

50 . )

184C

57 . )

185A

58 . )

185B

59 . )

185D

60 . )

189A

61 . )

189C

62 . )

190C

63 . )

190D

04 . )

192Q

65 . )

196A

66 . )

197A

67 . )

1981

68 . )

200C

= 0 a2 == ( 7 ¡ 2 )

obal height:

= 0 a2 == (-3 ; 12 )

obal height:

= 0 a2 == ( 1 ¡ 2 )

obal height:

= 0 a2 =


= 0 a2 == ( 0 ; 1 )

obal height:

= 0 a2= ( 0 ; 2 )

obal height:

= 1 a2 == ( 3 ; 2 )

obal height:

= 0 a2 == ( 4 ; 12 )

obal height:

= 0 a2 == (-1 ¡ 1 )

obal height:

= 0 a2 == (-3 ; 9 )

obal height:

= 1 a2 == ( 1 ; 2 )

obal height:

= 1 a2 == ( 13 ; 33

obal height:

= 0 a2 ■= ( 3 ; 2 )

obal height

= 0 a2 == ( 0 ; 1 )

obal height:

= 0 a2 == ( 1 ¡ 0 )

obal height:

= 1 a2 == (-1 ¡ 5 )

obal height:

= 0 a2 == (-1 i 1 )

obal height:

-1 a3 = 1

.332314998542

-1 a3 = 1

.040236666901

-1 a3 = 0

.087531915126

-1 a3 = 0

.051533618406

1 a3 = 0

.061565455601

-1 a3 = 1

.055139483611

0 a3 = 1

.712032645336

1 a3 = 1

.057028352204

0 a3 = 1

.031006094417

0 a3 = 1

.931621776106

1 a3 = 0

.065910740941

-1 a3 = 1)

.010284005728

-1 a3 = 0

.675801867206

-1 a3 = 0

.043017725483

0 a3 = 1

.069433995882

1 a3 = 0

.097521495699

1 a3 = 0

.146605513301

a4 =-143

a4 =-33

a4 = 3

a4 =-4

a4 = 0

a4 =-5

a4 =-4

a4 =-156

a4 =-3

a4 =-24

a4 = 2

a4 =-48

a4 =-4

a4 =-2

a4 =-5

a4 =-13

a4 =-3

a6 = 748

aó = 93

aó = 1

aó = 5

aó = 1

aó = ó

aó =-3

aó = 700

aó = 0

aó = 45

aó = 2

aó = 147

aó =-2

aó = 1

aó = 4

aó = 4

aó =-2



6. Lang's Conjectures. Silverman [9] used his above-cited examples of rank-one

elliptic curves E over Q to estimate the constants c,, c2 in S. Lang's Conjecture 2

(see [6]) about lower bounds for the Néron-Tate height 8 on nontorsion points in

£(Q). We wish to carry through a similar estimation with respect to Lang's

Conjecture 3 (see [6]) for Selmer's [8] rank-two elliptic curves E over Q.

In Section 3, Remark 2, we observed that the Néron-Tate height 5 is a quadratic

form on the rational point group E(Q). This property of 8 is tantamount to the fact

that the function

ß(P,Q) = HHPAQ)-8(P)-8(Q)}

Table 2

al = 0 a2 » 0 a3 - 0 a4 ■ 0 aó - -388800

PI » ( 76 / 1 $ 224 / 1 )

P2 - ( 124 / 1 I 1232 / 1)

The trans-formation with (risit(u) > ( 0 I 0 I 0 | 2 ) leads to

al - 0 a2 » 0 a3 » 0 a4 - 0 aó - -Ó075

Pl-< 19 / 1 | 28 / 1 )

p the local height decimal

2 (1/3 )*ln( 2 ) .2310490601803 < 13 / 12 )*ln< 3 ) 1.1901633127235 (1/3 )*ln( S ) .536479304144co -.220039705773

The global height is 1.73765197128

P2-( 31 / 1 i 134 / 1 )


2 (1/3 )*ln( 2 ) .2310490601863 (13/12 )*ln( 3 ) 1.1901633127235 ( 1 / 3 )*ln( 5 ) .536479304144<=° -.068619441325


Pl+P2=( 241 / 4 i-3689 / 8 )


2 ( 4 / 3 >*ln( 2 > .9241962407463 (13/12 )*ln< 3 ) 1.1901633127235 ( 1 / 3 )*ln( 3 ) .536479304144oo .18319182537


Regulator : 3.125459338543



Table 2 (continued)

al - 0 a2 = 0 a3 - 0 a4 = 0 aó » -26142912

PI » ( 26572 / 9 I 4329280 / 27 )

P2 - ( 61516 / 25 ( 15244064 / 125 )

The trans-formation with (rlsitju) - ( 0 I 0 I 0 ) 2 ) leads to

al » 0 a2 - 0 a3 - 0 a4 ■ 0 aó = -408483

Pl-( 6643 / 9 ; 541160 / 27 )

p the local height

2 (1/3 )#ln( 2 )3 (25/12 )*ln( 3 )41 (1/3 )*ln( 41 )ee


decimal

.2310490601862.2887756013911.237857355568.673826315477

P2»( 15379 / 25 ¡ 1905508 / 125 )

p the local height

2 (1/3 )*ln( 2 )3 (13/12 )*ln< 3 )5 (1/1 )*ln( 5 )41 (1/3 )*ln( 41 )OS


decimal

.2310490601861.1901633127231.6094379124341.2378573555Ó8.588854192712

Pl+P2=( 133393 / 784 ¡ 46655225 / 21952 )


2 (7/3 )*ln< 2 )3 (13/12 )*ln( 3 )7 (1/1 )*ln( 7 )41 (1/3 )*ln( 41 )


1.6173434213061.1901633127231.9459101490551.237857355568.039300793087

Regulator : 18.87131764437

for P, Q g £(Q) represents a symmetric bilinear form on P(Q). If E has rank two

over Q and P = Px, Q = P2 are two basis points of E(Q), the quantity

Ä=|det(j8(P,,P7)),._1> R

is called the regulator of the elliptic curve E over Q. In addition to the Néron-Tate

height of the basis points P,, P2 of the rank-two curves E in Selmer's tables [8], we

have also computed their regulator R. More detailed information about Selmer's

curves is to be found in [6]. To begin with, we list in detail two examples, namely the

curves with A = 30 and A = 246 in [8] (see Table 2).



In analogy to Silverman [9], we now use these Selmer curves to estimate the

constants in Lang's Conjecture 3. Suppose E over Q is given in Weierstrass normal

form

(E) y2 = x3 + ax + b (a,6eZ).

Following Lang [6], we define the height of E over Q to be the number

H(E) = max{|a|3, |è|2},

so that approximately

h(E)= -log/7(£)-6ftl,oo,

where again vx = - log | | denotes the additive archimedean valuation of Q. Let N

stand for the conductor of E over Q (see [11]).

Then we enunciate, in the case of rank-two curves,

Lang's Conjecture 3. There is a basis {PVP2} of E(Q) modulo torsion such

that 8(PX) 4 8(P2) and

8(PX) 4 cxH(E)l/2A ■ N«^2 ■ logiV -(2/f3)l/2,

8(P2) 4 c2H(E)1/l2 ■ NciN) • logJV • c

for some positive real constants c, cx, c2, where

lim e(N) = 0.N -»oo

Now the constants cx sind c2 in Lang's Conjecture 3 satisfy the inequalities

/ H(E)l/24 ■ N*»"2 ■ \ogN-(2/]/3)l/2Y1

c^[-m-j '

(H(E)l/12-N«N)-logN-cYl

C2>\ t{P2) j •

On choosing c = 1 and putting, in analogy to the example on p. 166 of [6],

e(N) = (\ogN-\oglogN)~1/2,

we obtain for the constants cx and c2 the estimates^

cx > 0.021,784,..., c2> 0.002,709,....

Here we let E range over the rank-two curves in [8] and take the maximal values for

cx said c2, which are attained at the curves with A = 246 and A = 30, respectively.

For the sake of completeness, we include here the numerical estimates of the

constants cx and c2 for all values of A in Selmer's table [8] in order to show how cx

and c2 oscillate as A varies (see Table 3).

1t This estimation is based on the assumption that the points in Selmer's table [8] are of minimal height.

We wish to thank M. Reichert for verifying this on a Siemens PC MX-2 for Selmer's curves with A = 30,

37, 65, 91, 110, 124, 126, 163, 182, 217, 254, 342, 468 and 469. Only for A = 254, the point P, + P2 is to

be taken instead of P-, since it has a slightly smaller height value.



Table 3

19

3037658691

110

12k126127I32153163182183201

203209210

217

218219246254271273282

309335342345348370379390397399407420433435436446^53462468469477497498

O.OO513126O.OI927030.00483937O.OO345923O.OO629273O.OO36522O.OI2277O.OO3742650.004330630.00414683O.OI830990.01053350.004581410.004450210.005247470.005116710.00954740.007217880.01260480.00308153O.OO3271990.005001260.02178430.005313650.003706660.004721230.01828640.00457362O.OO274443O.OO352578O.O1422140.01757280.00282848O.OO384730.01118180.003491610.0042891O.OO2675020.0121 164O.OO5185120.01394960.004828680.00418040.004157060.0108269O.OO388973O.OO339982O.OO744997O.OO5312930.0171181

0.001103890.002709030.0009623820.001825010.002279750.0004383370.001538550.001749930.001142670.0009409480.002447980.001111880.0004830740.0005170170.001995030.001209150.001164550.0008122240.00121204

O.OOO3I9989O.OO2OIO74O.OOI7I349O.OOI939O5O.OOO936257O.OOO782067O.OOO663O38O.OOI83782O.OO227242O.OOI65495O.OOIO3604O.OOI4I568O.OOI9I220.001179460.0007112330.0008590680.0005969490.0006248690.001186860.001075260.0005229350.001475410.001600660.001450410.001685460.0008779930.000981010.0003044540.00111861O.OOO907821O.OO14825

Fachbereich 9 Mathematik

Universität des Saarlandes

D-6600 Saarbrücken. West Germany

1. A. Bremner&J. W. S. Cassels, "On the equation Y2 = X2(X + p)," Math. Comp., v. 42, 1984,

pp. 257-264.2. J. P. Buhler, B. H. Gross&D. B Zagier, "On the conjecture of Birch and Swinnerton-Dyer for

an ellliptic curve of rank 3," Math. Comp., v. 44,1985, pp. 473-481.

3. J. W. S. Cassels, "Diophantine equations with special reference to elliptic curves," J. London

Math. Soc, v. 41, 1966, pp. 193-291.

4. H. G. FOLZ. Ein Beschränktheitssatz fur die Torsion von 2-defizienten elliptischen Kurven über

algebraischen Zahlkörpern, Ph.D. Thesis, Saarbrücken, 1985.



5. S. Lang, Elliptic Curves: Diophantine Analysis, Springer-Verlag, Berlin and New York, 1978.

6. S. Lang, "Conjectured Diophantine estimates on elliptic curves," Progr. Math., v. 35, 1983, pp.

155-171.7. S. Lang, Fundamentals of Diophantine Geometry, Springer-Verlag, Berlin and New York, 1983.

8. E. Selmer, "The Diophantine equation ax2 + by3 + cz3," Acta Math., v. 85,1951, pp. 203-362.

9. J. H. Silverman, "Lower bound for the canonical height on elliptic curves," Duke Math. J., v. 48,

1981, pp. 633-648.

10. J. T. Täte, "The arithmetic of elliptic curves," Invent. Math., v. 23,1974, pp. 179-206.

11. J. T. Täte, "Algorithm for finding the type of a singular fibre in an elliptic pencil," in Modular

Functions of One Variable IV, Lecture Notes in Math., vol. 476, Springer-Verlag, Berlin and New York,

1975, pp. 33-52.12. J. T. Täte, Letter to J.-P. Serre, Oct. 1,1979.

13. H. G. Zimmer, "On the difference of the Weil height and the Néron-Tate height," Math. Z., v. 147,

1976, pp. 35-51.14. H. G. Zimmer, "Quasifunctions on elliptic curves over local fields," J. Reine Angew. Math., v.

307/308,1979, pp. 221-246.15. H. G. Zimmer, "Torsion points on elliptic curves over a global field," Manuscripta Math., v. 29,

1979, pp. 119-145.

16. Modular Functions of One Variable IV (B. J. Birch & W. Kuyk, eds). Lecture Notes in Math., vol.

476, Springer-Verlag, Berlin and New York, 1975.


Computation of the Néron-Tate Height on Elliptic Curves€¦ · THE NERON-TATE HEIGHT ON ELLIPTIC CURVES 355 and k{6K - v(A)) A 4av 4 dv(2P) - 4dv(P) - M*2(P)) 4 -|a(, if2P*&, where

Documents

Computation of the Néron-Tate Height on Elliptic Curves€¦ · THE NERON-TATE HEIGHT ON ELLIPTIC CURVES 355 and k{6K - v(A)) A 4av 4 dv(2P) - 4dv(P) - M2(P)) 4 -|a(, if2P&, where