Magma 2010 Conference on 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. W¨ uthrich, 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) would agree 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 with the efficient (as well as effective) calculation of the mathematical objects 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?
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...
Explicit class field theory Theorem Given a global field K and an ideal m of O K , there is an abelian extension K [ m ] of K with Gal ( K [ m ] / K ) = { Ideals of O K prime to m } / (1 + m O K ) × , 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.
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 O K . 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 .
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
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?
The Birch and Swinnerton-Dyer conjecture Conjecture ( Birch and Swinnerton-Dyer) If L ( E / K , 1) = 0 , then there exists P K ∈ E ( K ) such that L ′ ( E / K , 1) = ˆ h ( P K ) · ( explicit period ) . The complex L -function does not seem to carry direct information about P K , only about its height. J. Silverman: a priori knowledge of ˆ h ( P K ) can be used to speed up the calculation of P K . 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 Σ K of Hecke characters of K is equipped with a natural p -adic analytic structure. Theorem (Katz) There is a p-adic analytic function ψ �→ L p ( ψ ) on Σ K such that = L ( ψ − 1 , 0) L p ( ψ ) , Ω ( p ) Ω ( ∞ ) ψ ψ for all ψ of type (1 + j , − k ) with j , k ≥ 0 . Here Ω ( p ) and Ω ( ∞ ) are appropriate p -adic and complex periods. ψ ψ
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 C p . Theorem (Rubin) Suppose that L ( ψ, 1) = 0 (and hence L p ( ψ ) = 0 ). There exists a global point P ∈ E ( Q ) and a differential ω ∈ Ω 1 ( E / Q ) such that ψ ) − 1 log 2 L p ( ψ ∗ ) = (Ω ( p ) ω ( P ) .
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 L p ( 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 L p ( E / Q , s ) and its derivatives. (To compute these to accuracy p − M takes time proportional to a polynomial in p and M .) Cf. M. Greenberg’s lecture.
Perrin-Riou’s conjecture Two p -adic L -functions: L p ,α ( E / Q , s ) and L p ,β ( E / Q , s ) x 2 − a p x + p = ( x − α )( x − β ) , ord p ( α ) ≤ ord p ( β ) . ord p ( β ) = 1: Kato-Perrin-Riou; Pollack-Stevens. (Cf. Bellaiche.) � 2 � 2 � 1 − 1 � 1 − 1 L p , † ( E , s ) := L p ,α ( E , s ) − L p ,β ( E , s ) . β α Conjecture (Perrin-Riou) If L p ,α ( E , 1) = 0 , there exists a point P ∈ E ( Q ) and ω ∈ Ω 1 ( E / Q ) such that p , † ( E , 1) = α − β L ′ [ ϕω, ω ] log 2 ω ( P ) .
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 X 0 (17) is supersingular at p = 3. X 0 (17) 193 : y 2 + xy + y = x 3 − x 2 − 25609 x − 99966422 � 915394662845247271 , − 878088421712236204458830141 � ( x , y ) = . 25061097283236 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.
p -adic Rankin L -functions Let f ∈ S 2 (Γ 0 ( 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 �→ L p ( f , ψ ) on Σ K such that = L ( f / K , ψ − 1 , 0) = L ( f ⊗ θ ψ , ∗ ) L p ( f , ψ ) , Ω ( p ) Ω ( ∞ ) Ω ( ∞ ) ψ ψ ψ for all ψ of type (2 + j , − k ) with j , k ≥ 0 .
A second analogue of Rubin’s formula Let χ triv be the norm character on K . It is of type (1 , 1) and hence lies outside the range of p -adic interpolation defining L p ( 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 ∈ Ω 1 ( E / K ) such that L p ( f , χ triv ) = log 2 ω ( 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) A η = associated CM abelian variety over K , End K ( A η ) ⊗ Q = K η , [ K η : K ] = dim( A η ) . 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 ∗ = χ triv . Then there exists a point P ∈ A η ( K ) and ω ∈ Ω 1 ( A η / K η ) K η such that η ) − 1 log 2 L p ( η ∗ ) = (Ω ( p ) ω ( P ) .
A sketch of the proof Choose a pair ( ψ, χ ) of types (1 , 0) and (1 , 1) such that 1 η ∗ = ψ − 1 χ 2 θ ψ ∈ S 2 ( N , ε ) with N = N ¯ N . 3 L ( ψ ∗ χ − 1 , 0) � = 0. 1. The p -adic Gross-Zagier formula, applied to f = θ ψ : L p ( θ ψ , χ ) = log 2 ω ( P ) , where P ∈ A ψ ( H χ ) corresponds to a point in A η ( K ). 2. A factorisation of p -adic L -series: L p ( θ ψ , χ ) = L p ( ψ − 1 χ ) L p ( ψ ∗− 1 χ ) ∼ L p ( η ∗ )Ω ( p ) η .
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 L p ( f , χ ) be the p -adic L -function of type II. Theorem (Bertolini, D (1997)) Suppose that N f = N ¯ N , so that L p ( f , χ triv ) = 0 . Suppose also that p || N and that p is inert in K. Then d 2 ds 2 L p ( f , χ triv χ s − ) s =0 ∼ log 2 ω ( P ) , for some P ∈ E ( K ) and ω ∈ Ω 1 ( E / K ) .
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