Heegner Points Modulo ℓ

William Stein

February 5, 2003

Contents

1 Introduction 1

2 Computation for N = 11 1 2.1 Remember the definition of Heegner points . . . . . . . . . . . . 1 2.2 Find Heegner points in E(K) using complex uniformization … 2 2.3 Reduce modulo ℓ for the first split ℓ and the first inert ℓ in K . . 2 2.4 Find model for X_{0}(11) coming from isogeny polynomial Φ_{11}(X, Y) =

0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.5 Find explicit birational map from Φ_{11} = 0 to E over ℚ . . . . . . 2 2.6 Find points on Φ_{11} = 0 corresponding to Heegner point . . . . . 2 2.7 What are these points? Could we compute them directly? …. 2 2.8 Lessons learned from this computation . . . . . . . . . . . . . . . 2

1 Introduction

2 Computation for N = 11

Let E be X_{0}(11).

2.1 Remember the definition of Heegner points

We do this in the special case X_{0}(11) so it is not necessary to worry about the modular parametrization. Let K = ℚ(\sqrt{D}) be a quadratic imaginary field such that D is squarefree and is a perfect square modulo 4·11. (Thus 11 splits in K.) These should be the fields so that 11 splits in K. The first few such fundamental discriminants D are:

−7, −8, −19, −24, −35, −39, −40, −43, −51, −52, −68, −79, −83, −84, −87, −95

Choose such a K. Fix an ideal I_{1} of the ring of integers O_{1} of K such that O_{1}/I_{1} = ℤ/11. For any positive integer λ let K_{λ} be the ring class field of K with conductor λ. The first few class numbers of the fields given above are:

1, 1, 1, 2, 2, 4, 2, 1, 2, …

so e.g., ℚ(\sqrt{−43}) has class number 1 (this is the last class number one case in the list). Note that K_{1} is the Hilbert class field of K. Suppose (λ, 11) = 1, and let O_{λ} = ℤ + λO_{1}, I_{λ} = I_{1} ∩ O_{λ}. Let z_{λ} be the point of X_{0}(11)(K_{λ}) corresponding to the isogeny ℂ/O_{λ} → ℂ/I_{λ}^{−1}.

Let

P_{1} = Tr_{K_{1}/K}(z_{1}) ∈ X_{0}(11)(K).

This is the Heegner point that we’re after.

2.2 Find Heegner points in E(K) using complex uniformization

2.2.1 D = −7

Using a program I wrote a while ago, we find that a floating point approximation for the Heegner point corresponding to D = −7 is approximately

(0.5000000000000000000236687070 + 1.322875655532295292727422603i, −2.000000000000000004347319996 + 5.291502622129181186268103643i).

We must recognize this as a point in E(ℚ(\sqrt{−7})). Let’s try LLL, i.e., PARI’s algdep: (note capital I)

? x=0.5000000000000000000236687070 + 1.322875655532295292727422603*I
? algdep(x,2)
%7 = x^2 - x + 2

The y-coordinate satisfies y^{2} + 4y + 32 = 0. View K as ℚ(x), where x = (1 + \sqrt{−7})/2 is one of the possible x-coordinates. Then y = 4x − 4. (We know, by factoring over K, that y = 4x − 4 or y = −4x, and can determine which by pluging into the Weierstrass equation.) so, the Heegner point is

(x, 4x − 4) ∈ E(ℚ(\sqrt{−7}))

where x^{2} − x + 2 = 0.

2.3 Reduce modulo ℓ for the first split ℓ and the first inert ℓ in K

2.4 Find model for X_{0}(11) coming from isogeny polynomial Φ_{11}(X, Y) = 0

2.5 Find explicit birational map from Φ_{11} = 0 to E over ℚ

This map is canonical up to ±1, since E has no CM.

2.6 Find points on Φ_{11} = 0 corresponding to Heegner point 2.7 What are these points? Could we compute them di-

rectly? 2.8 Lessons learned from this computation