STARK-HEEGNER POINTS ON ELLIPTIC CURVESDEFINED OVER IMAGINARY QUADRATIC FIELDS

MAK TRIFKovré

AhstractLet E be an elliptic curve defined over an imaginary quadratic field F ofclass number

1. No systematic construction ofglobal points on such an E is known. In this article,

we present a p-adic ana/ylie construction ofpoints on E, which we conjecture to he

global, defined over ring class fields ofa suitable relative quadratic extension K / F.

The construction follows ideas of DOI'mon ta produce an analog of Heegner points,

which is especially interesting since none ofthe geometly ofmodular parametrizations

extends ta this setting. We present sorne computational evidencefor our construction.

Contents1. Iotroduction...............2. Modular fonus on the upper half-space3. The p-adic construction .....4. Measure-valued modular symbols5. Numerical examplesReferences. . . . . . . . . . . . . . .

415429433438444452

1. IntrodnctionLet E be an elliptic curve defined over a number field F with conductor ideal .IV. IlsHasse-Weil L-series L(E/F , s) is defined by an Eulerproductthat converges for Res>3/2. Il is conjectured that this L-function has an analytic continuation to all SEC, afact known when F = <QI, thanks to results of Wiles [19] and others (see Theorem 2).Il therefore makes sense, at least conjecturally, to consider the leading term of theTaylor expansion of L(E/F, s) at s = 1. The Birch and Swinnerton-Dyer conjectureexpresses this tenu in tenus of algebraic iovariants of E /F. A weak version of theconjecture is given by the conjunction of the following sequence of statements, onefor each integer r :::: O.

DUKE MATHEMATICAL JOURNAL

Vol. 135. No. 3, © 2006Received 7 October 2005.

2000 Mathematics Subject Classification. Primary 14H52, 14Q05; Secondary l1R37, I1G15.

415

416 MAK TRIFKOVlé

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CONJECTURE 1 (BSD(r»)

If ord'~IL(E/F,s) = r, then the Z-rank of E(F) is equal to r. Moreovel; III(E/ F ) isfinite.

The full Birch and Swinnerton-Dyer conjecture asserts that the implication in BSD(r)

is in fact an equivalence and gives a precise formula for the leading term of the Taylor

expansion of L(E/F , s) at s = 1.The interest of the weaker statements BSD(r) lies in the fact thatthe best currently

known theoretical evidence for the Birch and Swinnerton-Dyer conjecture is the proof

of BSD(O) and BSD(1) when F = Q!, which follows from the work of Wiles [19],Gross andZagier [10], and Kolyvagin [II]. Zhang's generalization ofthe Gross-Zagier

formula in [20] to totally real fields leads to a similar proof of BSD(r), r :::: 1, for Ftotally real of odd degree.

Plainly, any approach to proving the Birch and Swinnerton-Dyerconjecture should

involve a method for constructing points on E. Indeed, all successful attacks onBSD(O) and BSD(l) to date use variants of the Heegner point construction, which isessentially the only one known. The goal of this article is to propose, and provide

numerical evidence for, a conjectural p-adic construction of so-called Stark-Heegnerpoints: p-adic analytic analogs of Heegner points in the first case where even BSD(O)is mysterious, namely, when the base field F is imaginary quadratic.

1.1. Heegner points over Q!We briefly recapitulate the construction and use of Heegner points in the most clas­

sical setting-when E is defined over Q! and parametrized by Xo(N)-the better toappreciate the difficulties that arise over an imaginary quadratic field.

1.1.1. Modularity and Heegner pointsLet N be a positive integer. The group ro(N) of matrices in SL;,(Z) which are uppertriangular modulo N acts on the extended upper half-plane .Yi" = .YI' U pl (Q!) by

Mobius transformations. The correspondence

r =>.Yé' <+ x, = (iC/(Z +Zr), (~))

extends to an isomorphism ro(N)\'yé'* ~ Xo(N)(iC), where Xo(N) is the modularcurve defined over Q! parametrizing pairs (E, C) of generalized elliptic curves with a

cyclic subgroup of order N.Let K be an imaginary quadratic field of discriminant D K. ArE K is said to

be of conductor c if the order @, = p. E KI!c(Z + Zr). C; Z + Zr} is equal to @e>the unique order of conductor c. The ring class field of conductor c is the Abelian

extension H,I K with Galois group Pic(@J corresponding by class field theory to thesubgroup KXiCX(@, @Zy of the idèle group of K. We fix an imaginary quadratic

field K satisfying the following hypothesis.

STARK-HEEGNER POINTS

THE HEEGNER HYPOTHESIS

Ali primes dividing N are split in K.

This hypothesis is necessary for the following definition to be nonvacuous.

417

Definition 1Let c be a positive integer prime to N and DK. A point (E, C) E Xo(N)(C) is called

a Heegner point of conductor c if

End(E) = End(EjC) = iD,.

Under the correspondence (1), this translates into the following definition.

Definition 2Let c be a positive integer prime to N and D K. A Heegner point of conductor c is anypoint x" where both T and NT have conductor c.

Heegner points are in fact defined over number fields. More precisely, we have the

following.

THEOREM 1Let c be prime to both N and DK. Then any Heegner point x, of conductor chascoordinates in H" the ring dass field of K ofconductor c.

Crucial to the proof ofthis theorem is the modular interpretation of points on Xo(N); inparticular, T as in Theorem 1 corresponds to a pair (E, C) with complex multiplication

by iD,. In the p-adic construction that is the subject of this article, the modularinterpretation is missing, which is probably a big obstacle to proving an analog of

Theorem 1.To transfer Heegner points from Xo(N) to an elliptic curve E/Q, we use the

modular parametrization of E, whose existence is the central claim of the Shimura­Taniyamaconjecture, proved byWiles [19], Taylor and Wiles [17], and Breuil, Conrad,

Diamond, and Taylor [2].

THEOREM 2

Let E/Q be an elliptic curve ofconductor N, with Néron dijferential WE. There existsa newform f E S2(fo(N)) with rational coefficients, a mOiphism ofalgebraic curves

<jJ : Xo(N) -+ E

defined Over Q, and a Manin constant c~ E Q such that

<jJ*WE = 2rric~f(z)dz

as dijferentials on fo(N)\/t'*.

We can now define Heegner points on E.

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418 MAK 1RIFKOVlé

Definition 3A Heegner point of conductor c on E is any point ofthe form P, = </>(x,) E E(He )

with x, a Heegner point of conductor con Xo(N).

In particular, when c = 1, that is, (1), = (1) K, we get a point P, defined over the Hilbert

class field H = HI of K.

Definition 4When P, has conductor 1, the point PK = trH/K</>(P,) E E(K) is calIed the basicHeegner point.

The basic Heegner point does not depend on the r chosen.

1.1.2. Complex-analylic construclionThe inspiration for our definition of p-adic Stark-Heegner points cornes from recastingthe above construction in purely complex-analytic terms. Let

j Y'

A E = {Znic. , f(z)dz 1 y E 'o(N»)

be the period lattice of E. Since E is an Abelian variety, the modular parametrization</> factors through the Abel-Jacobi map

AlXo(N) --+ Jo(N) --+ E

and can be expressed as the composition of a line integral and the Weierstrassparametrization:

n-+x = Znic. l' f(z)dz f-+ (S'J(x), S'J'(x»)."'0

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Il is best to think of this as a computational recipe for Heegner points whose properconceptual definition is modular.

1.1.3. Sign offunctional equationThe equality (Z) implies the equality of the L-series

L(E/Q, s) = L(f, s), (4)

so that L(E/Q, s) has an analytic continuation to aIl s E <C. The same is then true forthe function A(E/Q, s) = N'/2(Zn)-'r(s)L(E/Q, s), which alIows us to write down

STARK-HEEGNER POINTS

the functional equation for L(E/Q, s) in the particularly simple fOTIn

A(E/Q, s) = w(E, !QI)A(E/Q, 2 - s).

419

Here w(E,!QI) = ±I is called the sign of the functional equation of E/Q. In particular,when w(E,!QI) = -l, the special value L(E/Q, 1) vanishes, sa that the Birch andSwinnerton-Dyer conjecture predicts that E(!QI) has a point of infinite arder. This is

an instance of the general "yoga" of constructions of rational points; whenever signconsiderations force the L(E/F, s) ta vanish ta order t, we may hope, in light of theBirch and Swinnerton-Dyer conjecture, ta find an explicit construction of a rank-t

subgroup of E(F).

A similar functional equation is conjectured ta exist for E defined over an arbitrarynumber field F. When E/F has semistable reduction, the sign ofthe functional equation

of L(E/F , s) is conjectured ta be equal ta the sign w(E, F) given in the followingelementary fashion (see [14, Theorem 2]).

Definition 5Set w(E, F) = (-1)'+', where r is the number of Archimedean places of F (Le., realor pairs of complex conjugate embeddings) and sis the number of finite places whereE has split multiplicative reduction.

COROLLARY 1Let E/ F have a square-free conductor JV and hence semistable reduction. Let K / Fbe a quadratic extension ofdiscriminant prime ta JV. Consider the set S ofail places

v' of K satisfying the following conditions:(a) v' is Archimedean, or v' is fmite and divides JV" and

(b) if v is the place of F below v', then [K" : F,] = 2.Then w(E, K) = (-I)#s (which, in particulm; depends only on K, not on E).

Let us go back ta the case F = !QI, where K is an imaginary quadratic fieldand E/Q a curve with semistable reduction. Using the factorization L(E/K, s) =L(E/Q, s)L(Ei~K), s), one can show that L(E/K, s) has an analytic continuation anda functional equation whose sign is indeed w(E, K). Suppose in addition that Ksatisfies the Heegner hypothesis. Then Corollary 1 implies that w(E, K) = -1 and

that, therefore, L(E/K, 1) = O. Il is then natural ta ask about L'(E/K, 1).

THEOREM 3 (see Gross and Zagier [10])

There exists an exp/icit nonzero constant ex such tha!

L'(E/K, 1) = ahE(PK),

where hE is the canonical height on E(K).

Since the canonical height vanishes precisely on torsion points, we have the following.

420

COROLLARY2

If K satisfies the Heegner hypothesis, then

ord'~lL(E/K, s) = 1 {} PK is ofinfinite arder.

In other words, if PK is torsion, then ord'~lL(E/K, s) ::: 3.

MAK TRIFKOVlé

When L(E/K, s) vanishes to order exactly 1, the point PK of infinile order is theessential input for Kolyvagin's theory of Euler systems, which actually allows us toprove that rkzE(K) = 1 in this situation.

1.2. Stark-Heegner points over imaginary quadratic base fieldsWe now replace 1Q> as our base field with an imaginary quadratic field F, and we lookfor analogs of the ingredients of the classical Heegner point construction. We assume

for simplicity that F is of class number 1; the fields F that feature in our computationsare even Euclidean.

The equality of L-functions (4) is an instance of the general Langlands program,

which predicts the existence of an automorphic representation p of G~(AlF) such thatLCE/F, s) = L(p, s). Since the field F is still relatively simple, il tums out that we can

replace the abstract automorphic representation p with a concrete geometric object.By analogy with the upper half-plane Y/', we consider the the upper half-space

y/,(3) and ils completion by cusps y/,(3), given by

y/,(3) = lez, 1) 1 Z E C, 1 E lR~o}, y/,(3). = .!'t'(3) U 1I'1(F).

Note that the cusps now depend on the ground field F. There exists a natural actionof PG~(F) on y/,(3)•. The role of the level is played by an integral ideal JI' <; (r)F,

while the group lo(N) is replaced by

The datum of the automorphic representation p is equivalent ta a certain 10(%)­invariant harmonie differential

dz dl dz"'j = - fo(z, 1)- + h (z, 1)- + h(z, 1)-

1 1 1

on .!'t'(3)•. The function J = (Jo, h, h) ; y/,(3) ---* C3 plays the role of the modularform fez) over 1Q> (see Definition 6).

At this point the naive analogies with the situation over 1Q> break down. Thequotient space lo(A")\y/,(3). might seem a tempting substitute for Xo(N), except that

il is a three-dimensional real manifold and hence not a variety, let alone a modulispace. Moreover, there is no obvious parametrization 10(JlI')\y/,(3)* ---* E(IC).

STARK-HEEGNERPOINTS

Table 1

421

Complex p-adic

1. Archimedean place 00 1. Non-Archimedean place Jr IIAÎ

2. K IrQ imaginary quadratic 2. K / F quadratic, inert at ]f

(local degree 2 at (0) (local degree 2 at Jr)

3. Heegner hypothesis: 3. Stark-Heegner hypothesis:

Ail liN split in K Ali v\,.-,V, v i=- Jf split in K

4. Z-order (!) c K 4.I9F[~]-order(!) C K

5. Poincaré upper half-plane :if 5. Hyperbolic upper half-space J"t'(3)

(domain of f(z)dz) (damain of Wj)

6. Poincaré upper half-plane Jlf' 6. p~adic upper half-plane

(::::> quadratic irrationalities :tt. =Il'ICCp )-Il'I(F.)Kn:tt,<0) CC) K n YI'. ,< 0)

7. Weierstrass uniformization 7. Tate uniformization

WWei : CI AE -? E(C) cDTate: C;/q~ ---+ E(C p)8. Complex tine integral 8. Mixed multiplicative integral

f iT- f(z)dz E C, <l, <2 E J't'* WJ EC;,TJ,r2 E.Yt'rr.

" " 'r, SE P'(F)

In the absence of modularity, we might try to mimic the analytic construction of

Heegner points as in (3), but this is not trouble free either: it is unclear, for instance,

what points in J't'(3) should take on the role of quadratic irrationalities. A more serious

difficulty turns out to be that the differential fonn ùJ] has a single real period, from

which we cannot reconstruct the period lattice of E.Following the ideas of Darmon [6], we propose to resolve these conundrums

by working at a non-Archimedean prime 1[ dividing the conductor JV of J, rather

than at the infinite prime. Table 1 gives a useful heuristic dictionary hetween various

components of the complex and p-adic constructions; sorne concepts are defined

later on.

Remark 1Our construction is modeled on the conjectural p-adic construction of Darmon in [6].

Dannon considers an elliptic curve E defined over iQl and uses p-adic integrals to

construct points called the Stark-Heegner points, conjectured to be defined over ring

class fields of a real quadratic field K.

It is natural to attempt to extend Darmon's construction to our setting since the

quadratic extension K / Fis analogous to a real quadratic extension ofiQl in a number of

salient ways, most notably in that the group of units of (!) K is of rank one in both cases.

By now there is substantial computational, as weB as sorne theoretical, evidence for

Darmon's construction (see [7], [1]). The theoretical evidence of [1] relates the p-adic

Stark-Heegner points to classical Heegner points. While we show in this article that the

computational aspects of the construction go through for E defined over an imaginary

quadratic field, the analogs of the theoretical results of [1] lie deeper, inasmuch as

there are no "classical" Heegner points in our setting to begin with.

422 MAK TRIFKOVlé

Remark2The assumption n IlJV is essential for the existence of the Tate uniforntization as indictionary entry 7.

Remark3The Stark-Heegner hypothesis and Corol1ary 1 imply that the sign w(E, K) is -l,which (conjectural1y) forces L(E;K, s) = O. The Birch and Swinnerton-Dyer conjec­ture then predicts the existence of a point of infinite order in E(K). 'Our main goal isto propose a p-adic analytic construction of such a point and in fact of a whole Eulersystem in which it fits.

Remark4In the classical setting, ye plays two roles: it is the domain of the modular fonn, butit also contains the quadratic irrationalities that parametrize Heegner points. Over animaginary qnadratic field, the fonner role is played by the naive analog ye(3), while the(relative) quadratic irrationalities are now to be found in yen' Note that K n Jf'n # 0since Ir is assumed ta be inert.

Remark5Only the last entry in the dictionary requires extensive explanation, which is providedin Section 3. For now, suffice it to say that the integral in question is "mixed" in thesense that the first set of limits is p-adic, while the second consists of two rationalcusps, and it is "multiplicative" because it is defined as a limit of Riemann products

rather than sums and therefore satisfies the multiplicative analogs ofthe usual additivityproperties of integrals.

In order to get the correct statement of Shimura reciprocity, it tums out that itis essential to work with an "indefinite" version of this multiplicative integral. Weconjecture !hat we can find a rank-one subgroup Q <; <C~ with QI <; q~ for sorneinteger t and a function !hat to a single T E yen and a pair of cusps r, s E pl (F)

associates an e1ement

satisfying

fr ùJj/fr ùJj == fr ùJj (mod Q).

The existence of this indefinite mixed multiplicative integral is an analog of theconjecture of Mazur, Tate, and Teitelbaum. The success of our computations is astrong encouragement to explore this conjecture over imaginary quadratic fields.The prospect is tantalizing since the Mazur-Tate-Teitelbaum conjecture was provedby Greenberg and Stevens [9] using Hida families whose existence is unlikely formodular fonns on GL2(A F ).

rSTARK-HEEGNER POINTS

To a TEK n:/i'n we associate a Stark-Heegner point by the fonnula

423

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Here y, is the generator of the group ofunits in a suitable subring of M2x2((1) F[l/1f])fixing T, while l' E IP'I(F) is an arbitrary base point. Note that exponenliating by t wasnecessary to gel a point in C;/q~ rather than C;/Q-an unavoidable technicalitythat did Dccur in sorne of our computations.

This point is conjectured to be defined over a global field. To specify which one,we point out that since h(F) = l, the theory of orders in a quadratic extension K / Fis

virtually identical to the usual theory of quadratic orders over!QI. As suggested in entry4 ofthe p-adic column of the dictionary in Table l, we work with (1)F[I/1f]-ordersthroughout. Any (1)F[I/1f]-order (1) in K is equal to the order

(1), = (x E (1)K[~] lx =a (modC(1)K[~])for sorne a E (1)F[~]1

for sorne ideal c c (1) F, which we assume to be prime to 1f. Since F is principal, weoccasionally abuse language and call the conductor of (1) any generator of the ideal c.The subgroup

(6)

defines an Abelian extension HelK called the ring class field of conductor c. Sincewe work with (1)F[I/1f ]-orders, 1f splits in the class field defined by (6), and one mightexpect that H, is smaller than the ring class field we would have gotten by workingwith (1) F-orders instead. One of the lucky coincidences of working with a field ofclass number 1 is that the two ring class fields are equal since 1f is principal and iner!in K.

The goal ofthis article is to provide numerical evidence for the following conjecture.

CONJEcruRE 2

Let TEK n :/i'n. Assume that the (1) F[I/1fJ-lattices

and

L

bath have (1),for their ordel: Then the point J, that a priori lies in E(Cp ) is infactdefined Over the ring class field H, of K.

For a mOre precise version of this conjecture, giving the action of GHelK, see Conjec­ture 6.

We can define the basic Stark-Heegner point JK as before. By analogy withCorollary 2, we expect the following conjecture to hold.

424

CONJECTURE 3If h E E(K) is torsion, then rkzE(K) 2: 3.

MAK TRIFKovré

1.3. The computationsWe present the results of our computations for two curves, one for either value of thesign of the functional equation of L(E/F, s).

1.3.1. w(E, F) =-1Let F = Q(..;=TI) with a ring ofintegers generated bya = (1 + ..;=TI)/2. Considerthe curve

El :/+y=x3 +(l-a)x'-x,

which has prime conductor". = 2a+5 ofnorm 47. The reduction of El at". is nonsplit,sa we expect that the functional equation of L(EI / F , s) has sign w(EI , F) = -1 andthat rkzEI(F) = 1. Indeed, one easily checks that the obvious point P = (0,0)

generates EI(F).Since w(EI , F) = -1, for every choice of auxiliary quadratic extension K the

basic Stark-Heegner point h should in fact lie in El (F) and therefore be a multipleof P. (This is the analog of [6, Proposition 5.10]). We computed h ta 20 digits

of 47-adic accuracy and indeed found that it agrees with a multiple of P for every

K = F(.J8) in which". is inert and NF/l:é < 3000. A sample of these results ispresented in Table 2. When h = 0, we verify in many cases that rkzEI (K) = 3,

as predicted by Conjecture 3, using Denis Simon's program for descent over numberfields (see [15]). There are cases where the rank computation is inconclusive (denotedby "-" in Table 2), but we fully expect that this is due ta the great height of theMordell-Weill generators of EI(K).

When K = F(.J20a + 18), we find the basic Stark-Heegner point of greatestheight in the range of our computations: h = 26P. We expect III(EI / K ) ta have anelement of arder 13.

We can also get some sense of the Heegner points themselves, rather than justof their traces down ta K. Consider K = F(.JI3) of class number 5. Conjecture 2

predicts a Galois orbit consisting of five Stark-Heegner points J'i of conductor 1. Wecompute the polynomials satisfied by the x- and y-coordinates of the J'i 's. Modulo47'°, they are

fAT) = TS + (-2a + 2)T4 + (-Sa - 4)T3 - 9T' - aT + (-a + 2),

fy(T) = T S +(-a -1)T4 + (-a + 13)T3 +(-4a +32)T' +(a + II)T + (5a+5).

Bath ofthese polynomials define the Hilbert class field of K, as expected. This exampleis emphatically atypical; we found only a handful of class number 2 examples and one

class number 4 example in which fx(T) and fy(T) also have small integral coefficients.Table 3 contains more examples of polynomials satisfied by coordinates of conjecturalGalois orbits of J, 's.

STARK-HEEGNER POINTS

Table 2. Basic Stark-Heegner points on E 1/Q(.;:::T1)' JK = n[ü, 0]

425

L

8 n rk

-1 2 1

a-l -1 1

-a -1 1

-2 0 3a+l 1 1

a-2 -3 1

-3 -3 1

2œ-l -2 1

-2a -2 1

la -2 -2 1

-a-3 5 1

-a+4 1 1

2a + 1 -1 12a-3 -1 1

2a+2 0 32a -4 0 3

-a-4 -1 1

-a+5 -1 1

5 1 1

-3œ+4 3 1-3œ-l -3 1

a-6 0 3

a+5 0 3-6 0 33a-5 1 1

-3a -2 -1 1

4œ-2 -4 1

3a+3 -1 1

3a - 6 -1 12œ-7 0 3

-7 2 1

-4œ+5 0 34œ+ 1 -2 1

3a+4 0 3-3a +7 2 1

a-S 1 1-a-7 1 1

4œ -6 2 1

-2a+& 2 1-Za-G -2 1

4œ+2 2 1

-3a+8 -3 1-311' -5 -3 1

4œ+3 -1 1

-Sa +2 -1 14œ -7 1 1

-5œ+3 1 1

-501'+4 3 1Sa - 1 -1 1

-5œ+5 -1 1

8 n rk

5a 7 1

-5œ+7 -9 1

50:+2 1 1

-Zœ-& 2 1

-2a + 10 2 1

-4œ+9 3 1

a-IO -3 1-4œ-5 -3 1

a+9 1 1

3a+7 -1 1

-30' + 10 3 1

Ga -3 2 1

10 4 -

-6a+5 4 -Ga - 1 4 -

2a - 11 -1 1

Sa +4 1 1-2a-9 -3 1

-Sa+9 1 1a-lI -3 1

a+ JO 3 1

3et -11 -1 1

-6a+7 1 1

-3a-8 1 1

Ga + 1 3 1-Ga-2 -2 1

-6a+8 2 1la - 12 4 1

la + 10 4 1

-7a+2 -1 1

-7a +5 -1 1

7a - 1 -4 --Sa -6 0 3-7a 4 -

7a -7 0 3-6a-4 4 -

6a -10 0 3

7œ-S -7 1

2a -13 1 1-70'. - 1 -1 1-2a - 11 5 1

-3a + 13 -1 1

3œ+ 10 1 1-Sa - 7 0 3

Sa - 12 10 1

a+ 12 2 1

-Ct + 13 -4 --6a -5 0 3-6a+ll 0 3

4œ+9 0 3

8 n rk

-?a-2 0 3-4œ + 13 2 1

7a - 9 -2 113 -2 1

-8a+3 1 1-7a -3 -3 1

7a -10 1 1Sa -5 -1 1Sa -13 -1 1

5a+8 1 16a +6 4 -Ga - 12 4 -

-3et - 11 1 1-3et + 14 -3 1

Sœ-? 1 1-ct - 13 -1 1-a+ 14 -1 1

Sa -} 1 14œ -14 4 1

-7a + 11 1 17a +4 -II 1

-14 0 3

6œ+7 -2 1Ga -13 2 1

Sa +9 -1 1Sa - 14 1 1

-8a+9 1 1-80:-1 -3 1-30: + 15 3 1-30! - 12 -3 1-8a + 10 4 1

Sa +2 0 3-4œ - 11 -1 1-a+ 15 1 1

ct + 14 -1 14a-15 -1 1

6a+& 8 --Ga + 14 -4 1

9œ -4 -7 190! -5 1 115 1 19œ-7 1 19a - 2 -1 1

3et + 13 2 19a-& -2 1

-2a - 14 -14 12a-1G 2 17a -14 0 37a+7 4 -

~5œ + 16 1 1

426 MAK TRIFKOVlé

Table 3. x- and y-polynomials of Ir; E E I/Q("cm(H) defining the Hilbert class

field H of K = Q(.J=TI)(,ft)

8 J;of)'

-0:+4 fJT) - T2+ (-;~~~~ga + ~~~~~~~) T + (-I~:7a - t:J7)

fy(T) = T2 + (-~~~i~~~~a + ~~~~~:~~~~) T + (-~~~~~~~ct + ;i~~~~)

Sa - 23 !,(T) ~ T' + (-~. + 1) T' + (1. - 'j!) T + (1. + IIfy(T) = T 3 + (a +2)T2 + (~a + b) T + (ta - t)

5. !x(T) = T4 + (~~~~a - ~~;~~) T

3 + (?3~~6225a + i~~~~) T2

+ (- g~~Ct + :~~6~;~) T + ( /32:

6is 0: - 1138;~~)

fy(T) = T 4 + (- g~~j~~~ct - 152~7~~) T 3 + (- i~~~~m~cr - ~~~;~;~i~~) T 2

+(- ~~~~~~m~a+ :~~~~~~~~~) T + (- 3~~21Ii6~a+ ~113ff:is)

8a -7 !,(T) = T' + (11. - lO)T' + (28. + Il)T' + (3. + 26)T - Il

!,(T) ~ T' + (57. + 6)T' + (75. + 60)T2 + (37. + 51)T + (7. + 13)

Sa -1 Ix(T) = yS + (- ~12sg;~8711a + ~~~;~i~) T4+ (26~g5~8817 ct - 76~r:sOi;17) T 3

+ ( ~iil~;~Ci + ~;60~1;8813) T2 + (- ~22~~~80IScr + ~~6;~~~) T

+ (:;~8;821 ct - ~~~~~î)

29 fAT) = yS + (- ~i ct + ~rs~~) T4 + (;l~~~a - 37~~82S;) 13

+ (~~~~ga + ~~~~) T2 + (-~~:~~a + ~~~) T + L~9a - 1~8~)

fy(T) = TS + (122~3~3 a + ~~~~~~) T4 + (;~~~~a + ~g~~~~;) T 3

+ (;g~~~ a + ~g~:~~~) T2 + (~~~~i~~ a + ~~~~~~) T

+ (32212~OS~ a - 3~~~~9)

-16a+17 fx(T) = T 6 + (26J~1 a + i~:~) T S + (il~~~a + 2~~81) T4

+ (~~~~~ a - ~~~~~) T 3 + (~~:~i a - ;4~1) T 2

+ (- i48J:I a + ~:~:) T + (- ;19:S1 a + ;~~;)

f (T) - T6 + (- 139408711 a + 64295(014) TS + (_ 461302934 a + 1719072530) T4y - 124251499 124251499 124251499 124251499

+ (- i~~g-:~~a + 31322fi11~91:) T 3 + (i~~~~~~a + 3?2~26i~4~:) T 2

+ C~~~~:~a + ?it.lf581~:i) T + n~~:~~œ + 1522:is\o;~)

Sa - 43 fAT) = T9 + (- l~i8410a + liXii() T8 + (- ~cil~~Oii a _ I~~g~~j~g-t) T 7

+ (528il72

1:30a - 1~~:~~~j4) T6 + (3~n~i;~3 a + 2~~g~~~~~7) T5

+ (_ 310778902 a + 785509(56) T4 + (! 1955584855 a _ 2355419504) T37890481 7890481 71014329 71014329

+ (~~?4~2~0 a - 1~~~;~i~l) T 2 + (~:~î~:~ a + b~î~~) T+

+ (- ~~~~;a + ~~~~:n

.~

1i STARK-HEEGNER POINTS,:j

J.13.2. w(E, F) = 1Let F = Q(.)=3), and set a = (l + .)=3)/2. The curve

Ez : i + xy = x 3 + (a + l)xz + ax

427

(7)

has prime conductor lf = a + 8 of norm 73. The reduction at lf is split, saw(Ez, F) = 1. The Mordell-Weil group Ez(F) is generated by the point (-1,1)of arder 6.

We compnted the basic Stark-Heegner point h E E(K) ta 30 digits of 73-adicaccuracy, for fields K (./8), N F/Q8 < 1000, in which lf is iner!. The nnmerical resultshere are more interesting since we do not know a priori a generator of Ez(K) withwhich ta compare JK. lnstead, we use an algorithm that finds in sorne sense the bestpossible approximation in K ta the 73-adic coordinates of h. If the height of h isrelatively small, then the point in AZ(K) thus obtained often lies on E; we say wehave reeognized h as a global point.

The average height of basic Heegner points in this example seems ta be greaterthan when w(E, F) = -1. They therefore tend ta be difficult ta recognize accuratelyas global points. Of the 466 basic Stark-Heegner points computed, only 123 wererecognizable. Ofthose, the highest was found when K = F(fJ), fJz = 2a + 21:

1259988 126090782x = - 127165927

a + 127165927'

(2903147975024 11037094266063) (629994 63045391 )

y = 31646131095439a + 31646131095439 fJ + 127165927

a- 127165927 .

Amusingly, this is the only point among the global points we found whose coordinatesare not integral at lf, which means, since the Tamagawa nnmber en is 1, that h liesin the kemel of the reduction mod lf map on E(K).

lA. Computing with modular symbolsThe function <Pi' which ta a pair of cusps r, s E 1I'1(F) associates the integral

takes values in QiZ for a suitable real period Qi (see Proposition 1). Il is an exampleof a modular symbol on fo(%) with values in a right fo(.k")-module M, that is, afunction <1> : IP'1(F) x IP'1(F) --> M satisfying

<l>{r --> s} + <l>{s --> t} = <l>{s --> t), r, s, tE 1I'1(F),

<l>{yr --> ys} = <l>{r --> S}[y-l, r, SE 1I'1(F), Y E fo(%).

428 MAK TRIFKOVlé

The key technical step in computing the mixed multiplicative integral (5) involvesfinding a modular symbol <1> with values in measures on (9rr, the completion of (9 F atTC, satisfying

(8)

To define a modular symbol on fO(JV), it is enough to specify its values on the edgesof a fundamental domain for the action of f o(.Y) on .YE(3)'. These values cannot bechosen arbitrarily: each face of the fundamental domain imposes a Z[fo(JV)]-linearrelation among them. In the analogous situation for modular forms on .YI", this is nota serious obstacle to computations since the fundamental domain for the action offo(N) on .Yt" has only one face (namely, itself). By contrast, even the fundamentaldomain for the action of fo«(9 F) on .YE(3)' is a hyperbolic polyhedron with anywherefrom four faces (when F = Q>(H» to fourteen (when F = Q>(v'=2». For theaction of f o(JV) this number gets multiplied by roughly NF/Q%. Finding <1> as in (8)thus presents difficulties both algebraic, in terms of the number of relations the valuesof <1> have to satisfy, and geometric, in terms of the shape and incidence of faces ofthe fundamental domain.

The main computational innovation in this article, as weil as in Greenberg [8],is the observation that we do not have to set up large and complicated systems oflinear equations. The idea, elaborated in Section 4, is to consider ail functions onthe edges of a fundamental domain (or another large enough set of paths), includingthose not satisfying face relations. It is trivial to find such a "fake" measure-valuedmodular symbol <1>0 satisfying (8). We define the action of the Hecke operator Urr

on fake modular symbols. Repeatedly applying Urr to <1>0 produces a sequence offake modular symbols that get closer and closer to satisfying the face relations. In

the limit, we get an honest modular symbol <1>, which lifts (I/D.j)1>j as in (8). Giventhat the relatively small conductors we work with are already large enough for thenaive computation to be prohibitively slow, working with fake modular symbols wasessential to the feasibility of our computations.

1.5. Further directions

(1) The existence of indefinite mixed multiplicative integrals may seem like a tech­nicality, but at least in Darmon's theory over real quadratic fields (see [6]) they areintimately connected to Hida families. The success of our somewhat intricate com­putations, which assume the existence of indefinite mixed multiplicative integrals,suggests that Hida families of modular forms on GL2(AF ), although unlikely to existin a naive sense, merit further thought.

(2) The Stark-Heegner construction appears to work in two settings with a fairlytenuous formaI similarity: for E defined over Q>, producing points over ring class fieldsof a real quadratic K, and in the case presented in this article. The p-adic constructions

STARK-HEEGNER POINTS 429

seem fairly insensitive to the fine geometric structure of the situation, and it wouldbe interesting to see over which other fields they might work. A first step in this

direction would be a full-blown adèlic reformulation of the two existing constructionsso as ta avoid the ad hoc elementary treatments of modularity. While they facilitatecomputations, they may well be obscuring the conceptual picture.

(3) Finally, the prospects for actually proving Conjecture 2 are dim at present. Onemay hope, by analogy with the theory of complex multiplication, to find a modularinterpretation for the rnixed period integrals. A more promising avenue toward sorneimmediate theoretical confirmation would be ta consider degenerate cases (e.g., whenE is a base change from <QI) in the hope of relating this construction to classical

Heegner points.

2. Modular forms on the upper half-spaceWe give a quick survey of the geometry of the upper half-space .;ft(3) and associated

modular forms, which conjecturally correspond ta elliptic curves over imaginaryquadratic fields. Our account is elementary, avoiding automorphic representations, forthe twin reasons of accessibility and amenability to computations.

2.1. The action ofPGLz(!C)The upper half-plane has an action ofPGLi(Ift); il is slightly less obvious thatPGL2(!C)

acts on .;ft(3). Here are tbree ways of defining this action.

(1) Grarnm-Schmidt orthogonalization gives a one-to-one correspondence

y'f(3) <-+ PGL2(iC)/PSU2,

(z,1) <-+ (~ ~) PSU2,

(2)

which presents .;ft(3) as a homogeneous space for PGLz(iC). The subgroupPSU2 is the stabilizer of (0, 1).

We can think of .;ft(3) as the space of quaternions {z +tjlz E !C, t E Ift>o! c lHI.We define the action of PSL,(iC) by the formula analogous to the classical one:

Note that the order of multiplication matters since the quaternions are noncom­

mutative. To extend this action to all of PGLz(iC), we set

(~ ~) (z, 1) = (oz, 1011)·

430 MAK TRIFKOVlé

For (~ ~) E PGL2(F), this action is compatible with the standard action ofPGL2(F) on IP'I(F), giving an action of PGL2(F) on ail of .1(3),.

(3) Finally, and least enlighteningly, we can deduce from (2) an explicit formulafor the action. For (~ ~) E GL,(IC) and (z, t) E .1(3), set

8 = lez +dj2 + letl2• Then

( a b) (z, t) = ~ (az + b)(ez + d) + act2, lad - belt).

e d 8

2.2. Topology, metrie, and dijferentialsWe extend the topology on .1(3) to .1(3), by stipulating that the action of PGL2(IC)

be continuous and that the set V" = {(z, t) E .1(3)lt > h} be open. The V,,'s forma basis ofneighborhoods of the cusp ooj; by translation, a basis ofneighborhoods ofa cusp zEF consists of (Euclidean) open balls touching the "floor" IC x {O} of .1(3)

at z.A PGL,(iC)-invariant meltic is given by ds2 = (dzdz +dt2)/t2. The geodesics

are circles perpendicular to the floor (including verticallines).A basis of l-differentials is given by the eolumn vector

- 1 1 ( dz dt dZ)fJ = (fJo,!JI, fJ2) = - t' t' t .

The action of y E PGL,(iC) on fi is given by the formula

where the automorphy factor J(y; (z, t)) is

-2rs;'"

11'12_ Isl2

2rs;'"

with ;". = det(y), l' = ez + d, s = Ct. This expression is the analog of the automorphyfactor (ez + d)-2 for classical modular forms.

2.3. ModulaI' formsLet J= (Jo, fI, 12) : .1(3) --+ IC3 be a function with values in row vectors. We definethe action of ro(JV) on Jby the (weight 2) stroke operator given by

<fly)(z, t) = J(y(z, t))J(y; (z, t)).

STARK-HEEGNER POINTS 431

:s

L

Definition 6A cusp form of weight 2 for ro(.A') is a fuuction J = (Jo, fI, h) : .11'(3) ---> (;3 suchthat we have the following.(a) The dot product J .ft is a harmonic I-differential on .11'(3) invariant under

ro(JV) (i.e., Jly = h(b) For ail y E PS~((I)F), fCj@F Jly(z, 1) = 0 (i.e., the constant term of the

Fourier-Bessel expansion of J-see below-at the cusp y-1oo is zero).

The space of ail cusp forms for ro(JV) is denoted S2(JV).

By Definition 6, a cusp form J is invariant under (~ ~) E l'o(JV), X E (1)F, andthus has a Fourier-Bessel expansion of the form

- " - (4n l"lt) ("z)fez, t) = L.. c(,,)t2K lTi'iT 1fJ ln

O"oEmF v IDI v D

(see [13], [5]). Here 1fJ(z) = e4rriR'Z is a character of the additive group of F, and

where Ki(t), i = 0 or l, is the hyperbolic Bessel function that satisfies the differential

equation

d2K i IdKi ( 1)--+--- 1+--,- Ki=O

dt2 1 dl t2'

and decreases rapidly at infinity.The space S2(JV) cornes equipped with the action of Hecke operators. To each

prime element À of (1) F not dividing JV, we associate the operator TA:

We make essential use of the operator Urr for n I.AI' defined by

Hecke operators with composite indices are defined by recursions analogous to theclassical ones. There are a couple of minor differences with Hecke theory over Q.(a) The operator TA (resp., Urr ) actually depends on the prime element À (resp., n)

and not just on the prime ideal generated by À (resp., n).

432 MAK TRIFKOVlé

(b) A relaled feature is Ihat the matrix (~n, where E generates (r)~. acts as aninvolution on S2(%) and breaks it up into two eigenspaces with eigenvalue 1or -1:

2.4. Plusforms and the Shimura-Taniyama conjectureWe work exclusively with forms f E st(JI!"), referred to as plusforms. Their Fourier­Bessel coefficients satisfy CrEa) = c(a). The Fourier-Bessel expansion is then a sumover ideals of (r) F rather Ihan over individual elements:

f(z.t)= L c(a)t2K(4n 1a 1t) L1f!(EœZ).O#(.)<;@F v1DT <E@, Vl5

The action of TÀ and Un similarly depends only on the ideals O.) and (n). SincePGL;,«(r)F) = PSL2«(r)F)(~ ~), we mayas weil consider f to be invariant under thebigger subgroup

Newforms are defined as in the classical case. They are eigenforms for ail the operatorsTÀ • Je t JV. that are orthogonal to the space of forms coming from lower levels. Bymultiplicity l, they are also eigenforms for Ihe operators Un. nl%.

The Mellin transform of f E st(%) is given by

(4n)' ('XOMf. s) = #(r)~ IDFI Jo fi (O. t)t

2('-I) dt

= (2n)2(1-') IDFI'-I r(s)' L c(a)NF/<:!Ja)-'.(.)#0

(see [5]). Note that the sum here is over integral ideals. just as it is in the Dirichletseries L(E/F. s). This justifies our focus on plusforms and suggests a version of IheShimura-Taniyama conjecture.

CONJECTURE 4

There is a one-to-one correspondence between (cuspidal) newforms f E st (JI!") and

elliptic CUl1JeS E/F that do not have complex multiplication by F.

Elliptic curves E / F wiIh complex multiplication by F should correspond to Eisensteinseries. Sorne computational evidence for this conjecture can be found in [4] and [5].

STARK-HEEGNER POINTS 433

(9)

The modular symbol associated to Jassigns to any pair of cusps r, s E pi (F) the

path integral

The Mellin transform above is, up to a constant, simply iii]{O ---+ oo}.The modular symbol iii] {r ---+ s} satisfies a fundamental discreteness property

(see [11]).

PROPOSITION 1There exists a unique positive real number ri] (the period of J) such that the image

of iii] in IR is ri ]71.

We denote by

1 1'" "<Pf{r ---+ s} = - f . f3 E 7lri] ,

the integer-valued modular symbol associated to J.Unlike modular forms over iQi, J has only one real period, so there is no hope

of reconstructing from <P] the period laltice of the curve E associated to J by theShimura-Taniyama conjecture. A small technical benefit is that we do not need toconsider plus- and minus-modular symbols as in the case over iQi discussed in [7,Proposition 1.2].

3. The p-adic constructionWe briefly describe the construction of mixed multiplicative integrals that play therole of the classical Abel-Jacobi map in our theory. Our account is fairly terse, as thetheory in the imaginary quadratic case is closely parallel to the theory over iQi as givenin [6]. The reader is strongly encouraged to look at [6] for motivation and details.

We work overa fixed imaginary quadratic field F ofclass number 1. Let.IV = n Atbe an ideal of @F, with n a prime ofodd characteristic and n t At. Let J E Si(r0(.Ai'»

be a newform of level ,IV. Il is an eigenvector for the Urr-operator with eigenvalue

w = ±1.

3.1. Mixed multiplicative integrals

Consider the ring

We denote by f the image of R X in PGL2(@F[I/n]).

434 MAK TRIFKOVlé

A measure on jp'1(Frr ) is detennined by its values on compact open sets of thefonn U = au{f)rr for au E PGL,(Frr ). The matrix au is weil defined modulo thestabilizer of (f)rr for the action of PGL2(Frr), which is

The action of f on such compact open sets U is transitive; it is for this that weneed to invert 1f in the definition of R. The stabilizer of (f)rr in f is none otherthan

Notice how the divisibility and integrality conditions at 1f are imposed by f O(1f{f)rr)and away from 1f by f. Since the differential ] .1J is invariant under fo(.A'), therational integer

does not depend on the choice of the matrix au such that au{f)rr = U. An easycalculation starting with ]IUrr = w] shows that /.ijlr ---> s} is in fact a Z-valuedmeasure on jp'1(Frr ), which allows us to freely exponentiate by it in the following

definition.

Definition 7Let TI, r2 E ;YI'rr, r, s E jp'1 (F). We define the mixed multiplicative integral associatedto F by the fonnula

The limit is taken over unifonnly finer disjoint covers of jp'1 (<QJp) by sets of the fonnU = au{f)rr' The poinltu is an arbitrarily chosen test point in U.

This definition, which might seem unmotivated at first blush, is in fact inspiredby analogies with Hilbert modular fonns and the Poisson inversion fonnula of[18].

Our computations rely on the following conjecture, which is an analog ofDarmon's version of the Mazur-Tate-Teitelbaum conjecture (see [6]).

STARK-HEEGNER POINTS 435

CONJECTURE 5There exists a rank-one Z-lattice Q c C; commensurable with qZ with the following

property: there exists a unique function that to r E ;tfrr and r, s E pl (Frr ) assigns an

element

with the properties

(1)

(2)

fY'l Y

' fl'Wj = Wjy; ,

for y E r;

(3)

f'l' j'l' 1'1'Wjj Wj '" Wjr r Tl r

(mod Q).

3.2. Picard group tOl'sors

To fonnulate the Shimura reciprocity law for Stark-Heegner points, we need to con­struct geometric torsors for the Picard group of an order in a quadratic extension ofF. Let Kj F be a quadratic extension that satisfies the Stark-Heegner hypothesis: nis inert in K, and aIl vl.ft are split. We denote by (!)rr, Frr , and Krr the completionsat n of (!)F, F, and K, respectively. We fix an (!)[Ijn]-order (!) C Q with conductor(ideal) c <; (!)F relatively prime to .ft. We mayas weIl also assume that c is prime ton.

By the Stark-Heegner hypothesis, there exists an ideal 9Jl' of (!) K such thatN KjF9Jl' = é!t. Since (C1f, .ft) = l, the ideal 9Jl = 9Jl'(!)F[ljn] is equal to kerofor a unique surjective homomorphism of (!)F-algebras

Such a homomorphism is called an orientation of (!); we fix one, along with theassociated ideal 9Jl.

436 MAK TRIFKOVlé

Definition 8

An optimal embedding of 0 into R (an 0-optimal embedding for short) is an algebrahomomorphism W : K -> M2x2(F) snch that

An optimal embedding delines an orientation of 0", : 0 -> 0 FiJI! by sending a E 0to the lower right-hand entry of W(a). This is a ring homomorphism since, for a E 0,W(a) is an npper triangular matrix modulo 001.

Definition 9

We say that an optimal embedding W is oriented if 0", = o.

We denote by Emb(0, R) the set of all oriented optimal embeddings of 0 into R. Thegroup Î' acts on Emb(0, R) by conjugation.

Every ordered F-basis {Wh Wz} of K delines an embedding Ww ,."" : K ->M zx2(F) which sends a E K to the matrix of the multiplication by a expressed inbasis {W 1, W2}. For an 0 Fl1j n ]-Iattice a C K, we deline the order of a by

0. = (a E K 1 aa >; aJ >; 0K[~lWe have W;;i",,(R) = 0. n 0." where a and a' are 0 F [ljn]-lattices

An 0-optimal embedding WW "W2 is oriented if and only if a' = OO1a. If this equalityholds, then we say that the basis {WI, W2J is OO1-adjusted. Finally, set J/t'~.9J1 = {r E

JI!"" iW", is an oriented optimal embedding}.

PROPOSITION 2

There exist bijections

Pic(0) ~ Î'\Emb(0, R) ~ Î'\JI!"~,9J1.

Proo!We deline maps i : Pic(0) -> Î'\Emb(0,R) and j: Î'\Emb(0,R) -> Pic(0) andshow that they are inverses of each other.

To deline i, let a be a proper 0-ideal. Choose an OO1-adjusted 0 F[ljn]-basis

{WI, W2} for a, and set i(a) ta be the class of the embedding WW"W2 in Î'\Emb(0, R).Since bath a and OO1a are proper 0-ideals, we have W;;'"JR) = 0. n 0 9J1n = 0, so

,,-~

that Ww""" is optimal. Il is oriented by the choice of {WI, W2J. Changing {WI, W2} to

i : Pic(@) ~ r\Emb(@,R),

437

(10)

(Il)

3.3. The Stark-Heegner points

Let" E Jff'~,9Jl, and let '!T = '!Tl" be the corresponding oriented optimal embeddingof {9 into R. Let y, be the image under '!Tl" of a generator of (9x. Fix an integer t sothat Q' <; qZ, where Q is the lattice of Conjecture 5.

This proposition exhibits r\Jff'~,9Jl as a torsor for Pic({9),

Definition 10

The Stark-Heegner period J, and the Stark-Heegner point J, are given by

To construct the j inverse to i, pick an embedding '!T E Emb(@, R). There exist Wl, W2

such that '!T = '!Tw ,,",,' Define j('!T) to be the ideal class of the @F[I/n]-Iattice a

spanned by {Wl, W2}, To show that j is well defined, we need to check that a is aproper @-ideal. Since '!T is optimal and oriented, we have

so a is a proper {9-ideal. Since '!T is oriented, the basis (Wl, (2) is 9Jl-adjusted. Il isdetermined by '!T up to scaling, which does not change the ideal class of a.

Finally, the correspondence '!T = '!Tw, ,"" ++ W2/Wl is a r -equivariant bijectionbetween Emb((9, R) and Jff'~,9Jl, which induces the second bijection of the propo­

sition. 0

We have cond(@a)lcond(@) = c, and hence cond(@a) is prime to Jlt. This forces{99Jla <; {9a with the quotient annihilated by both Jlt and c. Since (c, Jlt) = 1, wemust have

another 9Jl-adjusted basis conjugates '!Tw "",, by an element of r, and scaJing a doesnot change '!Tw"",, at ail, so we do get a well-defined map

STARK-HEEGNER POINTS

We have the reciprocity homomorphism

rec : Pic({9) ~ Gal(HdK),

where He is the ring class field associated to the order {9 of conductor c. The principalconjecture of this article, best thought of as an analog of the theory of complexmultiplication, is then the following.

438 MAK TRIFKOVlé

CONJECTURE 6

The point l, E E(Kp ) is infact in E(H,). The action of Gal(He!K) is given by

The action of Pic((!)c) was defined in Section 3.2.

In the rest of the article, we present numerical evidence for this conjecture.

4. Measure-valued modular symbolsLet 1·1 be the absolute value on Cp normalized so that Inl = q-l, where q = #JFu. Ameasure 1-' on (!)u is determined by its moments Vot(t')};>o.

Definition 11Let !?ii((!)u) denote the space of all Cp-valued measures (Le., bounded distributions)on (!)u. By boundedness, there is a well-defined norm on !?ii((!)u) given by

111-'11 = sup 11-'(1')1.i?,:O

For a real number r, we define an (!)c,-submodule !?ii,((!)u) of !?ii((!)u) by !?ii,((!)u) =II-' E !?ii((!)u)1111-'11 :s q-'j.

The semigroup of matrices

acts on the right on IP'I (Fu) by

t \ (a b) = dt + bc d ct + a

As nie, this action preserves (!)u and therefore induces a left action on !?ii((!)u).

We recall the definition of modular symbols. Elements of the group !>o =Divo(IP'I(F)) can be thought of as paths in J'('(3l* by identifying the generator (s) - (r)

with the geodesic arc {r --+ sJ. The group ro(.;V) acts on Divo(IP'I(F)) on the left byfractionallinear transformations.

Let M be a right 2;0(n)-module (so, in particular, a ro(.;V)-module), with thegroup action denoted by (m, y) Ho mly.

Definition 12

The group of M -valued modular symbols is Symb(M) = Hom(!>o, M). Il has a rightaction of ro(.;V) given by

(<Ply){r --+ sj = (<p{yr --+ ys})ly.

,,[r+a r+s) 1 (1 a)(<I>IUrr ){r -+ s} = L." <1> ---;;- -+ ---;;- 0 n .aEIF,..

An element of SymbCOI"V)(M) can be thought of as a function ll"(F) x ll'1(F) -+ M

satisfying the properties (7) of the introduction (with r 0(.;V) replaced by ro(.;V)). TheHecke operator Urr acts on <1> E SymbcOIX)(M) according to the formula

439

An M-valued modular symbol on ro(.A~) is an element of

STARK-HEEGNER POINTS

In this and similar expressions, the SUffi is taken Qver any set of representatives a of

lFrr . An eigenvector for this Urr is referred to as an eigensymbol.

4.1. Fake modular symbols

Darmon and Pol1ack [7] present a polynomial-time algorithm for computing mixedmultiplicative integrals. The total integral map p : §((!)rr) -+ Cp is defined by

The key step in the Darmon-Pol1ack algorithm requires finding a §((!)rr )-valued eigen­symbol <l>i such that p(<I>i) = ,pi' The algorithm proceeds in three steps.(1) Lift,pi to a measure-valued modular symbol <1>0, so that the diagram

§((!)rr)~o/tp

ll"(F) x ll'l (F) -+ Cp"'1

commutes. Though,pi is an eigensymbol for Urr , <1>0 need not be.(2) Show that the limit

<l>i = lim <1>0 1 (~Urr)nIJ-HXl W

exists. Il plainly is a lift of,pi but now also is an eigensymbol for Urr with thesame eigenvalue w.

(3) Il is now a matter of simple algebra to calculate the values of the mixedmultiplicative integrals (5) from <l>i' relying crucial1y on the fact that it is an

eigenlift of ,pi'Let S = {eil C "'0 be a finite set of geodesic paths which generates "'0 as a Z[ro(.A~)]­

module. If F is Euclidean, then Manin's continued fraction algorithm expresses anypath as a sum ofpaths of the form y{O -+ 00), y E PGL2((!)F). We therefore may (butdo not have to) take S = {y(O -+ oo)} as y ranges over a set of representatives for

440 MAK TRIFKOVlé

r o(.AI')\PGL2 «(!iF). A modular symbol cp E Symbco(x)(M) is completely detenninedby its values on S; that is, there is an injection

SymbcoIX)(M) '-+ Map(S, M).

This suggests the following simple but useful tenninology.

Definition 13

A falee modular symbol is any map cp : S ---> M.

Lifting <Pl as in step (1) of the Darmon-Pollack algorithm, but to a Jake modularsymbol, is trivial. To adapt step (2), we must extend the action of Un fromSymbcoIJV)(M) to ail of Map(S, M). For this, we "disembody" Un by recordingenough infonnation about its action on S. Given e; E S and a E FJ(, we have

(12)

for some a;., E ro(AI') and (possibly not all distinct) ej E S. Let cp be inSymbcoIX)(M) for now. We compute

(cpIUn)(e,) = L L cp(ej+:., (~ :).aElF'", J

(13)

Once we fix a choice of the aL's, we can reconstruct (cpIUn)(e,) for e, E S (andtherefore cplUn itself) from the cp(ej)'s, ej E S. This makes it clear how to extendthe action of Un to ail of Map(S, M).

Definition 14

For cp E Map(S, M) (not necessarily coming from an honest modular symbol), definecplUn by the right-hand side of (13).

It is worth pointing out that the definition of Un does depend on the choiee of aL 's;nevertheless, as we see, the limit of repeated iterations of Un does not.

The following lemma describes the effect of the matrices constituting U; on thenann of a measure.

LEMMA 1

Let n E Z~o, and let r E lR. Assume that /.< E {JJ, satisfies 1/.<(1)1 ::; q-"-'. ThenJor

anya E (!iF,

STARK-HEEGNER POINTS

]roofWe calculate

441

The ith term in the last sum has norm at most Inn; /L(t;)1 ::; q-n;-, ::; q-n-,. Since by

assumption I/L(I)I ::; q-n-, also, the lemma is proved. 0

We present our lifting result in a slightly more general setting. Let 4> E SymbcoCf)((9c,)

be any integral eigensymbol of Un with eigenvalue wofnorm Iwl > q-l. (The readermay think of il as 4>j, the case ta which we apply the results of this section.) We can

also think of 4> as an element of Map(S, (9c,) fixed by V = (I/w)Un .

LEMMA2

Let <l> E Map(S, §-l «(9n)) he a lift of 4>. Then <l>1V also takes values ill §-l «(9n) alld

is a lift of4>.

Proof

Since (<l> lV)(r --->- s}(I) = (4) 1V){r --->- s) = 4>(r --->- s}, Ihe zeroth moment of <l> IV isin fact integral. For higher moments, it follows from (14) that

k '" 1 (a+r a+s)(<l>IV)(r --->- s}Ct ) = L.. ak-<l> -- --->- -- (1)w n n

aeJF'rr

As II/wl < q, the only possible nonintegers appearing in this sum are I/w in the firstterm and the higher moments of <l> in the second; by assumption, bath of them havenorm at most q and hence

1(<l>IV){r --->- s}(tk)1 ::; q.

We are now ready ta prove our main result on the computation of eigenlifts.

THEOREM4

Assume that Iwl > q-l Let <l>o E Map(S, §-l «(9n)) he any lift of 4>. Theil the limit

<l> = lim <l>o IVn

n~oo

o

442 MAK TRIFKOVré

exists in Map(S, !»-l((l)u)) and is infaet an honest modulaI" symbol; that is, <1> E

Symbro(A')(!»((l)u)). MOl"eoveI; <1> is the unique eigensymbol ofUu lifting </>'

ProofSet <1>" = <1>0 IV" .By Lemma 2, we see that <1>" E Map(S, !»-l ((l)u)) as well. To praye

the existence of <1>, we show that the sequence {<I>,,} is Cauchy. For k :::: 0, the zerath

moment of <l>k - <1>0 vanishes: (<I>olVk - <1>0)(1) = </>lVk - </> = O. We can then applyLemma 1 to <l>k - <1>0 to conciude that

As Illwllq < 1, by picking n large enough we can make the difference <I>"+k - <1>"

small for every k.The sequence {<I>,,} is thus Cauchy, and since !»-l is complete, the limit <1> E

Map(S, !»-l((l)u)) exists. Il is ciear that p(<I» = </> and that <l>1V = <1>; that is,

<l>IUu = w<l>.The only thing remaining to check is that <1>, a priori an element of

Map(S, !»-l((l)u)), actually belongs to Symbro(A')(!»-I((l)u)). Since S generates ~o

as a Z[['o(%)]-module, it is necessary and sufficient that <1> respect every relation

(Yi E ['o("Y) and ei E S not necessarily distinct), that is, that

(15)

For this, fix a positive integer n. By repeatedly applying (13), for every a E (l)Fln"we can find matrices fJ~.i E ['0(%) such that

(1 a) "j-lo n" ei = L.(fJa ) ej

]

and such that, moreover,

(<I>IU;)(ei) = L L <I>(ej) 1fJL G ;,,).OE(!)F/rcM j

For each i there exists a 8i E ['0(%) and a permutation ai of (l)Fln" such that

(1 G). = 8. (1 ai(G))o 11: 11 YI 1 0 nll

STARK-HEEGNER POINTS

for ail a E t!! F / 11:". Using the fact that el> IV" = eI>, we calculate

eI>(I:>,-l e,) = L[(eI>lV")(e,)]IY,j i

= ~" L L L eI>(ej) 113~,i Gj ae(!)F/rrn j

= ~" L L L eI>(ej) 1131.,8, Gi aef!JF/rrn j

(grouping ail the terrns with (b ~) on the right)

;,,)YiŒ,(a))

11:"

443

Denote by ~b the expression in parentheses, and apply p to it:

P(~b) = Lq,(L(13~,-I(b)ylej) 18ii j

Œ,-I(b)) ) "'( (1 b) '\' -1) 07fn ei = '+' 0 n" ~ Yi ei = .,

(16)

Since 11/wl < q, we can find an E > 0 such that 11/wl :<: ql-'. As P(~b) = 0,we can apply Lemma 1 ta (16) to conclude:

Since we started with an arbitrary n, we must have el> ( Li y,-I e,) = o.FinalIy, to check uniqueness, say that eI>' is another Urr-eigensymbol lifting q,.

Then p(eI> - eI>') = 0, so that by Lemma 1 for any n,

implying el> = eI>'. o

This proposition is an analog of [16, Theorem 7.1]. For another approach to the trickof ignoring face relations, see M. Greenberg's note [8].

444 MAK TRIFKOVré

4.2. Computing mixed period integrals

To compute the Stark-Heegner period (10), we assume that our base field F isEuclidean, and we specialize the methods of the preceding section to the Z-valuedeigensymbol <Pj with eigenvalne w = ± 1. An easy computation shows that theassignment

{r -> s} t-> 1g(-t)dJ.'j{r -> s}@,

gives a Urr-eigenlift of <Pj, which mnst equal <t>j by the uniqueness statement of

Theorem 4. In particular, since J is invariant under C;,t ~), we have <t>j{O -> co} =J.'j{O -> oo}.

This deceptively simple description is not an effective method for computing<t>j, but it does help us get a handle on mixed multiplicative integrals. By a series ofelementary steps-taking a p-adic logarithm of the defining expression for #,;'1: wïin Definition 7, using the Manin continued fraction algorithm to reduce to the caser = 0, s = 00 (here we use the fact that F is Euclidean), and expanding into a powerseries in t-we reduce the computation of a mixed multiplicative integral to that ofthe moments of the measure J.' j {O -> oo}:

(17)

(for more details of this reduction, see [7, Sections 1.3,2.1]). These numbers can becalculated from the fact that <t> j is an eigensymbol for Urr ;

(since d<t>j{b/n -> co) is supported on b + n(!Jrr)

= ~1 (a + nt)kd<t>j!::' -> co}W a+JTl9", JT

k

= ~L (k)ak-inid<t>j!::' -> OO}(ti).W Î=O l Ir

To compute the moment (17) to accuracy n M, it suffices to know the moments oftheeigenlift <t>j {a/n -> oo}(ti ) to accuracy n M - i . In computation, we store all measuresas such tapered sequences of moments.

5. Numerical examplesStarting with arE .Yt'~.9Jt with order (!J of conductor c, the algorithm described inSection 4.1 produces a point Jr whose coordinates belong to a p-adic field K rr . We

STARK-HEEGNER POINTS 445

conjecture that this point is defined over the global field He> the ring class field of K

of conductor c. Note that 1[ splits completely in H" so H, injects into K rr •

We find a set of representatives T], ... , Th of f\Yé'~·9Jt and compute the corre­sponding p-adic points J" = (Xi, y;) to accuracy 1[". If they indeed forrn a Galoisorbit defined over H" then they should pass the following two tests.

The trace test

Thesum

h=

1:,::

1

1

i

TiEf\Yf~·9JJ

should be a global point in E(K). We deem this test to be passed if we can find a(global) point in E(K) which agrees with the (p-adic) point h to the accuracy n ofthe computation.

The polynomial test

The polynomials

h

!x(T) = TI(T - Xi),;=1

h

!y(T) = TI(T - Yi)i=l

(18)

satisfied, respectively, by the x- and y-coordinates of J" = (Xi, y;), should be in K[T]

and should define (a subextension of) Hel K. To test this, we compute the polynomials(18) and approximate their coefficients by elements of K. If the resulting polynomialsindeed split over H" then we deem Conjecture 6 to have been verified.

Our calculations use modular symbols (Manin's continued fractions algorithm inparticular) and therefore require that we work over a Euclidean imaginary quadraticfield F = Q(J/5), which forces DE {-l, -2, -3, -7, -II}. We also assume thatthe conductor of E / F is prime, which simplifies the calculations without sacrificingmuch theoreticalinterest. Cremona's tables of curves over imaginary quadratic fields(see [4]) are unformnately much shorler than the ones over Q, and they containrelatively few curves of small prime conductor. In the exarnples in Sections 5.2and 5.3, the conductor 1[ is split over Q, so our p-adic calculations in fact take placeinQp.

5.1. Global appmximationThe two tests of Conjecture 6 reduce to a basic approximation problem: given a p-adicnumber fJ E Qp known to accuracy p", find an algebraic number b E Qi so that fJ == b(mod p"). There are of course infinitely many such b, but in any given finite subfieldF C Qi, one among them is optimal in the sense of having the smallest height (fewestdigits). Note that the approximation problem and its optimal solution are pure number

446 MAK TRIFKOVlé

theory-they make no reference to the elliptic curve E. A random approximation ofthe coordinates of h yields a global point in the plane which is unlikely to lie on E.Remarkably, however, if we choose the optimal approximation, then we get a globalpoint that often does lie on E, especially if the height of the approximation is smallrelative to the accuracy of the calculation. We observe a similar phenomenon in thepolynomial test: when the optimal approximations of the coefficients of j, or fy havesmall height, the resulting global polynomials tend to define the correct class fieldof K.

We briefly describe our method for finding the optimal approximation b, which isa variation on the one used in [7, page 14]. Since the conductor n in both ourexamplesis split over a rational prime p, we have the task of approximating an element of!QIpby an element of the imaginary quadratic field F. To be precise, let F = !QI(œ) for analgebraic integer œ, let fJ E Zp be the p-adic number that we wish to approximate,and fix a, b E Z so that

a == œ (mod pH), b == fJ (mod pH).

In practice, the integer b is the result of our p-adic calculations.Any approximation of fJ by an element of F is given by a triple of integers

(r, s, 1), p t 1, so that

ra +s-- == b (mod p").

1

Such triples lie in the lattice

L = ((r,s,/) EZ3Ira+s-Ib==0(mod pH)}

with basis (0, b, 1), (0, p", 0), (-1, a, 0). The LLL algorithm, implemented in PARI,is very efficient at finding the (almost) shortest-Iength vector 10 = (ro, so, 10) E L (see[3, Sections 2.6, 2.7.3] for details). Having found it, we take

roa + Sob = -'------'-

10

to be our preferred global approximation of fJ. This most certainly does not prove thatfJ is in F, much less that it actually equals b; but in practice, global points obtainedby applying this algorithm to the coordinates of the p-adic points J, often lie on E.

To get a sense of how this approximation algorithm behaves, it is instructiveto apply it to pi, thought of as a p-adic number given to accuracy p", n ~ i. Thealgorithm recognizes pi as

• pi for i approximately less than n/3;

• a random element of F for i between approximately n/3 and 2n/3;

• 0 for i > 2n/3.

STARK-HEEGNER POINTS 447

We thus have the most confidence when our approximations are of small height relativeto the accuracy of the computation. This, in tum, makes all the more interesting thefew examples with many digits which pass the tests anyway. Samples of both areincluded in Sections 5.2 and 5.3.

For a point (x, y) E E(<lJ!), the height of x is approximately two-thirds the heightof y. Il therefore sometimes happens that the x-coordinate is recognized correctly, asit lies just within the range of accurate approximation, but that the y-coordinate is justoutside the range and needs to be computed directly from the equation.

We retum to the IWo curves from the introduction.

5.2. w(E, F) =-1Recall the curve

E[ :l+y=x3 +(l-a)x2 -x

defined over F = <lJ!(yCTf), a = (1 + yCTf)/2. Ils conductor rr = 2a + 5 is primeof norm 47. The reduction at rr is nonsplit, so by Corollary l, w(E[, F) = -1. Asexpected, rkzE[(F) = 1; the obvious point P = (0,0) generates E[(F).

We computed the basie Heegner points 1K associated to quadratic extensionsK = F(-j8), N8 ::; 3000, in which rr is inert. The sign w(E[, F) being -l, weexpect that the basic Stark-Heegner point 1K in fact lies in E[(F) (analogously to[6, Proposition 5.10]). Since we know a generator of E[(F), we can dispense withthe global approximation algorithm and simply search for the 47-adic point 1K in thelist of (47-adie expansions of) multiples of P. A match was found in every examplewe tested. This method frees us from' the height restrictions of the approximationalgorithm, so that we can identify points as high as 26P (when K = F(.J20a + 18)).

The results for the first page worth of K's are displayed in Table 2. The firstcolumn contains 8 with K = F(.j8). The second column gives the integer n such thatPT = nP (at least to 20 digits of p-adie accuracy). The third column gives rkzE[ (K)(computed as rkzE[(K) = rkzE[(F) + rkzE~')(F)), computed where possible in areasonable time using Denis Simon's program for descent over number fields in [15].Our calculations confirm Conjecture 3: in every case where the rank computation issuccessful, we find that rkzE[(K) = 3 when 1K = 0 and rkzE[(K) = lotherwise.The cases where the raak computation is inconclusive are indicated by "-" in thethird column of the table.

As a warm-up for the polynomial test, consider the following three auxiliaryquadratic extensions of class number 2:

L

K2 = F(.J2a + 1),

448 MAK TRIFKOVlé

Our computations reveal that the three corresponding Stark-Heegner points Pi ofconductor 1 all have the same x-polynomial,

Jx(T) = T 2 + (-a + I)T - l,

with discriminant -a + 2, while their y-coordinates actually belong to F:

YI = -l, Y2 = Y3 = O.

Thus even though they a priori lie in Hilbert class fields of three different fields, infact ail three Pi'sare defined over Ko = F (.J-a + 2), P2 is in fact equal to P3, andthe compositum KoKi is the Hilbert class field of Ki for i = 1,2,3.

We found the most impressive suceess on the polynomial test for K =F(.J-31a - 13) of class number 11. Up ta 20 digits of 47-adic accuracy, we find thatthe x- and y-coordinates of Stark-Heegner points of conductor 1 satisfy, respective1y,the po1ynomia1s

f (T) - Til + (_ 1001 a + 17)T IO + (323272 a _ 424678)T9

x - 81 27 6561 2187

(1089383 171223) 8 (6204140 6960362) 7

+ 2187 a + 729 T + 6561 a + 2187 T

(23838260 2734360) 6 (14741863 21734605) 5+- a- T+ a- T

6561 2187 6561 2187

(31785055 945548) 4 (187616 345851) 3

+ 6561 a + 2187 T + - 243 a + 81 T

(1233710 39776) 2 (418849 404362)

+ - 6561 a + 2187 T + 2187 a + 729 T

(152569a _ 71186)

+ 2187 729'

J (T) _ Til + (_9040 a + 808)TIO + (27617002 a _ 9969382)T9

y - 729 243 531441 177147

+ ( 357040964 a _ 36661465)T8+( 190683592a _ 10652966)T7531441 177147 177147 59049

+ (2222665025 6043268) 6 (2659900916 239230063) 5

531441 a+ 177147 T + - 531441 a 177147 T

+ (994603849 49156820) 4 (153123922 422939471) 3

177147 a- 59049 T + 177147 a 59049 T

+ (_12_0.,..30,...1_5_5 7540141) 2 (2238101 2089850)

a- T + a- T19683 6561 6561 2187

+ ( 322343 a _ 222079).6561 2187

STARK-HEEGNER POINTS 449

Both of these do in fact cnt out the Hilbert class field of K. A sample of othercomputations for the polynomial test is displayed in Table 3. If fy is missing, it meansthat it was not recognized accurately.

ln the range of discriminants N8 < 3000, we successfully checked thatthe Stark­Heegner points of conductor 1 are indeed defined over the Hilbert class field of K fora total of 48 fields K of class number of at least 2, distributed according to the classnumber h as follows:

h

# of examples

i1

i!

5.3. w(E, F) = 1The smallest curve of prime conductor in Cremona's tables whose functional equationhas sign 1 is defined over F = 1Q(,J=3). We write a for Cl + ,J=3)/2. The curve inquestion is

E2 : yZ + xy = x 3+ (a + I)xz + ax,

with prime conductor n = a+8 ofnorm 73. The reduction atn is split, so w(E2 , F) =1, and in fact, Ez(F) is generated by the point (-1, 1) of order 6.

The situation here is arguably more interesting since we have no way a priori ofconstructing points even on Ez(K). The Stark-Heegner points tend to be of greaterheight than when w(E, F) = -l, in agreement with what has been observed in[7]. This manifests itself in the relatively small number, 123 of 466, of basic Stark­Heegnerpoints accurately recognized as global points. The points h for dise K :s 304are displayed in Table 4. Note that the 8 in the first column actually generates thediscriminant ideal of K = F(~)/F (i.e., includes a factor of4 where necessary). Thecoordinates of h are expressed in terms of the algebraic integer fJ, which generates(r) K over (r) F and satisfies

r Z - 8fJ2 + rfJ + -4- = O.

Here r E {O, l, a, a - 1) is determined by 8 == r2 (mod 4).Disappointingly, we found no examples of K in the discriminant range N 8 < 1000

for which the x- and y-polynomials of Stark-Heegner points of conductor 1 wereaccurately recognized (excluding, of course, the cases where h(K) = l, so that theStark-Heegner point is equal to the basic Stark-Heegner point). This is perhaps notsurprising given the relative scarcity of even the basic Stark-Heegner points correctlyidentified as global points.

450

Table 4. Basic Stark-Heegner points on Ez/Q(R)

MAK TRIFKOVlé

discK PK.+3 (-fi + (-a - I),(-a + I)fi + (-a + 2))4œ + 1 (fi, -fi - a)5 (. - I)fi, (-20 + 2)fi)

-7a+3 (-1atJ - ~a, (~a - ~)f3 +Ua + ~))--4a -3 (-.fi + (-a + I),2afi +(2a - 3))8et' -4 (-a + I)fi - I,.fi -.)

4œ -8 (ia -1, (-ia+ Dp+ (-~a+ i))-7a+7 (. + I)fi - 2, (-3. + 3)fi +4.)

Sa + 1 (' 3)fi (32 ") (" ')fi (SOO 1"'))490'+49 + ;j9a+;W' 343 a +343 + 3430:+ 343

7a+4 -~a-~,Ua-Dp+~)

Sa+7 ( 2164 3147 ) f3 (4507 5648 )-10609(t + 10609 + -10609 CX - 10609 '

( 107599 95386 ) fJ (16659 + 313835 ) )lO92727O! - 1092727 + 1092727Ct' 1092727

-7a-5 (_Ma _ 1548) f3 + (_1989 a _ 3779)4489 4489 4489 4489'

(1529460: _ 13017 ).8+ (1l7240 a + 125590))300763 300763 300763 300763

8a+4 ( ( 169 731 ) fi (1209 496 )2401 a - 24(Jl + - 2401 a - 2401 '

(30773 52510 ) fJ ( 45057 2232 ) )-117649 a + 117649 + 117649a + 117649

-4a + 12 «(-2. +3)fi + (3. - 5), (5. +3)fi +(-6. -7))

-11a+ll (lafi+ Ha -J), (-1. -l)fi+ (-1.+ 1))

8a+5 (( 136 869)fi (67 1814)2401 Ct - 2401 + - 2401 ex - 2401 '

(13178 36007 ) fi ( 22487 + 54662) )-1l7649 a + 117649 + 1l7649Q' 117649

-0:+ 12 [12 32) fi (6 33) (148 ")fi (In 83));wa - 49 + -:Net' - 49 ' -343 Ct' + 343 + -343 a - 343

-9a -4 4 20 (4 28)fi (" 62))-'j"a + 9' 'j"a+ Ti + Tia - Ti

12a + 1 (521 352)f3 (1860 955) (14714 2(051)f3 (44385 - 66455))-1849 a - 1849 + - 1849 a - 1849 ' 79507 a - 79507 + 79507 a - 79507

13 (la -I)fi+(. -2), Ha -j)fi+ H. -l))-4a - 11 (la - 2!)f3 + (.ill. a _ ID) (_698 a ~ 337) f3 + C26a _ 394))

5 25 25 25' 125 125 125 25

8a - 16 _J1 a -!!. C03a -.:!2)f3+(J1a +!l))2 4' 8 8 4 8

16a - 8 (136 258)f3 (243 712) (8462 6132)f3 (2669 7670))961 a - 961 + - 961 a - 961 ' - Y79l a + 29791 + 29791 a + 29791

16a -7 (a + 3)fi + (7a + 2), (-lia -7)fi + (-39a + 8»)-lia - 5 94 373 (905 353 ) f3 (1169 3091) )

- 361 a - 3IIT' 6859 a - 6859 + 6859 a + 6859

-16a + 5 (a-2,-afi+(-a+I))

STARK-HEEGNER POINTS

Table 4 (cant.). Basic Stark-Heegner points on E 2/ Q(HJ

451

discK PK

-120:' + 16 (4 4 (122 358)fi (' 2))-13er - 39' 1521 Cl - 'illï + na + 39

-16er + 13 ( (_ 136052a _ 132335),B + (_ 28291 a + 280S4)162409 162409 12493 12493'

(_ 32367650 Ci _ 16(940087) fi + (_ 12888595 Cl:' + 19591856) )65450827 6545OS27 5034679 5034679

16er -3 ( (8336 2809) fi (13611 14900)1932T0:' - 19321 + 19321 Cl:' - 19321 '

(1617354 1101729),B (3390197 4878612) )-2685619Q' + 2685619 + -2685619 a + 2685619

-8a - 11 ( (120202 a _ 13185) fi + (_31705 Cl:' + 243556)49729 49729 49729 49729'

(64005056 7296414) fJ (22812960 160690568) )11Ol\9567 a + 11089567 + -110895670:'+ 11089567

lIa +8 ( ( 435 "46 ) fi (493 9003 )3276[ Cl:' - 32761 + - 32761 Ct - 32761 '

(16099 125481O),B (1702799 2541963) )- 5929741 ex + 5929741 + - 5929741 ex + 5929741

19a-4 (e85 324)fJ (302 560) ( 11110 +2320).8+( 21533 +22120)169 U -169 + -169 U -169' -2197U 2197 -2197U 2197

-19u + 15 (-a +3. (-a - 2)fi + (-a - 1»)

-12a +20 (20 '( '" 151)fi (10 1)-WU - 57' -3249 U + 3249 + WU + 57

-ZOU + 8 (.6 50 (3694 1103) 13 (3 25)WU + 57' -3249 U + 3249 + -WU - 57

Computing the basic Stark-Heegner points appears ta be comparable ta descentas a practical method of producing points on E(K). Of the 123 basic Stark-Heegnerpoints successfully recognized. 68 were not found wilhin reasonable time by thedescent algorithm. Overall, descent was successful about twice as often. finding agenerator of E(K) in 254 of 466 cases. Interestingly, while in the descent methodthe height intervenes as an explicit limit on point searches, in the Stark-Heegnerconstruction it is the number of digits of accuracy in the eigenlift which effectivelylimils the height of points that can be recognized correctly. We made no attempt tafind comparable values for the limiting variables of the two methods; the comparisonbetween them is meant ta be illuslrative, not rigorous.

The main time expenditure of the Stark-Heegner method is in the eigenIift com­putation; getting 30 digits of accuracy on a 750 MHz machine took about one day. But,with the eigenlift in hand, any individual h can be computed in about one minute.The Stark-Heegner construction should theu be useful if one needs ta find points overa large number of K's on a single elliptic curve.

(More extensive tables for bath examples, as weil as the PARI programs used tagenerate them, can be downloaded from www.math.mcgill.ca/mak)

Acknowledgments. Above ail, l thank Henri Darmon for encouraging this work andbeing generous with time and ideas. l am also grateful ta Matthew Greenberg forsuggesting ta me that il may be possible ta avoid solving the face relations; ta Dominic

452 MAK TRIFKOVlé

Lemelin, whose programs for modular symbol over imaginary quadratic fields werethe starting point of the computations; and to Denis Simon for providing his programfor descent of elliptic CUrves over number fields.

References

[1] M. BERTüLINI and H. DARMüN, The rationality ofStark-Heegnerpoints over genus

fields alreal quadratic fields, ta appear in Ann. of Math. (2).

[2] C. BREUIL, B. CONRAD, F. DIAMüND, and R. TAYLOR, On the modularity ofellipticcurves ove,. Q: Wild 3·adic exercises, J. Amer. Math. Soc. 14 (2001), 843-939.MR 1839918

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Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West,Montréal, Québec H3A 2K6, Canada; mak@math.mcgill.ca