Elliptic curves and p-adic uniformisation

H. Darmon^{∗}

May 24, 2000

Elliptic curves. An elliptic curve is a curve of genus one with a distinguished rational point. It can be described by a homogeneous equation of the form

E : Y^{2}Z = 4X^{3} + aXZ^{2} + bZ^{3},  (1)

where the parameters a, b ∈ Z satisfy the condition

Δ := −2^{12}(a^{3} + 27b^{2}) ≠ 0.

The Diophantine theory studies the rational solutions (X, Y, Z) ∈ Q^{3} of equation (1). It is convenient to ignore the trivial solution (0, 0, 0) and to identify solutions if they differ by multiplication by a non-zero scalar. Solutions to (1) are thus viewed as points in the projective plane P_{2}(Q). Let E(Q) ⊂ P_{2}(Q) denote this solution set. More generally, if F is any field, let E(F) ⊂ P_{2}(F) be the corresponding set of solutions with values in F. It is identified with the set of (x, y) ∈ F^{2} satisfying the associated affine equation

y^{2} = 4x^{3} + ax + b,  (2)

together with the “point at infinity” corresponding to (X, Y, Z) = (0, 1, 0).

Among all the projective^{1} curves over Q, the elliptic ones are worthy of special consideration, because they alone are algebraic groups: the set E(Q) is equipped with a binary composition law

∗This is a transcription of the author’s Coxeter-James lecture given at the CMS Winter meeting in Kingston in December 1998. It is a pleasure to thank Massimo Bertolini and Adrian Iovita for many fruitful exchanges over the years, and the Canadian Mathematical Society for its invitation to deliver the Coxeter-James lecture.

1I.e., defined by a system of homogeneous equations.

E(Q) × E(Q) ⟶ E(Q)

defined by a system of polynomials with rational coefficients, making E(Q) into a commutative group, with identity element the distinguished point at infinity. The same set of polynomials endows E(F) with a natural addition law^{2}, admitting a simple geometric description in terms of the chord and tangent method: viewing points in E(F) as points on the affine plane by equation (2), one simply sets P + Q + R = 0 whenever P, Q, and R lie on the same line. (See for example [ST], ch. I.) The Diophantine study of E is facilitated and enriched by the presence of this extra structure.

The group E(C) is isomorphic to the quotient of C by a lattice Λ. For a suitable Λ, the inverse isomorphism sends z ∈ C to (℘(z), ℘′(z)) ∈ E(C), where

℘(z) = \frac{1}{z^{2}} + ∑_{λ∈Λ−0} (\frac{1}{(z − λ)^{2}} − \frac{1}{λ^{2}} )

is the Weierstrass ℘-function attached to Λ. The group law on E(C) corresponds to the usual addition law of complex numbers on C/Λ. This explicit analytic description yields the structure of E(C) and E(R): the former is a product of two circles, and the latter is either a circle or the product of a group of order 2 with a circle.

The structure of E(Q) lies deeper. In the case of the elliptic curve

E : y^{2} = x^{3} + 877x,  (3)

Bremner and Cassels [BC] showed that E(Q) is generated by the point (0, 0) of order 2 and the point of infinite order

(x,y)=(( \frac{612776083187947368101}{78841535860683900210} )^{2}, \frac{256256267988926809388776834045513089648669153204356603464786949}{78841535860683900210^{3}} ).

For the elliptic curve^{3}

y^{2} + xy + y = x^{3} − 20333x + 203852,  (4)

2provided that the equation (1) remains non-singular over F, which is the case for example if the characteristic of F does not divide Δ.

3its equation is not given in Weierstrass form as in equation (1), but can be brought to this form by a simple change of variables.

it turns out that E(Q) is generated by the six points P_{j} = (x_{j}, y_{j}) with

\begin{matrix}P_{1} = (−51, 1078), & P_{2} = (3, 376), & P_{3} = (165, 1078), \\ P_{4} = (−24, 835), & P_{5} = (−132, 835), & P_{6} = (136, 106). \end{matrix}

In fact, a point in E(Q) can be written uniquely as n_{1}P_{1} + ⋯ + n_{6}P_{6}, with n_{j} ∈ Z.

In general, how are the rational solutions to equation (2) calculated? As with many fundamental questions in number theory, the first progress dates back to Fermat, who introduced his famous method of infinite descent and used it to show that certain elliptic curves, related to the Fermat equation with exponent 4 and 3, have finitely many solutions. Fermat’s descent was later adapted by Mordell^{4} to prove the following general result about E(Q), which is suggested by the special cases (3) and (4).

Theorem The group E(Q) is a finitely generated abelian group, i.e.,

E(Q) ≃ T ⊕ Z^{r},

where T is a finite group (identified with the torsion subgroup of E(Q)).

The integer r is called the rank of E(Q): it represents the minimal number of solutions needed to generate a finite index subgroup of E(Q) by repeated application of the chord and tangent law. The rank depends in a subtle way on E, and can get quite large^{5}.

Unfortunately, the proof of Mordell’s theorem, based on Fermat’s descent, is not effective; it is not known whether Fermat’s descent procedure always terminates eventually. The following basic question remains open.

Question: Is there an algorithm to compute E(Q)?

4The proof was then further generalized by Weil to cover abelian varieties over number fields. An abelian variety is a projective (commutative) algebraic group; it is a natural higher dimensional generalization of elliptic curves. For this reason Mordell’s theorem is often referred to as the Mordell-Weil theorem, and E(Q) is called the Mordell-Weil group attached to E.

5It is expected that it can get arbitrarily large, although this is not proved. The record so far is an elliptic curve of rank ≥ 22 [Fe].

What is desired is a deterministic recipe which, given a and b in equation (1) (say) yields a description of E(Q). The torsion subgroup T can be calculated without difficulty; the key challenge arises in computing the rank r and a system of generators for E(Q).

The Birch and Swinnerton-Dyer conjecture. Further insights about E(Q) may be gleaned by studying E over other fields, such as the finite field F_{p} with p elements consisting of the residue classes modulo a prime p. The set E(F_{p}) is finite. A simple heuristic argument suggests that its cardinality N_{p} is roughly p + 1. Indeed Hasse proved that the “error term” a_{p} := p + 1 − N_{p} satisfies

|a_{p}| ≤ 2\sqrt{p}.

Reduction of solutions modulo p gives a natural map E(Q) ⟶ E(F_{p}). One might expect the presence of a large supply of rational points in E(Q) to have an impact on the size of E(F_{p}) on average. Compelled by this insight, Birch and Swinnerton-Dyer studied the asymptotic behaviour of ∏_{p<X} N_{p}/p as X gets large. On the basis of numerical experiments, they were led to conjecture that

∏_{p<X} N_{p}/p ≃ C_{E}(log X)^{r},  (5)

where C_{E} is a constant depending only on E. This striking conjecture asserts that the rank r - an a priori subtle global invariant of the arithmetic of E over Q - can be read off from the asymptotic behaviour of the N_{p}, reflecting information about E of a “local” nature.

Subsequently, following a suggestion of Davenport, Birch and Swinnerton-Dyer gave a more sophisticated formulation of the conjecture in terms of the Hasse-Weil L-function L(E, s) attached to E. Let s be a complex variable, and for p |̸Δ let

L(E, p, s) := (1 − a_{p}p^{−s} + p^{1−2s})^{−1}  (6)

be the local L-function attached to E at p. There is also a simple definition of L(E, p, s) for the finite set of primes dividing Δ, whose precise nature need not concern us here. (See for example [Si].) The L-function L(E, s) of E over Q is then defined by the “Euler product”

L(E, s) := ∏_{p} L(E, p, s),  (7)

where the product is taken over all the primes. The Hasse bound |a_{p}| ≤ 2\sqrt{p} implies that it converges absolutely in the right half-plane Re(s) > 3/2. In particular, the point s = 1 is outside the domain of absolute convergence. However, the fundamental Shimura-Taniyama conjecture, which will be discussed further below, implies that L(E, s) has an analytic continuation to all of C.

Noting the identity (for p |̸Δ)

L(E, p, 1) = \frac{p}{N_{p}} ,

and comparing it with the quantity occuring in (5), Birch and Swinnerton– Dyer were led to conjecture that the rank r should be reflected in the order of vanishing of L(E, s) at s = 1.

Conjecture BSD: The function L(E, s) satisfies

ord_{s=1}L(E, s) = r.

A more precise version of this conjecture expresses L^{(r)}(E, 1), the rth derivative of L(E, s) at s = 1, in terms of various quantities associated to E over Q, most notably a “regulator term” measuring the arithmetic complexity of a system of generators for E(Q), and the order of a conjecturally finite group known as the Shafarevich-Tate group of E, and denoted by the Cyrillic letter \underline{III}. The precise definition of this group would take the reader too far afield; suffice it to say that \underline{III}(E) measures the difficulty of computing E(Q) by Fermat’s descent method. In particular, its finiteness implies that Fermat’s descent terminates when applied to E. It is for this reason that the Shafarevich–Tate conjecture, which predicts the finiteness of \underline{III}(E) for all E, is widely viewed as the most important outstanding question in the arithmetic of elliptic curves.

Concerning the Birch-Swinnerton Dyer conjecture, Tate wrote [Ta1]

“This remarkable conjecture relates the behaviour of a function L, at a point where it is not at present known to be defined, to the order of a group \underline{III}, which is not known to be finite.”

This quote accurately summarized the state of knowledge (or perhaps, ignorance) on the question, until around 1987, when the results of Gross-Zagier and Kolyvagin, and then Wiles, led to dramatic breakthroughs^{6}.

The Shimura-Taniyama conjecture and Wiles’ theorem. The fact that a priori L(E, s) is not even known to be defined at s = 1 presents an obvious obstacle to tackling the Birch and Swinnerton-Dyer conjecture. In 1993, Wiles established the analytic continuation of L(E, s) for a large class of elliptic curves by relating E (and its L-function) to modular forms.

Given an integer N, let Γ_{0}(N) be the group of matrices in SL_{2}(Z) which are upper triangular modulo N. It acts as a discrete group of Mobius transformations on the Poincaré upper half-plane

H := {z ∈ C|Im(z) > 0}.

A cusp form of weight 2 for Γ_{0}(N) is an analytic function f on H satisfying the relation

f (\frac{az + b}{cz + d}) = (cz + d)^{2}f(z), for all ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(N),  (8)

together with suitable growth conditions on the boundary of H. For example, the invariance in equation (8) implies that f is periodic of period 1, and one requires that it can be written as a power series in q = e^{2πiz} with no constant term:

f(z) = ∑_{n=1}^{∞} λ_{n}q^{n}.

The Dirichlet series

L(f, s) = ∑ λ_{n}n^{−s}

is called the L-function attached to f. A direct calculation reveals that L(f, s) is essentially the Mellin transform of f:

Λ(f, s) := Γ(s)L(f, s) = (2π)^{s} ∫_{0}^{∞} f(iy)y^{s−1}dy.  (9)

6Prior to this one should also mention the work of Coates and Wiles establishing partial results towards the Birch and Swinnerton-Dyer conjecture for elliptic curves with complex multiplication – a restricted class, but one which has played an important role in the development of the theory.

The space of cusp forms of weight 2 on Γ_{0}(N) is a finite-dimensional vector space and is preserved by the involution W_{N} defined by

W_{N}(f)(z) = Nz^{2}f(\frac{−1}{Nz}).

Hecke showed that if f lies in one of the two eigenspaces for this involution (with eigenvalue w = ±1) then L(f, s) satisfies the functional equation:

Λ(f, s) = −wΛ(f, 2 − s).  (10)

In particular, L(f, s) has an analytic continuation to all of C.

The curve E is said to be modular if there exists a cusp form f of weight 2 on Γ_{0}(N) for some N such that

L(E, s) = L(f, s).

Taniyama and Shimura conjectured in the fifties that every elliptic curve over Q is modular. This important conjecture gives a framework for proving the analytic continuation and functional equation for L(E, s), and illustrates a deep relationship between objects arising in arithmetic, such as E, and objects, such as f, which are part of an ostensibly different circle of ideas – related to Fourier analysis on groups, and the (infinite-dimensional) representation theory of adelic groups, as described in the ambitious Langlands program.

The conjecture of Shimura-Taniyama, as refined by Weil, predicts that the integer N is equal to the so-called arithmetic conductor of E. This integer can be computed effectively in terms of an equation defining E, and is divisible only by the primes dividing Δ, but with different exponents in general. From now on, the letter N will be used to denote the conductor of E.

Thanks to the work of Wiles [Wi], Taylor-Wiles [TW] and its extensions [Di], [CDT], one now knows that E is modular, at least provided that E satisfies a mild technical restriction.

Theorem STW. If 27 does not divide N, then E is modular.

Complex uniformisation. The modularity of E can also be formulated as a statement about the complex uniformisation of the Riemann surface E(C). (Cf. [Ma2]).

Theorem STW_{∞}: If 27 does not divide N, then there is a complex analytic uniformisation

φ_{∞} : H/Γ_{0}(N) ⟶ E(C).

The classical uniformisation theorem of complex analysis states that every Riemann surface is expressible as a quotient of H by the action of some discrete subgroup Γ of SL_{2}(R). The above statement lies deeper. Its arithmetic content comes from the fact that it makes a precise statement about the nature of Γ, and relates it to the arithmetic of E over Q. Groups like Γ_{0}(N) which are defined by simple congruence conditions on the matrix entries, are examples of what are called called arithmetic subgroups of SL_{2}(Z).

Evidence for the Birch–Swinnerton-Dyer conjecture. As Mazur writes in [Ma1],

“ It has been abundantly clear for years that one has a much more tenacious hold on the arithmetic of an elliptic curve E/Q if one supposes that it is […] parametrized [by a modular curve].”

This sentiment is supported by the following result, following from the work of Kolyvagin [Ko] and earlier work of Gross and Zagier [GZ].

Theorem GZK. Let E be an elliptic curve over Q of rank r. Suppose that E is modular, and that ord_{s=1}L(E, s) ≤ 1. Then

ord_{s=1}L(E, s) = r,

and the Shafarevich-Tate conjecture is true for E.

The theorem (or rather, its proof) even supplies a procedure for computing E(Q), based on the theory of complex multiplication, which relies on the modularity of E and is more efficient that the descent method of Fermat.

Higher order zeroes. The following question remains as the ultimate challenge concerning the Birch and Swinnerton-Dyer conjecture.

Question: What if ord_{s=1}L(E, s) > 1?

In this case the relation between r and the order of vanishing of L(E, s) at s = 1 remains mysterious. It appears that the inequality

ord_{s=1}L(E, s) ≥ r  (11)

ought to be easier to prove than the reverse inequality^{7}. However, even this “easy half” of the Birch Swinnerton-Dyer conjecture seems out of reach for now. The process whereby the presence of “many” rational points in E(Q) forces higher vanishing of L(E, s) at s = 1 is simply not understood.

There are elliptic curves E for which the sign −w in the functional equation (10) is −1, and for which r ≥ 3, such as the elliptic curve

y^{2} = 4x^{3} − 28x + 25  (12)

of conductor N = 5077. In this case L(E, s) vanishes to odd order, and theorem GZK implies that L′(E, 1) = 0. Hence

ord_{s=1}L(E, s) ≥ 3.  (13)

But this is basically as far as one can go! Indeed the following question remains open:

Question: Is there an elliptic curve E over Q with ord_{s=1}L(E, s) > 3?

In his undergraduate summer project [Gh], Alexandru Ghitza evaluated the first few derivatives of L(E, s) at s = 1 for the curve of rank 6 given by equation (4). In this case L(E, 1) = 0 and the sign −w in the functional equation for L(E, s) is 1, so that L(E, s) vanishes to even order ≥ 2. Ghitza’s numerical calculations (performed on a high-speed computer with an accuracy of around four significant digits after the decimal point) produced

L′′(E, 1) ≃ −0.0000195,
L^{(4)}(E, 1) ≃ −0.00000027,
L^{(6)}(E, 1) ≃ 717.6663612.

7For example, it is known to hold in the function field case, by work of Tate[Ta2]. The reverse inequality seems inextricably linked with questions related to the Shafarevich–Tate conjecture.

This strongly suggests that L(E, s) vanishes to order 6 at s = 1, as predicted by the Birch and Swinnerton-Dyer conjecture, but it appears to be an extremely difficult theoretical problem to prove that L′′(E, 1) = 0, even for this specific curve!

Using the known elliptic curves with r ≥ 22, note that a proof of (11) would imply the existence of L-functions for which ord_{s=1}L(E, s) ≥ 22.

The work of Goldfeld. Producing L-functions L(E, s) with high order zeroes at s = 1 has a number of applications. For example, Goldfeld [Go] showed that the existence of a suitable^{8} elliptic curve E for which ord_{s=1}L(E, s) ≥ r implies the following asymptotic lower bounds on the growth of the class number h(D) of the imaginary quadratic field of discriminant D:

h(D) ≥ c(log |D|)^{r−2−ϵ}.

The importance of this estimate lies in the fact that the constant c is effective, and can be calculated in terms of the elliptic curve E. Goldfeld’s work, using the elliptic curve of equation (12), led to the unconditional estimate

h(D) ≥ \frac{1}{55}(log |D|)^{1−ϵ},

and thus to a solution of the celebrated class number problem of Gauss. Note the key role played in this estimate by equation (13), which is based in turn on theorem GZK.

More recently, Ram Murty has informed me that an analogue of a conjecture of Polya about the Riemann zeta-function ζ(s), which was subsequently shown to be false, ought to be true after replacing ζ(s) by L-functions L(E, s) admitting a high-order zero at s = 1. (Cf. [Mu].)

p-adic analysis. In the face of the difficulties associated with understanding the complex L-function, it has proved fruitful to replace the complex variable s by a p-adic one, and the classical Hasse-Weil L-function by a p-adic analogue.

In addition to the usual “archimedean” distance d_{∞}(x, y) = |x − y|, the rational numbers are equipped (for each prime p) with the p-adic distance d_{p}(x, y) = p^{−ord_{p}(x−y)}, according to which two rational numbers are declared to be close to each other if (the numerator of) their difference is divisible by a high power of p. Ostrowski’s theorem asserts that the usual absolute value and the p-adic ones, as p ranges over the primes, give a complete list of metrics (up to a suitable equivalence) which are compatible with the field structure on Q.

8by “suitable” it is meant that the number of primes dividing N to odd order should be odd, if w = 1, and even, if w = −1.

Just as R is the completion of Q with respect to the usual metric, the field Q_{p} is the completion of Q with respect to d_{p}. It has a greater arithmetic complexity that R, in the sense that its algebraic closure Q̄_{p} is of infinite degree over Q_{p}, unlike C over R. As a consequence, Q̄_{p} is not a complete field^{9}. The role of C is played by a larger field, denoted C_{p}, the completion of Q̄_{p} with respect to the p-adic metric.

The p-adic upper half plane H_{p} is defined as

H_{p} := P_{1}(C_{p}) − P_{1}(Q_{p}) = C_{p} − Q_{p}.

Note that replacing C_{p} by C, and Q_{p} by R, yields two copies of the usual Poincaré upper half plane. In the p-adic setting, C_{p} − Q_{p} does not split naturally into two disjoint connected pieces, so that it is more natural to work with H_{p} in its entirety.

The space H_{p} is endowed with a rich theory of “p-adic analytic functions” which mirrors the complex-analytic theory. By analogy with the complex case, it could be tempting to define an “analytic” function on H_{p} as a C_{p}- valued function which admits a power series expansion in each open disk. In the p-adic setting, however, two open discs are either disjoint or one is contained in the other! The space of “analytic functions” according to this definition turns out to be too large and not “rigid” enough to yield a useful theory: for example, the principle of analytic continuation fails.

A fruitful function theory, obeying many of the principles of classical complex analysis, is obtained by replacing open discs by so-called affinoid sets, which are made up of a closed p-adic disc with a number of open disks deleted. The affinoids cover H_{p} and can be used to define a sheaf of rigid analytic functions which enjoys many of the same formal properties as the sheaf of complex analytic functions on H.

9for example, if ζ_{n} is a primitive p^{n}th root of unity, then ∑_{n=1}^{∞} ζ_{n}p^{n} does not have a limit in Q̄_{p} even though its partial sums form a Cauchy sequence.

The group SL_{2}(Q_{p}) acts on H_{p} by fractional linear transformations, just as SL_{2}(R) acts on H. If Γ is a discrete subgroup of SL_{2}(Q_{p}), the quotient X_{Γ} := H_{p}/Γ inherits a p-adic topology and becomes a rigid analytic curve: it is the analogue, in the p-adic realm, of a Riemann surface.

The p-adic uniformisation theory of Mumford addresses the question of which curves X/C_{p} can be be written as a quotient H_{p}/Γ, for Γ ⊂ SL_{2}(Q_{p}). Unlike the complex case, not every curve over C_{p} can be so uniformized. Mumford identifies a simple necessary and sufficient condition^{10} for X to admit a p-adic uniformisation. Curves over C_{p} with this property are called Mumford curves. An elliptic curve over Q is a Mumford curve precisely when its conductor is exactly divisible by p. The p-adic uniformisation theory of elliptic curves with p||N was developped by Tate, and later generalized by Mumford to curves of higher genus.

Rigid analytic Shimura-Taniyama. Let E be an elliptic curve over Q with 27 |̸N, so that E is modular in the sense of theorem STW. The following result is a p-adic analogue of theorem STW_{∞}, and follows by combining the result of Wiles with earlier work of Eichler-Shimizu-Jacquet-Langlands, Shimura, and Cerednik-Drinfeld. (Cf. for example the work of Jordan-Livné [JL].)

Theorem STW_{p}. Suppose that p||N, so that E is Mumford curve over Q_{p}. Then there exists a discrete arithmetic subgroup Γ of SL_{2}(Q_{p}) and a rigid analytic uniformisation of E(C_{p}):

φ_{p} : H_{p}/Γ ⟶ E(C_{p}).

The key word in this theorem is the word arithmetic. It means that the groups involved in the uniformisation are analogous to Γ_{0}(N). The definition of these groups is somewhat more involved. Rather than provide a complete definition, here is an example which gives the flavour of the general case. Let

B := Q + Qi + Qj + Qk

10X should have a model over O (the ring of integers of C_{p}) whose special fiber is a union of projective lines intersecting transversally at ordinary double points.

be the ring of Hamilton quaternions with coefficients in Q, and let

R = Z[i, j, k, \frac{1 + i + j + k}{2} ]

be Hurwitz’s maximal order. If p is an odd prime, then B ⊗Q_{p} is isomorphic to the ring M_{2}(Q_{p}) of two by two matrices with entries in Q_{p}; after choosing such an isomorphism, the group

Γ = R[1/p]_{1}^{×}  (14)

of elements of norm 1 in R[1/p] can be viewed as a subgroup of SL_{2}(Q_{p}). This Γ is an example of a p-adic arithmetic subgroup of SL_{2}(Q_{p}); in fact, if E is an elliptic curve of conductor 2p, then E(C_{p}) is uniformized by H_{p}/Γ.

The pull-back to H_{p} of a suitable invariant differential ω on E yields a Γinvariant differential f(z)dz on H_{p}. The function f is a rigid analytic modular form of weight two on H_{p}, i.e., a rigid analytic function on H_{p} satisfying the transformation property analogous to (8)

f (\frac{az + b}{cz + d}) = (cz + d)^{2}f(z), for all ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ.  (15)

Schneider’s p-adic L-functions. By analogy with the construction of L(f, s), the following goal seems natural.

Goal. Attach to a rigid analytic modular form f a p-adic L-function L_{p}(f, s), by a process of p-adic Mellin transform.

What is desired here is a C_{p}-valued function of the variable s ∈ C_{p} which is rigid analytic, at least in a neighbourhood of s = 1. A definition along those lines was proposed by Schneider [Sch], by associating to f a p-adic measure μ_{f} on P_{1}(Q_{p}), the p-adic boundary of H_{p}. This measure behaves like the boundary measure attached to f, and indeed it satisfies the following analogue of the Poisson inversion formula [Te] which allows f to be recovered from μ_{f}:

f(z) = ∫_{P_{1}(Q_{p})} \frac{dμ_{f}(t)}{z − t}.

By analogy with formula (9) for the complex L-function, Schneider proposed defining

L_{p}(f, s) := ∫_{Z_{p}^{×}}⟨x⟩^{s−1}dμ_{f}(x),  (16)

where Z_{p}^{×} ⊂ P_{1}(Q_{p}) is the group of p-adic units and ⟨x⟩ = x/(lim_{n→∞} x^{p^{n}}).

This definition does not lead to a satisfactory theory of p-adic L-functions, because the definition of L_{p}(f, s) is sensitive to the identification of B_{p} with M_{2}(Q_{p}) used to make Γ act on H_{p} and P_{1}(Q_{p}). It appears that even the order of vanishing of L_{p}(f, s) depends on these choices, and so it is doubtful that a conjecture analogous to conjecture BSD can be formulated for Schneider’s L_{p}(f, s).

The Iovita-Spiess construction. A definition of a p-adic L-function which is modelled on Schneider’s approach, but does lead to a fruitful theory, was proposed by Adrian Iovita in a graduate course at McGill University, and independently by Michael Spiess. The Iovita-Spiess construction is best explained in the special case of the group Γ of equation (14). (The full details are given in [BDIS].) Let K be a maximal commutative subalgebra of the quaternion algebra B. It is isomorphic to a quadratic imaginary field in which the prime 2 is either inert or ramified. Let O be the ring of integers of K. Replacing K by a conjugate subalgebra, one may assume that

K ∩ Γ = (O_{K}[1/p])^{×}.

In fact, if the p-class group Pic(O[1/p]) is trivial, the subalgebra K with this property is unique up to conjugation by elements of Γ. Assume for simplicity that this is the case. The identification B_{p} = M_{2}(Q_{p}) yields an action of K_{p}^{×} := (K ⊗ Q_{p})^{×} on the boundary P_{1}(Q_{p}) of H_{p}, having at most two fixed points and acting transitively on the complement Ω. A choice of base point in Ω thus yields a continuous map

η : K_{p}^{×}/Q_{p}^{×} ⟶ Ω ⊂ P_{1}(Q_{p}).

Let μ_{f,K} := η^{∗}(μ_{f}) be the pullback of Schneider’s measure μ_{f} to a measure on K_{p}^{×}/Q_{p}^{×}, and let μ̄_{f,K} be the measure obtained by composing μ_{f,K} with complex conjugation on K_{p}^{×}. The invariance of μ_{f} under Γ translates into the invariance of μ_{f,K} and μ̄_{f,K} under the action of O[1/p]^{×}, and hence yields measures on the compact p-adic group G_{∞} := K_{p}^{×}/(Q_{p}^{×}O[1/p]^{×}). Letting μ_{f,K}^{(2)} be the convolution measure μ_{f,K} ∗ μ̄_{f,K}, define, in analogy with Schneider’s definition (16):

L_{p}(f, K, s) := ∫_{G_{∞}}⟨\frac{x}{x̄}⟩^{s−1}dμ_{f,K}^{(2)}(x).

Note the crucial role played by the quadratic imaginary field K in the definition of L_{p}(f, K, s). In fact, the measure dμ_{f,K}^{(2)} interpolates special values of the complex L-function L(f/K, s) of f over K. More precisely, if χ : G_{∞} ⟶ C_{p}^{×} is a non-trivial character of finite order, interpreted as an idèle class character in the usual way, there is the interpolation formula

∫_{G_{∞}} χ(x)dμ_{f,K}^{(2)}(x) = Ω_{p} \frac{L(f/K, χ, 1)}{Ω_{∞}},  (17)

where Ω_{p} ∈ C_{p} and Ω_{∞} ∈ C are suitable p-adic and complex periods, and L(f/K, χ, s) is the complex L-function of f over K twisted by the character χ. The complex number \frac{L(f/K,χ,1)}{Ω_{∞}} turns out to be algebraic and is viewed as an element of C_{p} by choosing an embedding of Q̄ in C_{p}. This interpolation formula follows from a generalization of a formula of Gross [Gr1] for special values of L-series. See [BDIS] for details.

The p-adic Birch and Swinnerton-Dyer conjecture. If E is an elliptic curve over Q satisfying the conclusion of theorem STW_{p}, so that it is attached to a rigid analytic modular form f on H_{p}, define

L_{p}(E, K, s) := L_{p}(f, K, s).

Even before the connection with Schneider’s approach was made explicit, the p-adic L-function L_{p}(E, K, s) could be constructed from a different and more general point of view, which does not rely on p-adic analysis and also allows the definition of L_{p}(E, K, s) in the good reduction case, where p |̸N. In this level of generality, the p-adic Birch and Swinnerton-Dyer conjecture for the function L_{p}(E, K, s) was formulated and studied in a series of articles [BD1], [BD2], [BD3], [BD4] and [BD5].

Because of the presence of the field K in the interpolation formula (17), it is natural to expect the order of vanishing of L_{p}(E, K, s) to be related to the rank r_{K} of the Mordell–Weil group E(K). In [BD1], it was conjectured that

ord_{s=1}L_{p}(E, K, s) ≥ r_{K}.

This p-adic variant of the “easy half” (11) of the Birch Swinnerton–Dyer conjecture has recently been proved in [BD6].

Theorem BD. The p-adic L-function L_{p}(E, K, s) vanishes to order at least r_{K} at s = 1.

The proof of theorem BD is based on two ingredients.

1. The theory of congruences between modular forms and the Jacquet-Langlands correspondence. This circle of ideas plays a crucial role in Wiles’ proof of theorem STW, and in Ribet’s earlier reduction [Ri] of Fermat’s Last Theorem to the Shimura-Taniyama conjecture.

2. Kolyvagin’s theory of the “Euler systems” of Heegner points, the principal ingredient in the proof of theorem GZK.

Thus, theorem BD relies crucially on the ideas of Gross-Zagier, Kolyvagin, Ribet, and Wiles, which have revolutionized the theory of elliptic curves through the proofs of theorems GZK and STW.

To conclude, here are two natural questions connected with the original Birch and Swinnerton-Dyer conjecture.

1. Theorem BD can be used to exhibit elliptic curves whose p-adic Lfunction L_{p}(E, K, s) satisfies

ord_{s=1}L_{p}(E, K, s) ≥ 22.

Does the existence of such p-adic analytic L-functions with high order zeroes have independent applications to other questions of number theory (or mathematics in general), as in Goldfeld’s solution of Gauss’s class number problem?

2. Is it possible to replace the rigid analytic L-functions by classical ones in the proof of theorem BD? The proof in [BD6] is based on congruences in an essential way and breaks down entirely when the prime p is replaced by the “place at ∞”. In this sense, it sheds no light on the original Birch and Swinnerton–Dyer conjecture, even on the “easy inequality”.

As Mazur writes in [Ma3],

“A major theme in the development of number theory has been to try to bring R somewhat more into line with the p-adic fields; a major mystery is why R resists this attempt so strenuously. ”

An explanation of the mysterious analogy between the complex and p-adic realms would surely lead to deep insights: it is an issue which lies at the heart of the tantalizing and elusive Birch and Swinnerton–Dyer conjecture.

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