SLIDE 1 Magma 2010 Conference
p-adic L-functions, (Stark-) Heegner points, and computer algebra
Henri Darmon
McGill University
February 22, 2010
SLIDE 2 Thank you
To the Magma group, and to
X.-F. Roblot,
uthrich, for organising this meeting and running it at the CRM.
SLIDE 3
Philosophies of mathematics
“Non-constructivists”. Abstract existence proofs. “Constructivists”, or “Kroneckerians”. Most mathematicians (and certainly most number theorists) would agree that a proof is more satisfying if it leads to an explicit (algorithmic) solution to a problem.
SLIDE 4 The “engineers”
There is a growing community of mathematicians concerned with the efficient (as well as effective) calculation of the mathematical
- bjects which arise in number theory.
One could refer to these mathematicians as “applied number theorists”, or “engineers”. Question Why should engineers care about p-adic L-functions?
SLIDE 5 Some engineering questions
I will focus on the following questions:
1 Explicit class field theory 2 Constructing units in number fields (global fields) 3 Calculating Mordell-Weil groups.
This narrow focus leaves out many aspects, such as the connections between p-adic L-functions and class groups, Selmer groups, and Iwasawa Theory. For these, see the lectures of W¨ uthrich, Washington, R. Greenberg, Kurihara, Matsuno, Liang, Coates...
SLIDE 6
Explicit class field theory
Theorem Given a global field K and an ideal m of OK, there is an abelian extension K[m] of K with Gal(K[m]/K) = {Ideals of OK prime to m}/(1 + mOK)×, called the ray class field of K of conductor m. Problem Given a global field K, and a modulus m, construct K[m]. The proofs of class field theory are constructive, and translate into algorithms that are implemented in MAGMA.
SLIDE 7
Kronecker-Weber and complex multiplication
Theorem (Kronecker-Weber) When K = Q the ray class field Q[m] can be generated by the m-th roots of unity Theorem (Main theorem of Complex Multiplication) When K is an imaginary quadratic field, the ray class field K[m] can be generated by the m-torsion points of elliptic curves with complex multiplication by OK. In these two very special cases, the resulting constructions of class fields are more efficient in practice, in addition to their greater aesthetic appeal. Hilbert’s twelfth problem asks if there are analogous constructions for other K.
SLIDE 8 Stark’s conjecture
Let K= number field, and m a modulus with S∞|m. ζ(K, A, s)= partial zeta-function of the class A modulo m. Conjecture (Stark) Suppose ζ(K, A, 0) = 0. Then there exists a unit u(A) of K[m] such that ζ′(K, A, 0) = log |u(A)|. The unit u(A) is called a Stark unit of K[m]. Magma computes the class fields of real quadratic fields by computing first derivatives of abelian L-series at s = 0.
- Cf. the talks of S. Dasgupta and H. Chapdelaine
SLIDE 9
Calculating Mordell-Weil groups
The following problem closely ressembles the calculation of units: Problem Given an elliptic curve E over a global field K, calculate the Mordell-Weil group E(K). Question Does the approach to the construction of units based on Stark’s conjecture have a counterpart for elliptic curves? The analogue of the L-functions ζ(K, A, s) is the Hasse-Weil L-series L(E/K, s). Can it be used to compute points in E(K) numerically?
SLIDE 10 The Birch and Swinnerton-Dyer conjecture
Conjecture ( Birch and Swinnerton-Dyer) If L(E/K, 1) = 0, then there exists PK ∈ E(K) such that L′(E/K, 1) = ˆ h(PK) · ( explicit period). The complex L-function does not seem to carry direct information about PK, only about its height.
- J. Silverman: a priori knowledge of ˆ
h(PK) can be used to speed up the calculation of PK. This approach works best for S-integral points, with S small.
SLIDE 11 The Katz p-adic L-function
Let K be an imaginary quadratic field, and suppose p splits in K. The space ΣK of Hecke characters of K is equipped with a natural p-adic analytic structure. Theorem (Katz) There is a p-adic analytic function ψ → Lp(ψ) on ΣK such that Lp(ψ) Ω(p)
ψ
= L(ψ−1, 0) Ω(∞)
ψ
, for all ψ of type (1 + j, −k) with j, k ≥ 0. Here Ω(p)
ψ
and Ω(∞)
ψ
are appropriate p-adic and complex periods.
SLIDE 12 Rubin’s formula
Let E/Q be an elliptic curve with complex multiplication by K. (In particular, K has class number one.) Deuring: L(E/Q, s) = L(ψ, s) = L(ψ−1, s − 1), where ψ is a Hecke character of type (1, 0). Over C: L(ψ, s) = L(ψ∗, s). This need not be true over Cp. Theorem (Rubin) Suppose that L(ψ, 1) = 0 (and hence Lp(ψ) = 0). There exists a global point P ∈ E(Q) and a differential ω ∈ Ω1(E/Q) such that Lp(ψ∗) = (Ω(p)
ψ )−1 log2 ω(P).
SLIDE 13 The Mazur-Swinnerton-Dyer p-adic L-function
Question Can p-adic L-functions be used to recover rational points on elliptic curves, in more general settings? Let E/Q be a (modular) elliptic curve. Mazur-Swinnerton-Dyer: There is a p-adic L-function Lp(E/Q, s) attached to E/Q, defined in terms of modular symbols. Pollack-Stevens: Their theory of overconvergent modular symbols leads to an efficient, polynomial time algorithm to compute Lp(E/Q, s) and its derivatives. (To compute these to accuracy p−M takes time proportional to a polynomial in p and M.) Cf.
SLIDE 14 Perrin-Riou’s conjecture
Two p-adic L-functions: Lp,α(E/Q, s) and Lp,β(E/Q, s) x2 − apx + p = (x − α)(x − β),
- rdp(α) ≤ ordp(β).
- rdp(β) = 1: Kato-Perrin-Riou; Pollack-Stevens. (Cf. Bellaiche.)
Lp,†(E, s) :=
β 2 Lp,α(E, s) −
α 2 Lp,β(E, s). Conjecture (Perrin-Riou) If Lp,α(E, 1) = 0, there exists a point P ∈ E(Q) and ω ∈ Ω1(E/Q) such that L′
p,†(E, 1) = α − β
[ϕω, ω] log2
ω(P).
SLIDE 15 The work of Kurihara and Pollack
- M. Kurihara and R. Pollack combine Perrin-Riou’s conjecture and
the Pollack-Stevens algorithm to compute rational points on elliptic curves p-adically, when p is a supersingular prime. Example: The curve X0(17) is supersingular at p = 3. X0(17)193 : y2 + xy + y = x3 − x2 − 25609x − 99966422 (x, y) = 915394662845247271
25061097283236
, −878088421712236204458830141
125458509476191439016
See M. Kurihara and R. Pollack, ‘Two p-adic L-functions and rational points on elliptic curves with supersingular reduction, L-Functions and Galois Representations (Durham, 2007), 300–332, London Math Society LNS 320.
SLIDE 16 p-adic Rankin L-functions
Let f ∈ S2(Γ0(N), ε) be a modular form of weight two and let K be an imaginary quadratic field. Theorem (Hida) There is a p-adic analytic function ψ → Lp(f , ψ) on ΣK such that Lp(f , ψ) Ω(p)
ψ
= L(f /K, ψ−1, 0) Ω(∞)
ψ
= L(f ⊗ θψ, ∗) Ω(∞)
ψ
, for all ψ of type (2 + j, −k) with j, k ≥ 0.
SLIDE 17 A second analogue of Rubin’s formula
Let χtriv be the norm character on K. It is of type (1, 1) and hence lies outside the range of p-adic interpolation defining Lp(f , −). Suppose that the form f corresponds to an elliptic curve E. Theorem (Bertolini, Prasanna, D) If N is the norm of a cyclic ideal of K, then there exists a Heegner point P ∈ E(K) and ω ∈ Ω1(E/K) such that Lp(f , χtriv) = log2
ω(P).
No p-adic period is involved in this formula.
SLIDE 18 A generalisation of Rubin’s formula
Let η = Hecke character of K of type (1, 0) Aη = associated CM abelian variety over K, EndK(Aη) ⊗ Q = Kη, [Kη : K] = dim(Aη). The Gross-Zagier type formula for p-adic Rankin L-functions can be used to prove a generalisation of Rubin’s formula. Theorem (Bertolini,Prasanna,D) Let η be a Hecke character of K of type (1, 0) satisfying ηη∗ = χtriv. Then there exists a point P ∈ Aη(K) and ω ∈ Ω1(Aη/Kη)Kη such that Lp(η∗) = (Ω(p)
η )−1 log2 ω(P).
SLIDE 19 A sketch of the proof
Choose a pair (ψ, χ) of types (1, 0) and (1, 1) such that
1 η∗ = ψ−1χ 2 θψ ∈ S2(N, ε) with N = N ¯
N.
3 L(ψ∗χ−1, 0) = 0.
- 1. The p-adic Gross-Zagier formula, applied to f = θψ:
Lp(θψ, χ) = log2
ω(P),
where P ∈ Aψ(Hχ) corresponds to a point in Aη(K).
- 2. A factorisation of p-adic L-series:
Lp(θψ, χ) = Lp(ψ−1χ)Lp(ψ∗−1χ) ∼ Lp(η∗)Ω(p)
η .
SLIDE 20 Other Rankin L-functions
p-adic Rankin L-functions come in two distinct flavors:
1 Type I: interpolate L(f , χ−1, 0) with χ of type (2 + j, −k)
with k, j ≥ 0.
2 Type II: interpolate L(f , χ−1, 0) with χ of type (1, 1).
From now on, let Lp(f , χ) be the p-adic L-function of type II. Theorem (Bertolini, D (1997)) Suppose that Nf = N ¯ N, so that Lp(f , χtriv) = 0. Suppose also that p||N and that p is inert in K. Then d2 ds2 Lp(f , χtrivχs
−)s=0 ∼ log2 ω(P),
for some P ∈ E(K) and ω ∈ Ω1(E/K).
SLIDE 21 Heegner points
More precisely, we have d2 ds2 Lp(f , χtrivχs
−)s=0 =
a⊂OK
d ds Lp(f , a, s)s=0
2
, where Lp(f , a, s) is a partial anti-cyclotomic p-adic L-function attached to f and the ideal class a. For each a, there is a Heegner point Pa ∈ E(H) with d ds Lp(f , a, s)s=0 ∼ logω(Pa). Remark: All the points that have been obtained so far by considering p-adic L-functions can, in one way or another, be
- btained from Heegner points.
SLIDE 22 Stark-Heegner points
When K is a real quadratic field in which p is inert, and p||N, one can still make sense of the expressions d ds Lp(f , a, s)s=0. Defining Pa to be the local points in E(Kp) such that d ds Lp(f , a, s)s=0 ∼ logω(Pa) leads to the prototypical example of Stark-Heegner points. These points arise from p-adic Rankin L-functions, and cannot be
- btained otherwise (as far as we know).
SLIDE 23
Practical calculations
Stark-Heegner points can be computed in practice, relying on the Pollack-Stevens algorithms to compute the relevant p-adic L-functions. Example: E = X0(17), K = Q( √ 197). > HP,P,hD := stark heegner points(E,197,Qp); − > Computing the Stark-Heegner points of discriminant 197 over the Elliptic Curve defined by y2 + xy + y = x3 − x2 − x − 14 over Rational Field. The calculation is being done in 17-adic field mod 17100.
SLIDE 24
Practical calculations, N = p = 17, D = 197.
The discriminant D = 197 has class number 1 1 Computing point attached to the binary quadratic form < 1, 13, −7 > Sum of the Stark-Heegner points (over Cp) = (−6310862856360707267736337884716982701323524464440413797 1682682084792961477237437503838942485130056788048891527 129603886134 + O(17100) : . . . : . . .)
SLIDE 25 Practical calculations, N = p = 17, D = 197.
This p-adic point is close to the global element 101482189978655728200356115
36889211724013911765602449 ,
. . .
which is indeed a global point on E(K).
SLIDE 26 Practical calculations, N = p = 37, D = 401.
> load shp37A; Loading "shp37A" Loading "../data/M.37.1.plus" The maximal accuracy currently available is 100 37
Please enter the desired accuracy 10
SLIDE 27 Practical calculations, N = p = 37, D = 401.
You are now set up to perform Stark-Heegner point calculations on E = Elliptic Curve defined by y2 + y = x3 − x over Rational Field working over Qp = the 37-adic field mod 3710 > HP, P, hD := stark heegner points(E,401,Qp);
- -> Computing the Stark-Heegner points of
discriminant 401 over the Elliptic Curve defined by y2 + y = x3 − x over Rational Field The calculation is being done in 37-adic field mod 3710
SLIDE 28
Practical calculations, N = p = 37, D = 401.
The discriminant D = 401 has class number 5 1 Computing point attached to <1,19,-10> 2 Computing point attached to <-2,19,5> 3 Computing point attached to <4,15,-11> 4 Computing point attached to <-4,17,7> 5 Computing point attached to <2,17,-14>
SLIDE 29 Practical calculations, N = p = 37, D = 401.
Sum of the Stark-Heegner points (over Cp) = (O(3710) : −1 + O(3710) : 1 + O(3710)) This p-adic point is close to [0, -1,1] (0 :
1) is indeed a global point on E(K). The polynomial satisfied by the x-ccordinates of the Stark-Heegner points is 81x5 − 219x4 + 1195x3 − 173x2 − 976x + 527.
SLIDE 30 Stark-Heegner points and Abel-Jacobi maps
By “staring long and hard” at the formulae for Pa, one gets the following geometric interpretation (in the simplest case where E has conductor p).
1 A modular form of weight two on Γ0(p) can be “reinterpreted”
as a “Hilbert modular form F of weight (2, 2)” on X := SL2(Z[1/p])\(Hp × H), i.e., a “regular two-form” ωF on this space.
2 To each ideal class a ⊂ OK, one can associate a topological
cycle ∆a ⊂ X of real dimension 1.
3 One then defines
Pa“ = ”
ωF ∈ K ×
p /qZ = E(Kp).
SLIDE 31 Magma and p-adic L-functions
Question better than they could without them? The settings in which Stark-Heegner points have been implemented so far are limited:
1 Pollack, D: E = elliptic curve over Q of conductor p, K =
real quadratic field.
2 Logan, D: E = elliptic curve over Q(
√ D) of conductor 1, K = ATR extension.
3 Trifkovic: E = elliptic curve over Q(
√ −D) of conductor p, K = quadratic extension.
4 M. Greenberg: Heegner points and Stark-Heegner points
arising from Shimura curves.
SLIDE 32 Abel-Jacobi maps
Problem Given a closed r-dimensional cycle ∆ which is null-homologous, and a closed differential ω, compute
ω ∈ (C or C×
p )/(periods).
The calculation of such Abel-Jacobi type invariants raises interesting challenges, both theoretical and practical. Thank you for your attention!
SLIDE 33
... and sorry for running late!
