STARK-HEEGNER POINTS ON ELLIPTIC CURVES DEFINED OVER IMAGINARY QUADRATIC FIELDS

STARK-HEEGNER POINTS ON ELLIPTIC CURVES DEFINED OVER IMAGINARY QUADRATIC FIELDS

MAK TRIFKovré

Ahstract

Let E be an elliptic curve defined over an imaginary quadratic field F ofclass number 1. No systematic construction of global points on such an E is known. In this article, we present a p-adicana/ylieconstruction of points on E, which we conjecture to he global, defined over ring class fields of a suitable relative quadratic extension K / F.

The construction follows ideas of DOI'mon ta produce an analog of Heegner points, whichis especially interesting since none ofthe geometly ofmodular parametrizations extends ta this setting. We present sorne computational evidencefor our construction.

Contents

1. Iotroduction...

2. Modular fonus on the upper half-space 3. The p-adic construction . . . . . 4. Measure-valued modular symbols 5. Numerical examples

References. . . .

415 429 433 438 444 452 1.Introdnction

LetE be an elliptic curve defined over a number field Fwith conductor ideal.IV.Ils Hasse-WeilL-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 allSEC,a fact known whenF

=

<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 the Taylor expansion ofL(E/F,s) at s

=

1.The Birch and Swinnerton-Dyer conjecture expresses this tenu in tenus of algebraic iovariants of E /F. A weak version of the conjecture is given by the conjunction of the following sequence of statements, one for each integerr :::: O.

DUKE MATHEMATICAL JOURNAL Vol. 135. No. 3,©2006

Received 7 October 2005.

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

415

416 MAK TRIFKOVlé

(1) CONJECTURE1 (BSD(r»)

Iford'~IL(E/F,s)

=

r, then the Z-rank ofE(F)isequal to r. Moreovel;III(E/F )is finite.

The full Birch and Swinnerton-Dyer conjecture asserts that the implication inBSD(r) is in fact an equivalence and gives a precise formula for the leading term of the Taylor expansion ofL(E/F ,s) at s

=

^1.

The interest of the weaker statementsBSD(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 ofBSD(r), r :::: 1, forF totally 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 on BSD(O) and BSD(l) to date use variants of the Heegner point construction, which is essentially the only one known. The goal of this article is to propose, and provide numerical evidence for, a conjectural p-adic construction of so-calledStark-Heegner points: p-adic analytic analogs of Heegner points in the first case where even BSD(O) is mysterious, namely, when the base fieldF 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 overQ! and parametrized by Xo(N)-the better to appreciate the difficulties that arise over an imaginary quadratic field.

1.1.1. Modularity and Heegner points

LetN be a positive integer. The groupro(N) of matrices in SL;,(Z) which are upper triangular modulo N acts on the extended upper half-plane.Yi" = .YI' Upl(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 modular curve defined overQ! parametrizing pairs(E,C) of generalized elliptic curves with a cyclic subgroup of orderN.

Let K be an imaginary quadratic field of discriminant DK. 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 extensionH,I K with Galois group Pic(@J corresponding by class field theory to the subgroup KXiC X(@,

@Zy

of the idèle group ofK. We fix an imaginary quadratic field K satisfying the following hypothesis.

STARK-HEEGNER POINTS

THE HEEGNER HYPOTHESIS

Ali primes dividing N are split inK.

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

417

Definition 1

Let 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 2

Let c be a positive integer prime to N andDK.A Heegner point of conductor c is any pointx"where bothT andNT have conductor c.

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

THEOREM 1

Letc be prime to both N and DK. Then any Heegner point x, of conductor chas coordinates in H" the ring dass field ofK of conductorc.

Crucial to the proof ofthis theorem is the modular interpretation of points onXo(N);in particular,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 modular interpretation 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 ofE, whose existence is the central claim of the Shimura- Taniyamaconjecture, proved byWiles [19], Taylor and Wiles [17], and Breuil, Conrad, Diamond, and Taylor [2].

THEOREM 2

LetE/Q be an elliptic curve of conductor N, with Néron dijferentialWE.There exists a newform f E S2(fo(N)) with rational coefficients, a mOiphism ofalgebraic curves

<jJ :Xo(N) -+ E defined OverQ,and aManinconstantc~ E Qsuch that

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

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

We can now define Heegner points onE.

(2)

418 MAK 1RIFKOVlé

Definition 3

A Heegner point of conductor c onE is any point ofthe form P,

=

</>(x,) E E(He )

withx, a Heegner point of conductor conXo(N).

Inparticular, when c

=

1, that is,^(1),

=

^(1)K,we get a pointP,defined over the Hilbert class field H

=

^HI ofK.

Definition4

When P, has conductor 1, the point PK

=

trH/K</>(P,) E E(K) is calIed the basic Heegner point.

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

1.1.2. Complex-analylic construclion

The inspiration for our definition of p-adic Stark-Heegner points cornes from recasting the above construction in purely complex-analytic terms. Let

j

^Y'

AE

=

{Znic. , f(z)dz ¹y E 'o(N»)

be the period lattice of E. Since E is an Abelian variety, the modular parametrization

</>factors through the Abel-Jacobi map

Xo(N) --+Al Jo(N) --+ E

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

n-+x

=

Znic.

l'

^{f(z)dz f-+}(S'J(x), S'J'(x»).

"'0

(3)

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

1.1.3. Sign offunctional equation

The 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 for the 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 forL(E/Q, s) in the particularly simple fOTIn A(E/Q,s)

=

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

419

Herew(E,!QI)

=

±I is called the sign of the functional equation ofE/Q. In particular, when w(E,!QI)

=

-l, the special value L(E/Q,¹⁾ vanishes, sa that the Birch and Swinnerton-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 sign considerations force the L(E/F, s) ta vanish ta order t, we may hope, in light of the Birch and Swinnerton-Dyer conjecture, ta find an explicit construction of a rank-t subgroup ofE(F).

Asimilar functional equation is conjectured ta exist for E defined over an arbitrary number fieldF.WhenE/Fhas 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 following elementary fashion (see [14,Theorem 2]).

Definition 5

Setw(E,F)

=

^{(-1)'+',wherer} is the number of Archimedean places ofF(Le., real or pairs of complex conjugate embeddings) and sis the number of finite places where E has split multiplicative reduction.

COROLLARY1

Let E/Fhave a square-free conductorJVand hence semistable reduction. Let K /F be a quadratic extension ofdiscriminant prime taJV.Consider the setSofail places v' of K satisfying the following conditions:

(a) v' isArchimedean, orv' isfmite and dividesJV"and (b)

if

visthe place ofF belowv',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 field and 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 and a functional equation whose sign is indeed w(E,K). Suppose in addition that K satisfies the Heegner hypothesis. Then Corollary 1 implies thatw(E, K)

=

-1 and that, therefore, L(E/K, 1)

=

O.Il is then natural ta ask aboutL'(E/K, 1).

THEOREM3 (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 {} PKis ofinfinite arder.

In other words,

if

PK is torsion, thenord'~lL(E/K, s) :::3.

MAK TRIFKOVlé

When L(E/K, s) vanishes to order exactly 1, the point PK of infinile order is the essential input for Kolyvagin's theory of Euler systems, which actually allows us to prove thatrkzE(K)

=

1 in this situation.

1.2. Stark-Heegner points over imaginary quadratic base fields

We now replace1Q>as our base field with an imaginary quadratic fieldF, and we look for analogs of the ingredients of the classical Heegner point construction. We assume for simplicity thatFis of class number 1; the fieldsFthat feature in our computations are even Euclidean.

The equality of L-functions (4) is an instance of the general Langlands program, which predicts the existence of an automorphic representationp ofG~(AlF)such that LCE/F,s)

=

L(p,s).Since the fieldFis still relatively simple, il tums out that we can replace the abstract automorphic representationp with a concrete geometric object.

By analogy with the upper half-planeY/', we consider the the upper half-space y/,(3)and ils completion by cuspsy/,(3), given by

y/,(3) =

lez,

1)1Z EC,1 ElR~o}, y/,(3).

=

^{.!'t'(3) U1I'1(F).}

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

while the grouplo(N) is replaced by

The datum of the automorphic representation p is equivalentta 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) ---* C³plays the role of the modular form

fez)

over1Q>(see Definition 6).

At this point the naive analogies with the situation over 1Q> break down. The quotient spacelo(A")\y/,(3).might seem a tempting substitute forXo(N),except that il is a three-dimensional real manifold and hence not a variety, let alone a moduli space. Moreover, there is no obvious parametrization10(JlI')\y/,(3)*---* E(IC).

STARK-HEEGNERPOINTS

Table 1

421

Complex p-adic

1. Archimedean place 00 1. Non-Archimedean placeJrIIAÎ 2. KIrQimaginary quadratic 2. K / Fquadratic, inert at] f

(local degree 2 at(0) (local degree 2 atJr) 3. Heegner hypothesis: 3. Stark-Heegner hypothesis:

AilliNsplit inK Aliv\,.-,V, vi=-^Jfsplit inK 4. Z-order(!) c K

4.I9F[~]-order(!)

^CK

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

(domain off(z)dz) (damain ofWj) 6. Poincaré upper half-planeJlf' 6.p~adicupper half-plane

(::::>quadratic irrationalities :tt. =Il'ICC_p)-Il'I(F.)

Kn:tt,<0) CC)KnYI'.,<0)

7. Weierstrass uniformization 7. Tate uniformization

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

f

^{- f(z)dzEC,}<l, <2 EJ't'*

iT

WJEC;,TJ,r2 E.Yt'rr.

" _r,

"

_SE' _P'(F)

Inthe 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 inJ'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 singlereal period, from which we cannot reconstruct the period lattice ofE.

Following the ideas of Darmon [6], we propose to resolve these conundrums by working at a non-Archimedean prime ^1[ dividing the conductorJV of

J,

^rather

than at the infinite prime. Table 1gives a useful heuristic dictionary hetween various components of the complex and p-adic constructions; sorne concepts are defined later on.

Remark 1

Our 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 theStark-Heegner points, conjecturedtobe defined over ring class fields of areal quadratic field K.

Itis natural to attempt to extend Darmon's construction to our setting since the quadratic extensionK /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

Remark2

The assumptionn

Il

JVis essential for the existence of the Tate uniforntization as in dictionary entry 7.

Remark3

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

Remark4

In the classical setting, ^ye plays two roles: it is the domain of the modular fonn, but it also contains the quadratic irrationalities that parametrize Heegner points. Over an imaginary qnadratic field, the fonner role is played by the naive analogye(3),while the (relative) quadratic irrationalities are now to be found inyen' Note thatK

n

Jf'n

#

0 sinceIris assumed ta be inert.

Remark5

Only the last entry in the dictionary requires extensive explanation, which is provided in Section 3. For now, suffice it to say that the integral in question is "mixed" in the sense that the first set of limits is p-adic, while the second consists of two rational cusps, and it is "multiplicative" because it is defined as a limit of Riemannproducts rather than sums and therefore satisfies the multiplicative analogs of the usual additivity properties of integrals.

In order to get the correct statement of Shimura reciprocity, it tums out that it is essential to work with an "indefinite" version of this multiplicative integral. We conjecture !hat we can find a rank-one subgroup Q <; _<C~ with QI <; _q~ for sorne integert and a function !hat to asingle 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 the conjecture of Mazur, Tate, and Teitelbaum. The success of our computations is a strong encouragement to explore this conjecture over imaginary quadratic fields.

The prospect is tantalizing since the Mazur-Tate-Teitelbaum conjecture was proved by Greenberg and Stevens [9] using Hida families whose existence is unlikely for modular fonns on GL2(AF ).

r

STARK-HEEGNER POINTS

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

423

(5)

Herey,is the generator of the group ofunits in a suitable subring ofM2x2((1) F[l/1f]) fixingT,whilel' E IP'I(F) is an arbitrary base point. Note that exponenliating byt was necessary to gel a point in _C;/q~ rather than C;/Q-an unavoidable technicality that 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 sinceh(F)

=

^l,the theory of orders in a quadratic extensionK / Fis virtually identical to the usual theory of quadratic orders over!QI. As suggested in entry 4 ofthe p-adic column of the dictionary in Table l, we work with (1)F[I/1f]-orders throughout. Any(1)F[I/1f]-order(1)inK is equal to the order

(1),

= (x

^E

_(1)K[~] lx

^=a

(modC(1)K[~])

^{for sornea E}

(1)F[~]1

for sorne idealc

c

^(1)F, which we assume to be prime to1f.Since F is principal, we occasionally abuse language and call the conductor of(1)any generator of the idealc.

The subgroup

(6) defines an Abelian extension Hel K called the ring class field of conductorc. Since we work with(1)F[I/1f]-orders,1f splits in the class field defined by (6), and one might expect that H, is smaller than the ring class field we would have gotten by working with (1)F-orders instead. One of the lucky coincidences of working with a field of class number1is that the two ring class fields are equal since1fis principal and iner!

inK.

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

CONJEcruRE 2

LetTEK

n

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

L

bath have (1),for their ordel: Then the pointJ, that a priori lies in E(C_{p )} is infact defined Over the ring class field H, ofK.

For amOreprecise version of this conjecture, giving the action of G^{Hel K,} see Conjec- ture 6.

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

424

CONJECTURE3

If

h

^E E(K)is torsion, thenrkzE(K) 2:3.

MAKTRIFKovré

1.3. The computations

We present the results of our computations for two curves, one for either value of the sign of the functional equation ofL(E/F, s).

1.3.1. w(E,F)=-1

LetF

=

Q(..;=TI) with a ring ofintegers generatedbya

=

⁽¹

⁺

..;=TI)/2. Consider the curve

El :/+y=x³+(l-a)x'-x,

which has prime conductor".

=

2a+5ofnorm 47. The reduction ofElat". is nonsplit, sa we expect that the functional equation ofL(EI / F ,s)has signw(EI ,F)

=

^{-1 and}

that rkzEI(F) = 1. Indeed, one easily checks that the obvious point P = (0,0) generatesEI(F).

Sincew(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 multiple of 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 ofP for every K = F(.J8) in which". is inert and NF/l:é < 3000. A sample of these results is presented in Table 2. When h = 0, we verify in many cases thatrkzEI (K) = 3, as predicted by Conjecture 3, using Denis Simon's program for descent over number fields (see [15]). There are cases where the rank computation is inconclusive (denoted by "-" in Table 2), but we fully expect that this is due ta the great height of the Mordell-Weill generators ofEI(K).

When K

=

^F(.J20a

+

18), we find the basic Stark-Heegner point of greatest height in the range of our computations: h

=

^{26P.We expectIII(E}I / K )ta have an element of arder 13.

We can also get some sense of the Heegner points themselves, rather than just of their traces down ta K. ConsiderK

=

^F(.JI3) of class number 5. Conjecture 2 predicts a Galois orbit consisting of five Stark-Heegner pointsJ'i of conductor 1. We compute the polynomials satisfied by thex- and y-coordinates of the J'i's. Modulo 47'°, they are

fAT)

=

T^S

+

(-2a

+

2)T⁴

+

(-Sa - 4)T^3- 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 ofK,as expected. This example is emphatically atypical; we found only a handful of class number 2 examples and one class number 4 example in whichfx(T)andfy(T)also have small integral coefficients.

Table 3 contains more examples of polynomials satisfied by coordinates of conjectural Galois orbits ofJ,'s.

STARK-HEEGNER POINTS

Table 2. Basic Stark-Heegner points on E_1/Q(.;:::T1)'J_K = n[ü, 0]

425

L

8 n rk

-1 2 1

a-l -1 1

-a -1 1

-2 0 3

a+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 1}

2a-3 -1 1

2a+2 0 3

2a -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 3

3a-5 1 1

-3a -2 -1 1

4œ-2 -4 1

3a+3 ^-1 1

3a - 6 -1 1

2œ-7 0 3

-7 2 1

-4œ+5 0 3

4œ+ 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 1

4œ -7 1 1

-5œ+3 1 1

-501'+4 3 1

Sa - 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 1

a - l I -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 1

la - 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 1

13 -2 1

-8a+3 1 1

-7a -3 -3 1

7a -10 1 1

Sa -5 -1 1

Sa -13 -1 1

5a+8 1 1

6a +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 1

4œ -14 4 1

-7a+11 1 1

7a +4 -II 1

-14 0 3

6œ+7 -2 1

Ga -13 2 1

Sa +9 ^-1 1 Sa - 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 1

4a-15 -1 1 6a+& 8 - -Ga + 14 -4 1

9œ -4 -7 1

90! -5 1 1

15 1 1

9œ-7 1 1

9a - 2 -1 1 3et+13 2 1

9a-& -2 1

-2a - 14 -14 1

2a-1G 2 1

7a -14 0 3

7a+7 4 -

~5œ+16 1 1

426 MAK TRIFKOVlé

Table 3.x-and y-polynomials ofIr; EEI/Q("cm(H)defining the Hilbert class fieldH ofK = Q(.J=TI)(,ft)

8 _J;of)'

-0:+4 fJT) - T

2

_{+(-;~~~~ga+~~~~~~~)T+(-I~:7a}

- t:J7)

fy(T)

=

T²+(-~~~i~~~~a+~~~~~:~~~~)T+(-~~~~~~~ct

+

;i~~~~) Sa - 23 !,(T)~T'+(-~.+1)T'+

(1. -

'j!)T+

(1.

+

II

fy(T)= T 3+(a+2)T²+(~a+

b)

^T+(ta -

t)

5. _!x(T)

=

T⁴+(~~~~a- ~~;~~)^T3+_(?3~~6225a+i~~~~)^T2

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

: 6

is

^{0: -} 1138;~~)

fy(T)= T⁴+ (-g~~j~~~^{ct -} 152~7~~)^T3+ (-i~~~~m~cr - ~~~;~;~i~~)^T2

+ (-_~~~~~~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)T3 + (~iil~;~^Ci+~;60~1;8813)^T2+ (-~22~~~80IS^cr+~~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)=T^S+(122~3~3^a+~~~~~~)^T4+(;~~~~a+~g~~~~;)^T3

+(;g~~~^a+~g~:~~~)^T2+(~~~~i~~^a+~~~~~~)^T +(32212~OS~a - 3~~~~9)

-16a+17 fx(T)=T⁶+_(26J~1a+_i~:~)T^S+_(il~~~a+ _2~~81)T⁴

+_(~~~~~a - ~~~~~)^T3+_(~~:~ia - ;4~1)^T2 + (-^i48J:Ia+ ~:~:)T+ (-^;19:S1a+ ;~~;)

f (T) - T⁶+ (-^139408711a+^64295(014)T^S+ (_^461302934a+ 1719072530)T⁴

y - 124251499 124251499 124251499 124251499

+ (-i~~g-:~~^a+31322fi11~91:)^T3+(i~~~~~~^a+ 3?2~26i~4~:)^T2

+_C~~~~:~a+ ?it.lf581~:i)^T+n~~:~~œ+ 1522:is\o;~) Sa - 43 fAT)=T9+ (-l~i8410a+

liXii()

T⁸+ (-~cil~~Oiia _ I~~g~~j~g-t)T⁷

+(528

il7²1:30a - 1~~:~~~j4)T6+(3~n~i;~3a+ 2~~g~~~~~7)T⁵

+ (_310778902a+785509(56)T4+(!^{1955584855a _} 2355419504)T3

7890481 7890481 71014329 71014329

+_(~~?4~2~0a - 1~~~;~i~l)^T2+_(~:~î~:~a+_b~î~~)T+ + (-~~~~;^a+~~~~:n

.~

1

i STARK-HEEGNER POINTS

,:j

J.

13.2.w(E, F)

=

1

LetF

=

Q(.)=3), and set a

=

(l

+

.)=3)/2. The curve E

z : i +

xy

=

x³

+

(a

+

l)x

z +

ax

427

(7) has prime conductor lf

=

a

+

8 of norm 73. The reduction at lf is split, sa w(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-adic accuracy, for fieldsK(./8),N F/Q8 < 1000, in which^lfis iner!. The nnmerical results here are more interesting since we do not know a priori a generator of Ez(K) with which ta compareJK. lnstead, we use an algorithm that finds in sorne sense the best possible approximation in K ta the 73-adic coordinates of

h.

Ifthe height of

h

is relatively small, then the point inAZ(K) thus obtained often lies on E; we say we havereeognized

h

as a global point.

The average height of basic Heegner points in this example seems ta be greater than whenw(E, F)

=

-1. They therefore tend ta be difficult ta recognize accurately as global points. Of the 466 basic Stark-Heegner points computed, only 123 were recognizable. Ofthose, the highest was found whenK

=

^{F(fJ), fJz}

=

^2a

+

21:

1259988 126090782 x

= -

^127165927a

+

127165927'

( 2903147975024 11037094266063) (629994 63045391 ) y= 31646131095439a

+

31646131095439

fJ +

127165927a

- 127165927 . Amusingly, this is the only point among the global points we found whose coordinates are not integral atlf,which means, since the Tamagawa nnmberen is 1, that

h

lies in the kemel of the reduction modlf map onE(K).

lA.Computing with modular symbols

The function

<Pi'

which ta a pair of cusps r, s E1I'1(F)associates the integral

takes values inQiZfor a suitablerealperiodQi(see Proposition 1). Il is an example of a modular symbol on fo(%) with values in a right fo(.k")-module M, that is, a function <1> :IP'1(F) xIP'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,SE1I'1(F), YE fo(%).

428 MAK TRIFKOVlé

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

TC,satisfying

(8)

To define a modular symbol onfO(JV),it is enough to specify its values on the edges of a fundamental domain for the action of fo(.Y) on.YE(3)'. These values cannot be chosen arbitrarily: each face of the fundamental domain imposes a Z[fo(JV)]-linear relation among them. In the analogous situation for modular forms on .YI", this is not a serious obstacle to computations since the fundamental domain for the action of fo(N) on.Yt"has only one face (namely, itself). By contrast, even the fundamental domain for the action of fo«(9F)on.YE(³)' is a hyperbolic polyhedron with anywhere from four faces (when F

= Q>(H»

to fourteen (when F

=

Q>(v'=2». For the action of fo(JV)this number gets multiplied by roughlyNF/Q%.Finding<1>asin (8) thus presents difficulties both algebraic,in terms of the number of relations the values of<1> have to satisfy, and geometric,in terms of the shape and incidence of faces of the 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 of linear equations. The idea, elaborated in Section 4, is to consider ail functions on the edges of a fundamental domain (or another large enough set of paths), including those not satisfying face relations.Itis trivial to find such a "fake" measure-valued modular symbol <1>0 satisfying (8). We define the action of the Hecke operator U_rr on fake modular symbols. Repeatedly applying ^Urr to <1>0 produces a sequence of fake 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 asin (8). Given that the relatively small conductors we work with are already large enough for the naive computation to be prohibitively slow, working with fake modular symbols was essential to the feasibility of our computations.

1.5. Further directions

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

(2) The Stark-Heegner construction appears to workin two settings with a fairly tenuous formaI similarity: forEdefined overQ>,producing points over ring class fields of a real quadraticK, andin 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, anditwould be 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 constructions so as taavoid the ad hoc elementary treatments of modularity. While they facilitate computations, they may well be obscuring the conceptual picture.

(3) Finally, the prospects for actually proving Conjecture 2 are dim at present. One may hope, by analogy with the theory of complex multiplication, to find a modular interpretation for the rnixed period integrals. A more promising avenue toward sorne immediate theoretical confirmation would betaconsider degenerate cases (e.g., when E is a base change from <QI) in the hope of relating this construction to classical Heegner points.

2. Modular formsonthe upper half-space

We 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 imaginary quadratic fields. Our accountiselementary, avoiding automorphicrepresentations,for the twin reasons of accessibility and amenability to computations.

2.1. The action ofPGLz(!C)

The upper half-plane has an action ofPGLi(Ift);ilis 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 subgroup PSU₂is 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 of PGL₂(F)onIP'I(F),giving an action ofPGL₂(F) on ail of.1(3),.

(3) Finally, and least enlighteningly, we can deduce from (2) an explicit formula for the action. For_{(~ ~)} E GL,(IC) and(z,t) E .1(3),set

8

= lez +dj2 + letl

²•Then

( a b) (z, t)

= ~

^(az

⁺

^b)(ez

⁺

^d)

⁺

^act2,lad - belt).

e d 8

2.2. Topology, metrie, and dijferentials

We 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 form a basis ofneighborhoods of the cuspooj;by translation, a basis ofneighborhoods of a cuspzEFconsists of (Euclidean) open balls touching the "floor" IC x {O} of.1(3) at z.

A PGL,(iC)-invariant meltic is given byds²

=

(dzdz

+

dt²)/t2. The geodesics are circles perpendicular to the floor (including verticallines).

A basis of l-differentials is given by theeolumnvector

- 1 1 ( dz dt dZ)

fJ

=

(fJo,!JI, fJ2)

= - t ' t' t .

The action ofy E PGL,(iC) on

fi

is given by the formula

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

-2rs;'"

11'1

^2_

Isl

²

2rs;'"

with ;". = det(y),l' = ez

+

^{d,s = Ct.}This expression is the analog of the automorphy factor(ez

+

^d)-2 for classical modular forms.

2.3.ModulaI' forms

Let

J =

(Jo, ^fI,

12) :

^.1(3) --+ IC³be a function with values in row vectors. We define the action ofro(JV)on

J

by 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 6

A cusp form of weight 2 for ro(.A') is a fuuction

J =

(Jo,

fI, h) :

^.11'(3) --->(;3 such that 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 denotedS2(JV).

By Definition 6, a cusp form

J

is invariant under (~ ~) ^{E l'o(JV),X E (1)F,} and thus has a Fourier-Bessel expansion of the form

- " -(4n ^l"lt) ("z)

fez,

t)

=

L.. c(,,)t²K lTi'iT 1fJ l n

O"oEmF v

IDI

vD

(see[13],[5]). Here1fJ(z)

=

^e4rriR'Zis a character of the additive group ofF,and

whereKi(t),i

=

0 or l, is the hyperbolic Bessel function that satisfies the differential equation

d²Ki 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 primeelementÀof(1)F not dividing JV,we associate the operatorTA:

We make essential use of the operator^Urr forn

I.AI'

defined by

Hecke operators with composite indices are defined by recursions analogous to the classical ones. There are a couple of minor differences with Hecke theory over Q.

(a) The operatorTA (resp.,U_{rr )} actually depends on the prime elementÀ(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 an involution onS2(%)and breaksitup into two eigenspaces with eigenvalue 1 or-1:

2.4. Plusforms and the Shimura-Taniyama conjecture

We work exclusively with forms

f

^E

st(JI!"),

referred to as plusforms. Their Fourier- Bessel coefficients satisfyCrEa)

=

c(a). The Fourier-Bessel expansion is then a sum over ideals of (r) F rather Ihan over individual elements:

f(z.t)=

L

^c(a)t2K(4n

¹

^a

¹

^t)

L1f!(EœZ).

O#(.)<;@F

v1DT

^<E@,

Vl5

The action of T_À and Un similarly depends only on the ideals O.) and (n). Since PGL;,«(r)F)

=

PSL2«(r)F)(~ ~),we mayas weil consider f to be invariant under the bigger subgroup

Newforms are defined as in the classical case. They are eigenforms for ail the operators TÀ •Je

t

JV. that are orthogonal to the space of forms coming from lower levels. By multiplicity l, they are also eigenforms for Ihe operators Un.nl%.

The Mellin transform of

f

^E

st

^(%)is given by (4n)'

('XO

Mf. s)=

#(r)~

^IDFI

Jo

^{fi(O.t)t2('-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 Dirichlet seriesL(E/F. s).This justifies our focus on plusforms and suggests a version of Ihe Shimura-Taniyama conjecture.

CONJECTURE 4

There is a one-to-one correspondence between (cuspidal) newforms

f

^E

st

(JI!")and ellipticCUl1JeSE/_F that do not have complex multiplication by F.

Elliptic curvesE/F wiIh complex multiplication byFshould correspond to Eisenstein series. Sorne computational evidence for this conjecture can be found in [4] and [5].

STARK-HEEGNER POINTS 433

(9)

The modular symbol associated to

J

assigns to any pair of cuspsr,s Epi(F)the path integral

The Mellin transform above is, up to a constant, simplyiii]{O---+ oo}.

The modular symbol

iii]

^{{r ---+} s} satisfies a fundamental discreteness property (see [11]).

PROPOSITION1

There exists a unique positive real numberri] (the period of

J)

such that the image of

iii]

inIRis ri]71.

We denote by

1

1'" "

<Pf{r ---+ s} = -

f . f3

E 7l ri] ,

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 the}

Shimura-Taniyama conjecture. A small technical benefit is that we do not need to consider plus- and minus-modular symbols as in the case over iQi discussed in [7, Proposition 1.2].

3. The p-adic construction

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

We work over a fixed imaginary quadratic fieldFofclass number 1. Let.IV

=

nAt be an ideal of@F,withn a prime of odd characteristic andn

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 ofR^X inPGL₂(@F[I/n]).

434 MAK TRIFKOVlé

A measure on jp'1(F_{rr )} is detennined by its values on compact open sets of the fonn U

=

^au{f)rr for au E PGL,(F_{rr ).} The matrixau is weil defined modulo the stabilizer 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 we need to invert 1f in the definition of R. The stabilizer of (f)rr in

f

is none other than

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'), the rational integer

does not depend on the choice of the matrix au such that au{f)rr

=

^U. An easy calculation starting with ]IUrr

= w]

shows that /.ijlr ---> s} is in fact a Z-valued measure on jp'1(F_{rr ),} which allows us to freely exponentiate by it in the following definition.

Definition 7

LetTI, r2 E ;YI'rr, r, s E jp'1(F).We define the mixed multiplicative integral associated toF by the fonnula

The limit is taken over unifonnly finer disjoint covers ofjp'1(<QJp) by sets of the fonn U

=

^au{f)rr'The poinltu is an arbitrarily chosen test point inU.

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

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

STARK-HEEGNER POINTS 435

CONJECTURE5

There exists a rank-one Z-lattice Q

c C;

commensurable with qZ with the following property: there exists a unique function that tor E ;tfrrand r,s E pl(Frr ) assigns an element

with the properties (1)

(2)

f

^Y

'l

^{y;Y' Wj =}

fl'

^{, Wj fory E}

^r;

(3)

f'l' j'l' 1'1'

^{r Wjj r Wj '" Tl r Wj (mod Q).}

3.2. Picard group tOl'sors

To fonnulate the Shimura reciprocity law for Stark-Heegner points, we need to con- struct geometric torsors for the Picard group of an order in a quadratic extension of F.Let Kj F be a quadratic extension that satisfies the Stark-Heegner hypothesis: n is inert in K,and aIl vl.ftare split. We denote by (!)rr, _{Frr ,}and Krr the completions atn of(!)F, F,andK, respectively. We fix an (!)[Ijn]-order (!) C Qwith conductor (ideal) c<; (!)F relatively prime to.ft.We mayas weIl also assume that c is prime to n.

By the Stark-Heegner hypothesis, there exists an ideal 9Jl' of (!)^K such that NKjF 9Jl'

=

^é!t. Since (C1f,.ft)

=

l, the ideal 9Jl

=

9Jl'(!)F[ljn] is equal to kero for a unique surjective homomorphism of (!) F-algebras

Such a homomorphism is called an orientation of (!); we fix one, along with the associated ideal9Jl.

436 MAK TRIFKOVlé

Definition 8

Anoptimal embeddingof 0 into R (an0-optimal embeddingfor short) is an algebra homomorphismW : K -> M2x2(F)snch that

An optimal embedding delines an orientation of0", : 0 -> 0

Fi

JI!by sendinga ^E 0 to the lower right-hand entry ofW(a).This is a ring homomorphism since, fora E 0, W(a)is an npper triangular matrix modulo 001.

Definition 9

We say that an optimal embeddingW is oriented if0",

=

^o.

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

Every ordered F-basis {WhWz} of K delines an embedding Ww ,."" : K ->

M zx2(F) which sends a E K to the matrix of the multiplication bya expressed in basis {W1,W2}. For an 0

Fl

1jn]-IatticeaC K,we deline the order ofaby

0.

=

(a E K ¹aa >;

aJ

>;

0K[~ l

We haveW;;i",,(R)

=

^0.

n

0." whereaanda'are 0_F[ljn]-lattices

An 0-optimal embedding WW "W2 is oriented if and only ifa'

=

^OO1a.Ifthis equality holds, then we say that the basis {WI,W2JisOO1-adjusted.Finally, setJ/t'~.9J1

=

{r E JI!""

iW",

is an oriented optimal embedding}.

PROPOSITION2 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) and show that they are inverses of each other.

To deline i, let abe a proper 0-ideal. Choose an OO1-adjusted 0_F[ljn]-basis {WI, W2} fora,and seti(a)tabe the class of the embeddingWW "W2 inÎ'\Emb(0, R).

Since bathaandOO1aare proper 0-ideals, we haveW;;'"JR)_,,-~

=

^0.

n

0 9J1n

=

^{0, so}

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