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.
Local root numbers (Deligne, Rohrlich). Let be an elliptic curve, and let be a quadratic character of . Write for the local root number of . Then: 1. if has good reduction at and is unramified, then and ; 2. at the archimedean place, , and for every quadratic , since the archimedean root number depends only on the Hodge structure, which the twist preserves; 3. if is unramified at (equivalently ) and has conductor exponent at , then where records the value of the unramified character on a uniformizer.
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.
Sign of the functional equation. Let have conductor , and let be imaginary quadratic of discriminant with satisfying the Heegner hypothesis for . Then , hence .
Proof. 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.
Gross–Zagier, rank-one form. Let be a modular elliptic curve of conductor with modular parametrization , and let be imaginary quadratic satisfying the Heegner hypothesis for . There is a canonically constructed Heegner point with where is an explicit constant built from the Petersson norm of the newform of , the discriminant , and the degree of .
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
Lemma (structure of $C_{\mathfrak a}$). The subgroup is cyclic of order , and it is stable under the CM action of .
Proof. 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.
Main theorem of complex multiplication. Let be the Hilbert class field of , and let be the Artin reciprocity isomorphism, . Then: 1. every elliptic curve with CM by has a model over , and generates over ; 2. the Galois conjugate is isomorphic to , compatibly with all torsion, meaning there is an isogeny inducing on torsion points after the class-field-theoretic identification of Frobenius.
From this, we deduce the field of definition and Galois action of the CM points on .
Shimura reciprocity for Heegner points. Each Heegner point is defined over the Hilbert class field , and for , In particular acts simply transitively on the set of Heegner points.
Proof. 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.
Quasi-quadraticity of the naive height. For a symmetric ample on an abelian variety, the naive height satisfies, uniformly in ,
Existence and quadraticity of the Néron–Tate height. The limit exists for every , differs from by a bounded function, and defines a quadratic form: the pairing is bilinear, and for all .
Proof. 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.
Northcott finiteness. For every real and integer , the set of algebraic numbers with and (absolute logarithmic Weil) height is finite. Consequently, for a fixed number field , the set is finite.
Proof. 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.
Torsion criterion. For , one has if and only if is torsion. Equivalently, is positive definite on .
Proof. 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 ).
Néron local heights. There is a unique family of local symbols , defined on pairs of degree-zero divisors on with disjoint support, that is: 1. biadditive and symmetric; 2. continuous in each variable for the -adic topology; 3. normalized by Weil reciprocity: whenever , where for .
Global height as a sum of local heights. For represented by degree-zero divisors with disjoint support, the Néron–Tate pairing is the sum over all places of , and the right side is independent of the chosen representatives.
Proof of well-definedness. 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.
Local intersection multiplicity as proximity. Suppose two sections meet at a closed point of the special fiber, and choose a local coordinate so that with and for some . Then, the intersection multiplicity is
Proof. 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.
Existence and uniqueness of the vertical correction. Let the special fiber of be with irreducible components and multiplicities . Let be the closure of a degree-zero divisor on the generic fiber. Then there is a vertical -divisor , unique up to adding a rational multiple of the whole fiber, such that for every . The corrected local symbol is independent of the model and satisfies the Néron axioms.
Proof. 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 .
Archimedean local symbol. For degree-zero divisors and with disjoint support, the archimedean local symbol is and this is symmetric in and independent of .
Proof. 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.
Serre–Tate deformation theory. Let be an elliptic curve over a perfect field of characteristic , and let be a complete local Artinian ring with residue field that field. The deformations of over are governed by the deformation of its -divisible group; in the ordinary case the deformation functor is prorepresentable by a formal torus (the Serre–Tate canonical coordinate ), and a homomorphism of the reductions lifts to level if and only if it is compatible with the canonical coordinates to that order. In the supersingular case the analogous statement holds with the endomorphism action of the quaternion order, and the lifting of a given to successive infinitesimal orders is obstructed by the quasi-canonical lifting filtration, so that lifts modulo precisely for up to an explicit level determined by the discriminant of the CM order relative to .
Local intersection as a homomorphism count. Let be CM elliptic curves with the same reduction modulo . Then the local intersection multiplicity of the corresponding Heegner sections, restricted to the degree- Hecke component, is
Proof. 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.
Deuring correspondence (cited input). Fix a prime and let be the definite quaternion algebra over ramified exactly at and , with a maximal order ; for level prime to use an Eichler order of level . Then: 1. the supersingular elliptic curves over with -level structure are in bijection with the right ideal classes of ; 2. for two such curves corresponding to ideal classes with connecting ideal , the module of homomorphisms is identified with , and the degree of a homomorphism corresponds to the reduced norm divided by ; 3. the CM by on a supersingular curve corresponds to an embedding , optimal in the sense of Section 4.5.
Eichler optimal embedding numbers (cited input, explicit form). Let be the Eichler symbol of at (equal to if splits in , if is inert, if ramifies). The mass of optimal embeddings of into an Eichler order of level in , meaning the sum over the ideal classes of the number of optimal embeddings up to unit conjugacy weighted by with , factors as a product of local embedding numbers: where and . The local factor at is , which is when is inert, when ramifies, and when splits; thus supersingular CM points, and hence finite local heights, occur exactly at primes nonsplit in . Under the Heegner hypothesis each factor at is .
The lifting length attached to each embedding is governed by the theory of quasi-canonical liftings, which we also cite in explicit form.
Quasi-canonical lifting length (Gross, cited input). Let be a supersingular curve over with , and let be an optimal embedding. For each there is a quasi-canonical lift of level , an elliptic curve in characteristic zero with CM by the order of conductor , defined over the ring class field of that conductor, with Two CM points whose relative position is measured by an endomorphism of conductor exactly have order of contact (intersection multiplicity) modulo equal to the level at which their quasi-canonical lifts first diverge, and the -adic valuation of the difference of their Serre–Tate parameters is exactly .
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.
Finite local height at a nonsplit prime, explicit form. For nonsplit in and , the finite local height contribution to at is where counts those whose associated CM order has conductor exactly , so that each such carries the quasi-canonical lifting weight . Equivalently, grouping by the conductor of the order generated by , where is the number of optimal embeddings of the conductor- order compatible with a degree- homomorphism, computed by the explicit Eichler numbers with replaced by .
Proof. 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.
Radial reduction of the Laplacian. For a radial function with ,
Proof. 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.
Legendre form of the free Green function. A radial solution of solves Legendre’s differential equation whose solution decaying as is the Legendre function of the second kind . Consequently is a free-space Green function for , holomorphic in for , .
Proof. 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
Diagonal singularity. As , equivalently , so has the logarithmic singularity required of a Green function, with .
Proof. 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
Convergence of the automorphic Green function. For and for all , the series converges absolutely and locally uniformly, defines a -invariant function of and of with a logarithmic singularity along the diagonal of , and satisfies away from the -orbit of .
Proof. 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