Heegner points, heights, and L-functions

Introduction

This post is a tour of the Gross–Zagier theorem, whose core assertion is that the Néron–Tate height of a Heegner point equals, up to an explicit nonzero constant, the first derivative of an -function at the center of its functional equation: Here, is a modular elliptic curve, an imaginary quadratic field satisfying a Heegner hypothesis, a Heegner point, and the Néron–Tate height. The left side is arithmetic and measures the size of a rational point; the right side is the leading Taylor coefficient of an -series at its center. Astonishingly, these two real numbers coincide, and the proof shows why: both decompose into local contributions, one per place, which agree place by place. The chain of ideas is Throughout, write for the Tate–Shafarevich group, for a global root number, for the quadratic Kronecker character of , and for the hyperbolic measure.

1. From BSD to Heegner points

1.1 The rank problem

Let be an elliptic curve. By the Mordell–Weil theorem, the group of rational points is finitely generated, and the integer is the Mordell–Weil rank. The Birch–Swinnerton-Dyer conjecture predicts analytically through the Hasse–Weil -function where and is the conductor. The Euler product converges for , and by modularity continues to an entire function with functional equation , center . The conjecture asserts the right side being the analytic rank. Gross–Zagier addresses analytic rank one by producing an explicit point and measuring it.

1.2 Passing to an imaginary quadratic field

It is technically advantageous to work over an imaginary quadratic field of discriminant . The base-changed curve has the -function where is the quadratic character cutting out and is the -function of the quadratic twist by . This factorization is the restriction to of the Galois representation of : .

The center is . When the global root number , the completed -function is odd under , so its central value vanishes and the first candidate invariant is the derivative. A natural hypothesis on results in this sign.

1.3 The Heegner hypothesis and the sign

An imaginary quadratic field of discriminant satisfies the Heegner hypothesis for if every prime splits in . When , this is equivalent to being a square modulo , and it guarantees an ideal with .

Indeed, each splits iff is a nonzero square mod (with the analogous condition at ); assembling these by the Chinese remainder theorem gives with , and pieced together over has cyclic quotient.

To determine the sign, we record the local root numbers, using the theory of epsilon factors.

Statement (3) is the local incarnation of the global twisting formula: it says that twisting by a character unramified at multiplies the local root number by the value of the character on the local conductor. Assembling the local values gives the global sign.

By the factorization of Section 1.2, , and since we have . We compute the ratio place by place using the local root numbers, noting that and that has conductor coprime to .

At a finite prime : both and have good reduction and is unramified, so by (1) the local factors are and contribute nothing to the ratio.

At the archimedean place: by (2), , so the ratio contributes .

At a prime (so , and has good reduction there while is ramified): the twist by a ramified quadratic character at a good prime multiplies the local root number by the local root number of at , and the product of these local character root numbers over , together with the archimedean sign of , assembles into the global root number of , which for the quadratic character of an imaginary quadratic field equals times a product of split local signs. We will track this below via the aggregate identity.

At a prime (so , unramified at ): by (3),

Multiplying the finite ratios and folding in the aggregated ramified and archimedean contributions of , one obtains the closed form of the twisting formula Now evaluate. Since , the character is odd, so . Since every prime splits in , the decomposition law gives for each such , hence . Therefore Consequently, the completed -function satisfies . Setting gives , so , and since the archimedean factor is nonzero at , also .

The vanishing of the central value is what makes the derivative the first informative quantity, and it is this derivative that Gross–Zagier equates to a height.

The immediate consequence is the equivalence whose rightmost link is the torsion criterion proved in Section 3. In what follows, we describe the construction of and prove the formula.

2. Modular curves, CM points, and the construction of Heegner points

2.1 The modular curve as a moduli space

Fix . The open modular curve is the coarse moduli space of pairs consisting of an elliptic curve and a cyclic subgroup of order , and two pairs are identified when related by an isomorphism of elliptic curves carrying one subgroup to the other. Adjoining finitely many cusps compactifies to the smooth projective curve . Over the complex numbers, there is the uniformization where and adjoins the cusps. The dictionary sends to the lattice , the curve , and the cyclic subgroup ; two points give isomorphic pairs iff with .

The curve has a model over whose noncuspidal points classify pairs . By modularity, of conductor receives a nonconstant -morphism normalized with , with for a nonzero rational , where is the newform of and . Albanese functoriality gives a surjection Rational points on arise by forming degree-zero divisors from algebraic points on and pushing through .

2.2 CM points

Let be imaginary quadratic with ring of integers . A fractional ideal is a rank-two -lattice in , so is an elliptic curve, and multiplication by preserves , giving ; this is an equality when has multiplier ring , so has CM by . The class determines up to isomorphism ( via multiplication by ), so there are curves with CM by .

2.3 Adding level structure: the Heegner points

To lift to a point of , we need a cyclic subgroup of order . Fix once and for all an ideal with , which exists by the Heegner hypothesis. For each fractional ideal , set

Since is an inclusion of rank-two lattices, the quotient is finite. Multiplication by is an -linear isomorphism so we analyze . The multiplication map is well-defined because for , and it is -bilinear. It is a perfect pairing: for fixed nonzero there is with because , and symmetrically, so neither side has a nonzero radical. A perfect pairing of finite abelian groups identifies each factor with the Pontryagin dual of the other, and the Pontryagin dual of the cyclic group is cyclic of order . Hence , and therefore , is cyclic of order .

Stability under is immediate: for , multiplication by carries into itself and into itself, so it preserves the quotient .

The pair defines a point the Heegner point of discriminant attached to and . In the upper-half-plane model, a basis of adapted to represents by satisfying with and , the condition giving -compatibility. These are the classical Heegner points.

2.4 Fields of definition and the Galois action

The arithmetic of the is governed by complex multiplication, whose relevant statement we cite.

From this, we deduce the field of definition and Galois action of the CM points on .

The point records the pair . By part (1) of the cited input, is defined over . The subgroup is cut out by the ideal , which is a fixed rational structure independent of the ideal class; applying part (2), the isogeny realizing carries -torsion to -torsion, hence carries to . Therefore, which is the moduli-theoretic statement . Because is indexed by and multiplication by acts simply transitively on the class group, the induced action on Heegner points is simply transitive. In particular each has exactly Galois conjugates, all Heegner points, so its field of definition is the fixed field of the trivial subgroup, namely itself when is nontrivial, and when .

2.5 The Heegner divisor and the Heegner point

Form the divisor on , of degree . By Shimura reciprocity the Galois group permutes the summands transitively, so is fixed by and therefore defined over . The cusp is defined over , so subtracting it produces a degree-zero divisor class over , Pushing through the modular parametrization gives the Heegner point on , using . Equivalently, working with a single class for the trivial ideal class and setting , one has since is Galois-equivariant and the orbit of is . The rest of the post computes .

3. Heights as arithmetic intersection numbers

3.1 The canonical height exists and is quadratic

Let be a number field and an abelian variety with symmetric ample ; for take , so the naive height is . We cite the following consequence of the theorem of the cube.

By the cited estimate with , there is a constant with for all . Consider the sequence . For , Hence, for , so is Cauchy and the limit exists. Taking and gives , so is bounded.

The canonical height inherits a parallelogram law. Applying the cited relation to and , dividing by , and letting , the term is killed by the factor , leaving Also, directly from the definition, and because is symmetric. A real-valued function on an abelian group satisfying the parallelogram law above and is a quadratic form: define by the displayed polarization; then is equivalent to biadditivity of in each variable. Symmetry is inbuilt, and . Biadditivity gives for all integers .

3.2 Positivity and the torsion criterion

Positivity of the height, and the certification of infinite order, rest on Northcott finiteness.

Let be algebraic of degree with conjugates and minimal polynomial , where is the -th elementary symmetric function of the conjugates. The Mahler measure dominates each symmetric function: by the triangle inequality applied to the coefficients, . Thus, each coefficient of is a rational integer (when is an algebraic integer; in general one clears a bounded denominator similarly bounded by the height) of absolute value at most a fixed bound depending only on . There are finitely many integer polynomials of degree with coefficients so bounded, hence finitely many possible , hence finitely many .

For the elliptic curve statement, each has of degree and height ; by the first part there are finitely many possible -coordinates, and each corresponds to at most two points . Hence, the set is finite.

If is torsion, for some , so and .

Conversely, suppose . Then for all . Since is bounded, for all , so the infinite set lies in the finite set of Northcott’s theorem. Therefore two distinct multiples coincide, with , giving , so is torsion. Positive-definiteness on follows because is a quadratic form vanishing exactly on the torsion subgroup, which becomes zero in .

3.3 Local decomposition: Néron’s local heights

The global height pairing on the Jacobian decomposes into local terms. We state Néron’s theorem, take its existence for granted, and prove the well-definedness we need through Weil reciprocity and the product formula.

For a smooth projective geometrically connected curve with Jacobian , and for each place of with normalized absolute value , write for the completion and for the local normalization ( the residue cardinality at finite , and the usual archimedean normalization at ).

We prove independence of the choice of representatives, which is the crux; identifying the resulting biadditive symmetric function with the Néron–Tate pairing is then the cited characterization of the latter as the unique quadratic form inducing the canonical class.

Suppose and both represent , so for some rational function . Fix with support disjoint from both. Then by biadditivity and the Weil reciprocity property, Now has degree zero, so is a well-defined element of (the degree-zero condition removes the ambiguity of scaling ). By the product formula for the number field , Therefore, the total sum is unchanged when is replaced by , and by symmetry also when is varied. Hence, depends only on the classes , is biadditive and symmetric, and is continuous. Therefore, it agrees with the Néron–Tate pairing after normalizing the sign so that the archimedean contributions make the induced quadratic form positive definite, which is the cited final identification.

3.4 Finite local heights through intersection theory

At a finite place the local symbol is an intersection number on a regular model. Let be finite with valuation ring , uniformizer , residue field of cardinality , and valuation , and choose a regular proper flat model of . A point extends by properness to a section , a horizontal prime divisor meeting the special fiber in one closed point.

The ideals are and , so . The quotient is whose length as an -module is . Geometrically, the two sections agree to -adic order exactly , so intersection multiplicity measures the order of -adic contact.

For general degree-zero divisors, one extends by biadditivity and repairs model-independence by a vertical correction, a morsel of linear algebra we carry out.

Let be the intersection matrix of the special fiber. Two structural facts about come from the geometry of a proper regular arithmetic surface: first, the fiber class satisfies for all , because is a fiber of a proper flat map and is algebraically equivalent to a multiple of any other fiber, so it meets each vertical curve in degree zero; hence where , and is singular with in its kernel. Second, by the theorem of Zariski (the arithmetic Hodge index theorem for the special fiber), is negative semidefinite with kernel exactly the line .

We seek with , where , . A vector is in the image of the symmetric matrix if and only if it is orthogonal to . Compute using that equals the degree of on the generic fiber (a horizontal divisor meets a fiber in its generic degree) and . Thus , so and a solution exists; it is unique modulo , that is, modulo adding a multiple of the fiber.

Adding a multiple of the fiber to changes by , so the corrected symbol is independent of that ambiguity, hence model-independent after the standard comparison of models by blow-ups. Biadditivity and symmetry are inherited from the intersection product, and the Weil reciprocity normalization is checked by taking and using that principal divisors have trivial corrected self-intersection with horizontal classes, which recovers . Thus the corrected symbol satisfies the Néron axioms and agrees with the local height.

3.5 Archimedean local heights through Green functions

At an archimedean place the local symbol is computed by a Green function on the Riemann surface . Fix a smooth positive -form with .

A Green function for a point is a smooth function on with the properties so that near in a local coordinate one has .

Set . Because , the defining equation gives , so is a genuine potential for with no -dependence, and the normalization integral drops out. Evaluating at gives . Symmetry is Green’s reciprocity: for degree-zero with disjoint support, which follows from Stokes’ theorem applied to , whose integral vanishes because the two currents and are integrated against the complementary potentials symmetrically. Independence of is the same degree-zero cancellation. This is the local height at the archimedean place.

On modular curves, this Green function is the automorphic one, built from the hyperbolic resolvent. Its Legendre-function form is derived in Section 5.

4. Local heights at finite places: the quaternionic computation

Gross and Zagier compute the whole family , matched coefficient by coefficient against an analytic kernel. We reduce each finite local height to a count of quaternionic data, down to the deformation-theoretic and quaternionic inputs of the status note.

4.1 Hecke operators as correspondences

The Hecke operator acts on divisors of by summing over cyclic -isogenies. On moduli, the sum over the finitely many cyclic -isogenies from whose kernel meets trivially; equivalently is the correspondence with its two projections to . For a Heegner point, when has norm the isogeny realizes such an isogeny, so Hecke translates of CM points are again CM points, which is the source of the arithmetic below.

4.2 Reduction of CM points and the collision condition

Fix a rational prime and a place of above , and extend to a regular model over the ring of integers, so Heegner points reduce to the special fiber over . Two components and contribute to the finite local height at exactly when their reductions coincide, the contribution being the intersection multiplicity of Section 3.4.

The reduction of a CM elliptic curve modulo depends on the splitting of in :

  • if splits in , the reduction is ordinary, with an order in containing ; distinct Heegner points then reduce to distinct ordinary points unless related by a genuine -power isogeny, and the collisions are controlled by the ordinary deformation theory;
  • if is inert or ramified in , the reduction can be supersingular, and is a maximal order in the definite quaternion algebra ramified exactly at and .

The supersingular case carries the essential arithmetic: there the endomorphism ring is noncommutative, and embeds into it in finitely many ways, each recording a way for CM points to collide.

4.3 Intersection multiplicity as a lifting length

The intersection multiplicity of two sections agreeing modulo is the length of their largest common infinitesimal deformation, computed by the following input.

By Section 3.4 the intersection multiplicity at equals of the parameter measuring how far the two sections stay equal, which by definition is the largest such that the two deformations agree modulo . A degree- isogeny between the generic CM curves reduces to a homomorphism of degree between the reductions; conversely, by Serre–Tate, such a lifts to an isogeny modulo exactly when it is compatible with the deformation coordinates to order . Counting, for each order , the homomorphisms that persist to that order and summing over therefore reproduces the total order of contact, since a homomorphism contributing contact exactly to order is counted once for each , giving total weight , which is the intersection multiplicity it produces. The factor removes the double count from the involution , which give the same unoriented contact. The level-structure compatibility restricts to those carrying to , matching the Hecke condition.

4.4 Passing to the quaternion algebra: the Deuring correspondence

At a supersingular prime the homomorphism modules live in a quaternion algebra, computed by the Deuring correspondence and Eichler’s embedding theory.

The lifting length attached to each embedding is governed by the theory of quasi-canonical liftings, which we also cite in explicit form.

4.5 Optimal embeddings and the closed formula

Let be an order in a quaternion algebra over and let be an order in an imaginary quadratic field . An embedding with is an optimal embedding of into ; that is, is exactly the elements of that land in .

To turn the count into an explicit arithmetic function, fix the two optimal embeddings coming from the two colliding CM points, and let be the connecting ideal, so that with . The homomorphisms compatible with the CM structure are those in the orthogonal complement of under the reduced-trace pairing; on that rank-two complement (a rank-two lattice, since has rank four and the CM condition imposes two linear conditions) the reduced norm restricts to a positive-definite binary quadratic form of discriminant up to units. Write for its representation numbers.

By Section 4.3, the local multiplicity of two colliding CM points is , up to the factor absorbed into the embedding-number normalization. By the Deuring correspondence a degree- homomorphism is an element with , and compatibility with the two CM structures places in the complement on which the reduced norm is the binary form ; so the homomorphisms of degree are counted by . By the quasi-canonical lifting input, a given lifts modulo precisely when its associated CM order has conductor divisible by ; the largest such , the order of contact this produces, is the -adic conductor exponent of . Therefore contributes weight equal to that exponent, and summing the indicator “lifts mod ” over recovers the exponent as the total weight. Writing for the number of degree- compatible whose conductor exponent is at least , the sum of weights is , which telescopes as displayed. Each is itself an Eichler optimal embedding number for the order of discriminant , hence given explicitly by the product formula with the symbol ; this makes every term computable. Finally each supersingular closed point has residue cardinality a power of , so , and the aggregate is the stated multiple of .

4.6 The finite side as a generating series

Assembling over , the finite local heights are the Fourier coefficients of a generating series By the explicit local height formula, the coefficient at is summed over nonsplit , where counts the degree- homomorphisms intertwining the two CM structures whose associated order has conductor at least . These intertwiners form a rank-two lattice (the space intertwining two embeddings of into the quaternion algebra is a rank-two -module, since the centralizer of a maximal subfield is the field itself), and the reduced norm restricts to a positive-definite binary quadratic form on , so is a weight-one theta series attached to the imaginary quadratic order, in fact a linear combination of the very theta series of Section 6.1 for the conductor- order. The finite generating series is therefore the -linear combination of the nonconstant parts of these weight-one theta series, and the sum over lifting levels is exactly the coefficientwise -derivative that arises when the weight-one Eisenstein series is differentiated at the center, since of the Euler factor at contributes the grading by with weight . On the analytic side the nonarchimedean Fourier coefficient of at index is the -derivative of a product of local factors , each a local Whittaker function whose value is the local density of representations of by the reduced-norm form; the logarithmic derivative produces the same weight and the same grading by as . Matching the two coefficientwise is thus an identity of local representation densities against local Whittaker derivatives, and it is the nonarchimedean half of the comparison. The archimedean half is the content of the next section.

5. Local heights at the archimedean place: the hyperbolic Green function

The archimedean local height of Heegner divisors is a sum of values of the Green function on , which we derive as the resolvent kernel of the hyperbolic Laplacian, identify with a Legendre function, and match to the analytic kernel.

5.1 The hyperbolic Laplacian on radial functions

On with coordinates , the hyperbolic Laplacian is invariant under . The geodesic distance between two points is encoded by the point-pair invariant through . Write . On a radial function the Laplacian acts as an ordinary differential operator.

By -invariance of both and , it suffices to compute at a convenient configuration and with fixed. Place and use geodesic polar coordinates centered at , in which the hyperbolic metric is and the Laplacian is A radial function is independent of , so Now change variables to , with and . Then and Also . Adding,

5.2 The free-space Green function and the Legendre equation

We want the resolvent kernel solving off the diagonal, with a log singularity on it and decay at infinity. By the radial reduction, this is an ODE.

Substituting the radial reduction into gives that is, . Writing so that turns this into the standard Legendre equation of degree (the sign of the zeroth-order term matches after ). Legendre’s equation has a two-dimensional solution space spanned by and ; the first grows and the second decays as , with . The integral representation converges for and , is holomorphic in , and one checks by differentiating under the integral that it solves Legendre’s equation of degree ; it is the decaying solution. Hence solves off the diagonal. The normalizing factor is fixed by the singularity computed next.

5.3 The logarithmic singularity

The Legendre function of the second kind has the classical logarithmic behavior near , where is the digamma function and Euler’s constant; this follows from the hypergeometric expansion with analytic at and . Now , so Thus as claimed. Since , the current has the delta along the diagonal with the correct normalization, plus a smooth remainder coming from the term and the factor.

5.4 Automorphizing over

By Section 5.2, as , so . The number of with grows like (the hyperbolic lattice-point count, since is a lattice and balls have exponentially growing volume ). Therefore so the series converges absolutely and locally uniformly. Invariance in follows by reindexing, and the differential equation and single log singularity (from ) follow termwise. The physical Green function of Section 3.5 is recovered by removing the constant-eigenfunction contribution, which produces a simple pole at , and taking the finite part.

5.5 Evaluation at Heegner points and the archimedean matching

The archimedean contribution to at a place of over is, by Section 3.5 and the automorphic form of the Green function, a sum of Legendre functions at the point-pair invariants of CM points related by degree- maps. The analytic side produces the same object.

Compute the -Fourier expansion of the free Green function and compare with the Bessel coefficient of Section 6.3. Fixing and , so , the -th Fourier coefficient of in is This has a classical closed form in modified Bessel functions: for , with , , the product of the two independent solutions (regular at ) and (decaying at ) of the modified Bessel equation of order . This is the standard evaluation of the resolvent kernel of the hyperbolic Laplacian by separation of variables: for fixed , both and its coefficient satisfy, in , the ordinary differential equation obtained from by inserting , namely whose two solutions are and ; the Green-function boundary conditions (regularity as from below, decay as ) select the product , and the Wronskian fixes the constant.

Now compare with the Eisenstein coefficient from Section 6.3: the -th non-constant Fourier coefficient of is proportional to times the divisor-sum . The archimedean part of the analytic local term at index is therefore the value at the CM point of exactly the same Whittaker factor that appears in . Differentiating in at the center commutes with all of this and produces on both sides. Hence, term by term in the Fourier index , and summing over gives the equality of the archimedean local height with the archimedean analytic local term. The Legendre and Bessel forms are the same kernel written in the distance and Fourier variables.

Thus, for every and every place , the local height equals a specific local coefficient of an analytic kernel. It remains to build that kernel, organize the local identities into an identity of modular forms, and extract the newform.

6. The analytic side: Rankin–Selberg integrals and the Eisenstein kernel

This section builds the object whose central derivative is : the unfolding identity, the Eisenstein Fourier expansion, the -function, and the derivative at the center.

6.1 The newform and the theta series

Let be the normalized newform of . For an ideal class define the weight-one theta series where . Summing over classes gives , the number of integral ideals of norm , with Dirichlet series . The Rankin–Selberg convolution of against these theta series recovers , which the unfolding below makes precise.

6.2 The unfolding identity

For the Eisenstein series converges absolutely and decays rapidly, so we may interchange summation and integration: In the -th term substitute . Since by -invariance, is -invariant, and , the term becomes As ranges over a set of representatives for , the translated fundamental domains tile a fundamental domain for , without overlap and covering everything, because the coset space reindexes exactly the -translates. Hence the sum of integrals is a single integral over : A fundamental domain for is the vertical strip , since is generated by . Writing and integrating over extracts the zeroth Fourier coefficient:

We assemble the -function by recording the equal-weight Rankin–Selberg identity and then specializing; the mixed-weight case of (weight ) against (weight ) folds the extra weight into the Eisenstein series.

The function is -invariant, because under the automorphy factors from , from , and from the power of the imaginary part cancel; and decays rapidly at every cusp because is cuspidal. Its zeroth Fourier coefficient at is since the -integral is . By the unfolding lemma of Section 6.2, The inner integral is a Gamma integral: substituting , Collecting the -dependence gives the stated Dirichlet series times .

For the Gross–Zagier kernel the pairing is against with a weight-one Eisenstein series; the same unfolding replaces by , giving for one ideal class using . Summing over replaces by , the number of integral ideals of norm : The Dirichlet series factors as an Euler product because is multiplicative (Hecke) and is multiplicative with ; the convolution is the base-changed -function divided by a zeta factor from the diagonal overcount, and the denominator is the normalizing factor of , so clearing it leaves a completed -function. With the archimedean Gamma factor, the completed function is the two Gamma factors coming from the two copies at the complex place of , and it satisfies the functional equation by Section 1.3, so is the center and .

6.3 Fourier expansion of the Eisenstein series

The Fourier expansion of comes by Poisson summation. We do the level-one computation in full; the level- case differs only by finitely many Euler factors and the choice of cusp.

Coset representatives for are indexed by the bottom rows with up to sign, and . Hence Separate the terms with (forcing ), which contribute , from the terms with . For group by (using the symmetry) and remove the gcd condition by Möbius, writing every pair with as with ; this yields the factor after summing over the common divisor. Concretely, where the inner sum is now over all . Fix and apply Poisson summation to , : Substituting and then , The last integral is a classical Bessel integral: for , and for , Summing over and : the terms give which is the second constant-term piece. The terms, after collecting the sum over and with fixed (the sum over divisors producing ), give Assembling the three pieces yields the stated expansion.

For and the weight required by Gross–Zagier (weight one, with the theta series absorbing the second weight, and a possible nebentypus), the same Poisson computation applies cusp by cusp; the constant term acquires the level- scattering coefficients and the non-constant coefficients acquire the local factors at , but the archimedean Bessel and the shape of the expansion are unchanged. The key structural output survives: the non-constant Fourier coefficients are products of a divisor-sum (the finite/nonarchimedean local data) and a Whittaker function (the archimedean local data).

6.4 Differentiating at the center

Since the root number is , the completed -function vanishes at , so the derivative is the informative quantity. Differentiating the integral representation moves onto the Eisenstein series: where . The differentiated Eisenstein series is the analytic kernel. Its Fourier coefficients are the -derivatives of the coefficients in Section 6.3: Two features appear. The factor , and the logarithmic derivatives in the divisor-sums, are the analytic counterpart of the intersection multiplicities and lifting lengths of Section 4: differentiating an Euler factor produces the matching the arithmetic multiplicity. The factor is the archimedean Whittaker derivative, equal by Section 5.5 to the Green function . Hence the Fourier coefficients of are the total local analytic terms, matching the local heights of Sections 4 and 5.

7. Holomorphic projection and modularity of the height series

The height generating series is holomorphic, but the analytic kernel is real-analytic. Holomorphic projection, which we prove through Poincaré series, bridges the two.

7.1 Poincaré series and their Petersson pairings

For weight , level , index , and large, define the Poincaré series where . It converges for , transforms with weight , and its Petersson pairing against any weight- automorphic function unfolds.

Insert the definition of , conjugate, and unfold exactly as in Section 6.2. The pairing becomes an integral over , using and combining with the weight factor to leave the invariant measure. On the strip write ; integrating over against extracts the term . What remains is as claimed.

7.2 The holomorphic projection lemma

First take holomorphic, so . By Section 7.1, Setting gives , so the holomorphic Poincaré series represents the -th Fourier coefficient functional up to the constant . Since Fourier coefficients determine cusp forms, the span .

Now for general , Section 7.1 at gives Define with as displayed; then by the holomorphic computation above, where the normalizing constant is chosen exactly so that the two pairings against every agree; since the span , this forces for all . Uniqueness of inside follows because a cusp form orthogonal to all of is zero. For the integral sits at the edge of convergence when grows; one restores convergence by inserting , computing for , and continuing to , which is legitimate because the divergent boundary contribution is a constant (the non-cuspidal part) orthogonal to and hence invisible to the pairing. The continued value defines and .

7.3 Modularity of the height generating series

Applying holomorphic projection to the weight-two analytic kernel , whose -th Fourier coefficient is the total (archimedean and nonarchimedean) analytic local term at index before projection, produces a holomorphic weight-two cusp form the case of the holomorphic projection formula, whose -th Fourier coefficient is the holomorphically projected total analytic local term for index . On the geometric side define the height generating series

By Sections 4 and 5 the local heights match the local analytic terms place by place: for each , The holomorphic projection was needed precisely so that the archimedean and non-holomorphic contributions on the analytic side are replaced by the coefficient of a genuine weight-two form, and the matching identities of Sections 4 and 5 are exactly the statement that after this projection the coefficient equals the local height sum. Since and have identical Fourier coefficients for all and , the series is that same cusp form. In particular is modular of weight two; this is the analytic incarnation of the geometric fact, established independently and in sharper form by Gross, Kohnen, and Zagier, that Heegner divisor classes vary modularly.

8. The comparison and extraction of the newform

Both and lie in and are equal. Pairing with isolates the height on one side and on the other, using Hecke self-adjointness (proved next) and multiplicity one (cited).

8.1 Hecke operators are self-adjoint for the height pairing

The Néron–Tate pairing on is for the symmetric ample class giving the canonical principal polarization . For an endomorphism , the adjoint with respect to this pairing is the Rosati involution , characterized by ; this is a formal consequence of the functoriality of the height pairing under isogenies and the definition of as the dual morphism.

It remains to show for . Realize as the correspondence on given by the curve parametrizing cyclic -isogenies , with the two projections to . The transpose correspondence swaps and , that is, it parametrizes the dual isogenies. For the dual of a cyclic -isogeny is again a cyclic -isogeny, and the involution sending an isogeny to its dual is implemented on by an automorphism compatible with the level- structure, so as correspondences. Under the Albanese functoriality that defines the action on , the transpose of a correspondence induces the Rosati adjoint of the induced endomorphism, since the canonical polarization is defined by the theta divisor and correspondences act compatibly with it. Therefore , giving self-adjointness.

8.2 Extracting the height and the derivative

On the geometric side, decompose into Hecke eigenforms; spans a one-dimensional eigenline. Let be the -isotypic projection of , with . Self-adjointness of (Section 8.1) gives, for every , where the superscript denotes the component of the sequence landing in the -eigensystem, extracted because eigenforms with distinct systems are height-orthogonal. Hence the -isotypic part of the generating series is a scalar multiple of , and by Petersson orthogonality It remains to relate to . Since and the trace has the same -isotypic projection as up to the Galois averaging already built into , the projection formula for the pushforward and its dual gives The composite acts on the -isotypic line of as multiplication by , because is the optimal quotient onto up to the Manin constant and on . Therefore the factor from the relation between the full Heegner divisor and the single class under the trace. Combining,

On the analytic side, since is the holomorphic projection of the differentiated kernel and holomorphic projection preserves Petersson pairings against cusp forms, At the center the completed -function vanishes, so the -derivative lands entirely on , giving a nonzero multiple of with the explicit archimedean constant times the value of the zeta normalization at the center.

Since by Section 7.3, their Petersson pairings against coincide, . Substituting the two explicit evaluations from Section 8.2, where is the power of two from the trace normalization and is the nonzero archimedean-and-zeta constant obtained by differentiating the completed factor at the center (using , so only the derivative of survives). Multiplicity one for (cited) makes the -isotypic line one-dimensional, so this is a genuine scalar identity rather than a proportionality up to unknown vectors. Solving, Every factor is positive: , , , and is the product of the positive archimedean Gamma factor and the positive value of the zeta normalization at the center. Hence , giving the equivalence of infinite order. Matching and against the explicit constants of Sections 6 and 8 yields the closed form recorded in Section 9.

9. The precise formula and its constant

For definiteness we record the identity with its constant assembled. Let be the newform of with Petersson norm let , let , and let be the parametrization with . The Gross–Zagier identity in its Petersson-normalized form is so that The Manin constant and exact power of two depend on the normalization of ; the structural content is the strict proportionality with a positive constant, which feeds the regulator into the rank-one Birch–Swinnerton-Dyer formula.

10. Kolyvagin’s Euler system and the rank-one BSD consequence

Gross–Zagier gives one inequality: forces . Kolyvagin’s Euler system of Heegner points pins the rank at one and bounds , citing only the two structural inputs of the status note.

10.1 Heegner points over ring class fields

For a squarefree product of Kolyvagin primes (rational primes inert in with and for the working modulus ), CM by the order gives Heegner points over the ring class field of conductor , with , exactly as in Section 2. Two relations hold.

Both relations come from the Eichler–Shimura congruence and the moduli interpretation of Hecke operators. The trace sums the Galois conjugates of over , which by Shimura reciprocity for the order are the Heegner points of conductor lying above ; these are exactly the -isogenous CM points counted by , so their sum is applied to , which equals since lies in the -isotypic part where acts by . Relation (2) is the reduction of this identity modulo a prime above : by Eichler–Shimura, , and evaluating the CM points modulo collapses the conductor- point to the Frobenius image of the conductor- point, giving the stated congruence.

10.2 Derived classes

Kolyvagin’s derivative operators turn this norm-compatible family into cohomology classes. For each set The operator satisfies the telescoping identity , where .

We check -invariance of modulo prime by prime. For , By norm compatibility, (viewed in after the natural inclusion), while is the other term. Since is a Kolyvagin prime, and , so both terms vanish modulo . Hence for every , and since these generate , the class is -invariant modulo .

Now use the inflation-restriction sequence The invariant point gives a class in , which maps to a -invariant element of via the Kummer map; invariance and the vanishing of the obstruction group (which holds because has no -invariants under the working hypotheses, a consequence of the irreducibility of the mod- representation) lift it uniquely to . The local conditions follow from the Euler system relations: at primes not dividing the class is unramified because is, and at the congruence relation (2) computes the ramified part in terms of the Frobenius image, giving the controlled local behavior.

10.3 Bounding the Selmer group

The classes satisfy the Euler system local conditions: is unramified away from , and at each its ramified component is nonzero and computes the localization of at via relation (2).

Suppose for contradiction that the -Selmer group has rank exceeding one. Choose a nonzero class not proportional to the class of . By the global duality reciprocity law, for any global class the sum of local invariants of the cup product vanishes: We choose the Kolyvagin prime by Chebotarev so that the localizations and are simultaneously controlled: since the mod- representation on has large image, there is a positive-density set of primes (inert in , Kolyvagin) at which the Frobenius conjugacy class realizes any prescribed pair of local behaviors for and for the Heegner class. For such , all local invariants in the reciprocity sum vanish except the one at , which pairs the ramified part of (nonzero, by the infinite order of feeding relation (2)) against the localization . Vanishing of the total sum forces . Ranging over enough such forces to be locally trivial everywhere, hence globally trivial by the injectivity of the localization map on the relevant Selmer quotient, contradicting .

Therefore has rank at most one for all , so its projective limit, the -adic Selmer group, has corank at most one. The exact sequence then gives ; combined with the infinite order of this forces , and it forces to be finite because the corank-one Selmer group is exhausted by the free part. Running the argument at all gives finiteness of .

10.4 Descent to

Choose auxiliary imaginary quadratic satisfying the Heegner hypothesis for with (such exist by nonvanishing theorems for quadratic twists). Then , so ; by Gross–Zagier has infinite order, and by Kolyvagin with finite. Decomposing into -eigenspaces, the plus part is and the minus part ; since , the twist has rank zero, placing all of the rank in the plus part. Hence and is finite.

11. Comparing both sides

Both sides are assembled from the same input, the CM data of on the modular curve, and matched place by place: at finite primes, intersection multiplicities of CM points are counts of quaternionic embeddings matched to derivatives of local Whittaker functions counting the same reduced-norm solutions; at the archimedean place, both the Green function and the differentiated Eisenstein coefficient are at the same point-pair invariants. We summarize this as follows:

Geometric side Analytic side
CM points on theta series attached to
Heegner divisors Fourier coefficients indexed by norms
Néron–Tate height central derivative of
finite local intersections derivatives of local Whittaker functions
optimal embeddings into quaternion orders local representation densities
lifting lengths (Serre–Tate) logarithmic derivatives of Euler factors
archimedean Green functions differentiated Eisenstein coefficients

The theorem thus equates the arithmetic height of CM cycles with the central derivative of the automorphic object they generate, place by place.

12. Generalizations

Gross–Zagier is the first instance of a principle relating heights of algebraic cycles to derivatives of -functions. The Gross–Kohnen–Zagier theorem shows that Heegner divisor classes of varying discriminant in are the Fourier coefficients of a weight- modular form, so Heegner points vary modularly. The Kudla program generalizes the geometric side to special cycles on orthogonal and unitary Shimura varieties, predicting that their generating series are modular and their arithmetic intersection pairings compute derivatives of Eisenstein series, with Gross–Zagier the one-dimensional case. The arithmetic Gan–Gross–Prasad program relates heights of diagonal cycles on products of Shimura varieties to central derivatives of Rankin–Selberg -functions; the work of Yuan, Zhang, and Zhang gives a general Gross–Zagier formula for Shimura curves over totally real fields, reorganized through automorphic representations; and Waldspurger’s formula, expressing central -values through toric periods, sits at the analytic-rank-zero end. The unifying expectation is that central derivatives of -functions are heights of special cycles.

Denouement

Let us summarize this spectacular proof in words. The Heegner hypothesis makes and . The ideal classes of give CM elliptic curves, which the auxiliary ideal equips with cyclic -structure to produce Heegner points on , defined over and permuted by the class group. Summing them and pushing through gives . Its Néron–Tate height decomposes into local terms: intersection numbers on regular models, computed by the deformation theory of CM points and the quaternionic counting at nonsplit primes, and archimedean Green-function values, computed through the Legendre resolvent of the hyperbolic Laplacian. The same CM data builds the theta series whose Rankin–Selberg convolution with is ; unfolding, differentiation at the center, and holomorphic projection turn its central derivative into a weight-two cusp form. Place by place, the local heights match the local analytic terms, so the two weight-two forms agree, and pairing with through Hecke self-adjointness and multiplicity one yields with . Combined with Kolyvagin’s Euler system, this gives the Birch–Swinnerton-Dyer conjecture in analytic rank at most one, and it is the blueprint for the principle that central derivatives of -functions are heights of special cycles.

References

  • Benedict H. Gross and Don B. Zagier, Heegner points and derivatives of -series, Inventiones Mathematicae 84 (1986), 225–320.
  • Benedict H. Gross, Heegner points on , in Modular Forms (Durham, 1983), Ellis Horwood, 1984.
  • Benedict H. Gross, Winfried Kohnen, and Don B. Zagier, Heegner points and derivatives of -series II, Mathematische Annalen 278 (1987), 497–562.
  • Victor A. Kolyvagin, Euler systems, in The Grothendieck Festschrift II, Progress in Mathematics 87, Birkhäuser, 1990.
  • Henri Darmon, Rational Points on Modular Elliptic Curves, CBMS 101, American Mathematical Society, 2004.
  • Joseph H. Silverman, The Arithmetic of Elliptic Curves, GTM 106, Springer.
  • Serge Lang, Elliptic Functions, GTM 112, Springer.
  • David A. Cox, Primes of the Form , Wiley.
  • Daniel Bump, Automorphic Forms and Representations, Cambridge University Press.
  • Henryk Iwaniec, Spectral Methods of Automorphic Forms, GSM 53, American Mathematical Society.
  • Serre–Tate, deformation theory of abelian varieties, for the local deformation input.
  • Xinyi Yuan, Shou-Wu Zhang, and Wei Zhang, The Gross–Zagier Formula on Shimura Curves, Annals of Mathematics Studies 184, Princeton University Press, 2013.
  • Shou-Wu Zhang, Gross–Zagier formula for , Asian Journal of Mathematics 5 (2001), 183–290.

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