magma 2010 conference on p-adic l-functions...magma 2010 conference on p-adic l-functions p-adic...

Magma 2010 Conferenceon p-adic L-functions

p-adic L-functions, (Stark-) Heegner points,and computer algebra

Henri Darmon

McGill University

February 22, 2010

Thank you

To the Magma group, and to

M. Greenberg,

X.-F. Roblot,

M. Watkins,

C. Wuthrich,

for organising this meeting and running it at the CRM.

Philosophies of mathematics

“Non-constructivists”. Abstract existence proofs.

“Constructivists”, or “Kroneckerians”.

Most mathematicians (and certainly most number theorists) wouldagree that a proof is more satisfying if it leads to an explicit(algorithmic) solution to a problem.

Philosophies of mathematics

“Non-constructivists”. Abstract existence proofs.

“Constructivists”, or “Kroneckerians”.

Most mathematicians (and certainly most number theorists) wouldagree that a proof is more satisfying if it leads to an explicit(algorithmic) solution to a problem.

The “engineers”

There is a growing community of mathematicians concerned withthe efficient (as well as effective) calculation of the mathematicalobjects which arise in number theory.

One could refer to these mathematicians as “applied numbertheorists”, or “engineers”.

Question

Why should engineers care about p-adic L-functions?

The “engineers”

Question

The “engineers”

Question

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 theconnections between p-adic L-functions and class groups, Selmergroups, and Iwasawa Theory.

For these, see the lectures of Wuthrich, Washington, R. Greenberg,Kurihara, Matsuno, Liang, Coates...

Explicit class field theory

Theorem

Given a global field K and an ideal m of OK , there is an abelianextension 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 intoalgorithms that are implemented in MAGMA.

Theorem

Problem

Theorem

Problem

Kronecker-Weber and complex multiplication

Theorem (Kronecker-Weber)

When K = Q the ray class field Q[m] can be generated by them-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 withcomplex multiplication by OK .

In these two very special cases, the resulting constructions of classfields are more efficient in practice, in addition to their greateraesthetic appeal.

Hilbert’s twelfth problem asks if there are analogous constructionsfor other K .

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 bycomputing first derivatives of abelian L-series at s = 0.

Cf. the talks of S. Dasgupta and H. Chapdelaine

Conjecture (Stark)

ζ ′(K ,A, 0) = log |u(A)|.

Conjecture (Stark)

ζ ′(K ,A, 0) = log |u(A)|.

Conjecture (Stark)

ζ ′(K ,A, 0) = log |u(A)|.

Conjecture (Stark)

ζ ′(K ,A, 0) = log |u(A)|.

Conjecture (Stark)

ζ ′(K ,A, 0) = log |u(A)|.

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 theMordell-Weil group E (K ).

Question

Does the approach to the construction of units based on Stark’sconjecture have a counterpart for elliptic curves?

The analogue of the L-functions ζ(K ,A, s) is the Hasse-WeilL-series L(E/K , s). Can it be used to compute points in E (K )numerically?

Problem

Question

Problem

Question

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) = h(PK ) · ( explicit period).

The complex L-function does not seem to carry direct informationabout PK , only about its height.

J. Silverman: a priori knowledge of h(PK ) can be used to speed upthe calculation of PK .

This approach works best for S-integral points, with S small.

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 naturalp-adic analytic structure.

Theorem (Katz)

There is a p-adic analytic function ψ 7→ 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.

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 naturalp-adic analytic structure.

Theorem (Katz)

There is a p-adic analytic function ψ 7→ Lp(ψ) on ΣK such that

Lp(ψ)

Ω(p)ψ

=L(ψ−1, 0)

Ω(∞)ψ

Here Ω(p)ψ and Ω

(∞)ψ are appropriate p-adic and complex periods.

Rubin’s formula

Let E/Q be an elliptic curve with complex multiplication by K . (Inparticular, K has class number one.)

Deuring: L(E/Q, s) = L(ψ, s) = L(ψ−1, s − 1), where ψ is aHecke 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 aglobal point P ∈ E (Q) and a differential ω ∈ Ω1(E/Q) such that

Lp(ψ∗) = (Ω(p)ψ )−1 log2

ω(P).

Rubin’s formula

Theorem (Rubin)

Lp(ψ∗) = (Ω(p)ψ )−1 log2

ω(P).

Rubin’s formula

Theorem (Rubin)

Lp(ψ∗) = (Ω(p)ψ )−1 log2

ω(P).

Rubin’s formula

Theorem (Rubin)

Lp(ψ∗) = (Ω(p)ψ )−1 log2

ω(P).

The Mazur-Swinnerton-Dyer p-adic L-function

Question

Can p-adic L-functions be used to recover rational points onelliptic curves, in more general settings?

Let E/Q be a (modular) elliptic curve.

Mazur-Swinnerton-Dyer: There is a p-adic L-functionLp(E/Q, s) attached to E/Q, defined in terms of modular symbols.

Pollack-Stevens: Their theory of overconvergent modular symbolsleads to an efficient, polynomial time algorithm to computeLp(E/Q, s) and its derivatives. (To compute these to accuracyp−M takes time proportional to a polynomial in p and M.) Cf.M. Greenberg’s lecture.

Question

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), ordp(α) ≤ ordp(β).

ordp(β) = 1: Kato-Perrin-Riou; Pollack-Stevens. (Cf. Bellaiche.)

Lp,†(E , s) :=

(1− 1

Lp,α(E , s)−(

1− 1

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).

Lp,†(E , s) :=

(1− 1

Lp,α(E , s)−(

1− 1

Lp,β(E , s).

L′p,†(E , 1) =α− β

[ϕω, ω]log2

ω(P).

Lp,†(E , s) :=

(1− 1

Lp,α(E , s)−(

1− 1

Lp,β(E , s).

L′p,†(E , 1) =α− β

[ϕω, ω]log2

ω(P).

The work of Kurihara and Pollack

M. Kurihara and R. Pollack combine Perrin-Riou’s conjecture andthe Pollack-Stevens algorithm to compute rational points onelliptic curves p-adically, when p is a supersingular prime.

Example: The curve X0(17) is supersingular at p = 3.

X0(17)193 : y 2 + xy + y = x3 − x2 − 25609x − 99966422

(x , y) =(

91539466284524727125061097283236 , −878088421712236204458830141

125458509476191439016

See M. Kurihara and R. Pollack, ‘Two p-adic L-functions andrational points on elliptic curves with supersingular reduction,L-Functions and Galois Representations (Durham, 2007), 300–332,London Math Society LNS 320.

X0(17)193 : y 2 + xy + y = x3 − x2 − 25609x − 99966422

(x , y) =(

91539466284524727125061097283236 , −878088421712236204458830141

125458509476191439016

X0(17)193 : y 2 + xy + y = x3 − x2 − 25609x − 99966422

(x , y) =(

91539466284524727125061097283236 , −878088421712236204458830141

125458509476191439016

p-adic Rankin L-functions

Let f ∈ S2(Γ0(N), ε) be a modular form of weight two and let Kbe an imaginary quadratic field.

Theorem (Hida)

There is a p-adic analytic function ψ 7→ 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

p-adic Rankin L-functions

Let f ∈ S2(Γ0(N), ε) be a modular form of weight two and let Kbe an imaginary quadratic field.

Theorem (Hida)

There is a p-adic analytic function ψ 7→ 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

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-adicinterpolation 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 Heegnerpoint P ∈ E (K ) and ω ∈ Ω1(E/K ) such that

Lp(f , χtriv) = log2ω(P).

No p-adic period is involved in this formula.

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 canbe 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) satisfyingηη∗ = χtriv. Then there exists a point P ∈ Aη(K ) andω ∈ Ω1(Aη/Kη)Kη such that

Lp(η∗) = (Ω(p)η )−1 log2

ω(P).

Lp(η∗) = (Ω(p)η )−1 log2

ω(P).

Lp(η∗) = (Ω(p)η )−1 log2

ω(P).

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 = NN.

3 L(ψ∗χ−1, 0) 6= 0.

1. The p-adic Gross-Zagier formula, applied to f = θψ:

Lp(θψ, χ) = log2ω(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 .

1 η∗ = ψ−1χ

3 L(ψ∗χ−1, 0) 6= 0.

1 η∗ = ψ−1χ

3 L(ψ∗χ−1, 0) 6= 0.

1 η∗ = ψ−1χ

3 L(ψ∗χ−1, 0) 6= 0.

1 η∗ = ψ−1χ

3 L(ψ∗χ−1, 0) 6= 0.

1 η∗ = ψ−1χ

3 L(ψ∗χ−1, 0) 6= 0.

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 = NN, so that Lp(f , χtriv) = 0. Suppose alsothat p||N and that p is inert in K . Then

ds2Lp(f , χtrivχ

s−)s=0 ∼ log2

ω(P),

for some P ∈ E (K ) and ω ∈ Ω1(E/K ).

ds2Lp(f , χtrivχ

s−)s=0 ∼ log2

ω(P),

ds2Lp(f , χtrivχ

s−)s=0 ∼ log2

ω(P),

ds2Lp(f , χtrivχ

s−)s=0 ∼ log2

ω(P),

ds2Lp(f , χtrivχ

s−)s=0 ∼ log2

ω(P),

Heegner points

More precisely, we have

ds2Lp(f , χtrivχ

s−)s=0 =

∑a⊂OK

dsLp(f , a, s)s=0

where Lp(f , a, s) is a partial anti-cyclotomic p-adic L-functionattached to f and the ideal class a.

For each a, there is a Heegner point Pa ∈ E (H) with

dsLp(f , a, s)s=0 ∼ logω(Pa).

Remark: All the points that have been obtained so far byconsidering p-adic L-functions can, in one way or another, beobtained from Heegner points.

Heegner points

ds2Lp(f , χtrivχ

s−)s=0 =

∑a⊂OK

dsLp(f , a, s)s=0

Heegner points

ds2Lp(f , χtrivχ

s−)s=0 =

∑a⊂OK

dsLp(f , a, s)s=0

Stark-Heegner points

When K is a real quadratic field in which p is inert, and p||N, onecan still make sense of the expressions

dsLp(f , a, s)s=0.

Defining Pa to be the local points in E (Kp) such that

dsLp(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 beobtained otherwise (as far as we know).

dsLp(f , a, s)s=0.

Practical calculations

Stark-Heegner points can be computed in practice, relying on thePollack-Stevens algorithms to compute the relevant p-adicL-functions.

Example: E = X0(17), K = Q(√

> HP,P,hD := stark heegner points(E,197,Qp);

− > Computing the Stark-Heegner points ofdiscriminant 197 over the Elliptic Curve defined byy 2 + xy + y = x3 − x2 − x − 14 over Rational Field.

The calculation is being done in 17-adic field mod17100.

Example: E = X0(17), K = Q(√

Practical calculations, N = p = 17, D = 197.

The discriminant D = 197 has class number 1

1 Computing point attached to the binary quadraticform < 1, 13,−7 >

Sum of the Stark-Heegner points (over Cp) =

(−6310862856360707267736337884716982701323524464440413797

1682682084792961477237437503838942485130056788048891527

129603886134 + O(17100) : . . . : . . .)

1 Computing point attached to the binary quadraticform < 1, 13,−7 >

Sum of the Stark-Heegner points (over Cp) =

(−6310862856360707267736337884716982701323524464440413797

1682682084792961477237437503838942485130056788048891527

129603886134 + O(17100) : . . . : . . .)

This p-adic point is close to the global element

(10148218997865572820035611536889211724013911765602449 , . . .

which is indeed a global point on E (K ).

> load shp37A;

Loading "shp37A"

Loading "../data/M.37.1.plus"

The maximal accuracy currently available is 100 37-adic digits

Please enter the desired accuracy

> load shp37A;

Loading "shp37A"

> load shp37A;

Loading "shp37A"

> load shp37A;

Loading "shp37A"

You are now set up to perform Stark-Heegner pointcalculations on E = Elliptic Curve defined byy 2 + 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 ofdiscriminant 401 over the Elliptic Curve defined byy 2 + y = x3 − x over Rational Field

The calculation is being done in 37-adic field mod3710

1 Computing point attached to <1,19,-10>

2 Computing point attached to <-2,19,5>

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 : 1) is indeed a global point on E(K).

The polynomial satisfied by the x-ccordinates of theStark-Heegner points is

81x5 − 219x4 + 1195x3 − 173x2 − 976x + 527.

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 : 1) is indeed a global point on E(K).

The polynomial satisfied by the x-ccordinates of theStark-Heegner points is

81x5 − 219x4 + 1195x3 − 173x2 − 976x + 527.

Stark-Heegner points and Abel-Jacobi maps

By “staring long and hard” at the formulae for Pa, one gets thefollowing geometric interpretation (in the simplest case where Ehas 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 topologicalcycle ∆a ⊂ X of real dimension 1.

3 One then defines

Pa“ = ”

∫∂−1∆a

ωF ∈ K×p /qZ = E (Kp).

X := SL2(Z[1/p])\(Hp ×H),

3 One then defines

Pa“ = ”

∫∂−1∆a

X := SL2(Z[1/p])\(Hp ×H),

3 One then defines

Pa“ = ”

∫∂−1∆a

X := SL2(Z[1/p])\(Hp ×H),

3 One then defines

Pa“ = ”

∫∂−1∆a

Magma and p-adic L-functionsQuestion

Can computer algebra systems like Magma use p-adic L-functionsto calculate class fields and Mordell-Weil groups, better than theycould without them?

The settings in which Stark-Heegner points have beenimplemented 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.

Abel-Jacobi maps

Problem

Given a closed r-dimensional cycle ∆ which is null-homologous,and a closed differential ω, compute∫

∂−1∆ω ∈ (C or C×p )/(periods).

The calculation of such Abel-Jacobi type invariants raisesinteresting challenges, both theoretical and practical.

Thank you for your attention!

Abel-Jacobi maps

Problem

Abel-Jacobi maps

Problem

... and sorry for running late!

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