Model-theory of elliptic curves

David Pierce preliminary version: 2005.04.28

These notes are intended for a talk to be given at Bilgi University, Istanbul, April 29, 2005.

Contents

0 Introduction 1 1 Powers of sets 1 2 Structures 2 3 Formulas 4 4 Curves 5 5 Function-fields (optional) 7 6 Elliptic curves 8

0 Introduction

Model-theory: mathematics done ‘self-consciously’—with an eye on language and interconnections. General question: When are two structures mathematically the same (that is, isomorphic)? Necessary model-theoretic condition: when they are elementarily equivalent. When is this condition sufficient? Is it sufficient when the structures are function-fields of curves over an algebraically closed field? —Yes, unless the curves are elliptic curves with complex multiplication.

1 Powers of sets

ω = {0, 1, 2, … }
= closure of {∅} under A ↦ A ∪ {A}
= {∅, {∅}, {∅, {∅}}, … }.

Then 0 = ∅ and 1 = {0} = {∅}. n ∈ ω means n = {0, 1, 2, …, n − 1}, so n ⊂_{f} ω. Let I ⊂_{f} ω, let M ≠ ∅, and define

M^{I} = {functions from I to M}.

Typical element: (a_{i} : i ∈ I) or i ↦ a_{i} or a⃗ . Special case: Elements of M^{n} are also (a_{0}, a_{1}, …, a_{n−1}), and M^{n} itself is

\underbrace{M × ⋯ × M}_{n} .

M^{0} = {0} = 1. M^{1} = M. Each 𝒫(M^{I}) is a Boolean algebra, equipped with:

(0) the operations ∩, ∪ and ^{c};

(1) the distinguished elements ∅ and M^{I}; and (2) the relation ⊆. Special case:

𝒫(M^{0}) = 𝒫({0}) = {0, {0}} = {0, 1} = 2.

One can think of 2 as {F, T} and identify the study of 𝒫(M^{0}) with propositional logic. 𝒫(M^{2}) contains {(a, a) : a ∈ M}, the diagonal Δ_{M}. Let also J ⊂_{f} ω, and let α : I → J. This induces

M^{J} ⟶^{α^{∗}} M^{I}
(b_{j} : j ∈ J) ⟼ (b_{α(i)} : i ∈ I)

and hence

𝒫(M^{J}) ⟶^{α^{∗} or (α^{∗})′′} 𝒫(M^{I})
B ⟼ {α^{∗}(b⃗ ) : b⃗ ∈ B}

as well as

𝒫(M^{I}) ⟶^{α_{∗} or (α^{∗})^{−1}} 𝒫(M^{J})
A ⟼ {b⃗ : α^{∗}(b⃗ ) ∈ A}.

For example, Δ_{M} = α^{∗}M when α : 2 → 1. Also, let ι be the inclusion of n in n + 1. Then

ι^{∗}(b_{0}, …, b_{n−1}, b_{n}) = (b_{0}, …, b_{n−1}).

If B ⊆ M^{n+1}, then ι^{∗}B = {a⃗ ∈ M^{n} : (a⃗ , b) ∈ B for some b in M}. If A ⊆ M^{n}, then ι_{∗}A = A × M.

2 Structures

M becomes a structure 𝔐 when equipped with some (or no):

(0) maps f^{𝔐} from M^{n(f)} to M for some n(f) in ω ∖ {0}; then f^{𝔐} is an n(f)-ary operation on M;

(1) distinguished elements c^{𝔐} of M;

(2) subsets R^{𝔐} of M^{n(R)} for some n(R) in ω∖{0}; then R^{𝔐} is an n(R)-ary relation on M.

Then the signature L of 𝔐 consists of the various symbols f, c and R,— names for the corresponding operations, elements and relations. M is the universe of 𝔐, and 𝔐 is an L-structure. For example, ℝ is a structure with signature {+, −, ·, 0, 1, ⩽}. Structures with more than one universe are possible, e.g. vector-spaces. Different structures can have the same signature. Any ordered field is a structure with the same signature as ℝ. Since M = M^{1} and 1 = M^{0}, elements of M are 0-ary operations on M. Any n-ary operation f on M is identified with the (n + 1)-ary relation

{(a⃗ , f(a⃗ )) : a⃗ ∈ M^{n}}.

Hence the operations, distinguished elements and relations of 𝔐 correspond to certain elements of various 𝒫(M^{n}): the primitive relations of 𝔐. Suppose D_{I}^{𝔐} ⊆ 𝒫(M^{I}) and ∐_{I} D_{I}^{𝔐} is the smallest subset X of ∐_{I⊂_{f}ω} 𝒫(M^{I})

such that:

(0) X contains Δ_{M} and each primitive relation of 𝔐;

(1) X ∩ 𝒫(M^{I}) is a sub-algebra of 𝒫(M^{I});

(2) if α : I → J, then α_{∗}(X ∩ 𝒫(M^{I})), α^{∗}(X ∩ 𝒫(M^{J})) ⊆ X. The elements of ∐_{I} D_{I}^{𝔐} are the definable relations of 𝔐.

L-structures 𝔐 and 𝔑 are isomorphic,

𝔐 ≅ 𝔑,

if there is a bijection from M to N taking each primitive relation of 𝔐 to the corresponding relation of 𝔑. 𝔐 and 𝔑 are elementarily equivalent,

𝔐 ≡ 𝔑,

if there is an isomorphism from ∐_{I} D_{I}^{𝔐} to ∐_{I} D_{I}^{𝔑} taking each primitive relation of 𝔐 to the corresponding relation of 𝔑. Then

𝔐 ≅ 𝔑 ⟹ 𝔐 ≡ 𝔑.

Example. All algebraically closed fields of the same characteristic are elementarily equivalent. Their definable sets are the constructible sets over the prime field.

3 Formulas

Every definable relation X of the L-structure 𝔐 has a non-unique name ϕ: a string of symbols from

L ∪ {x_{n} : n ∈ ω} ∪ {=, ∧, ¬, ∃}.

Also symbols from {∨, →, ↔, ∀} can be used. Then ϕ is a formula (of firstorder logic), and X is the interpretation

ϕ^{𝔐}

of ϕ in 𝔐. Dictionary: (Here, n = n(f) = n(R), and α : n → I.)

s s^{𝔐}
x_{k} a⃗ ↦ a_{k} = ι^{∗}(a⃗ ), where ι : {k} ⊆ I
fx_{α(0)} ⋯ x_{α(n−1)} a⃗ ↦ f^{𝔐}(α^{∗}(a⃗ ))
Rx_{α(0)} ⋯ x_{α(n−1)} α_{∗}(R^{𝔐})
= Δ_{M}
∧ ∩
¬ c
∃x_{k} ϕ ι^{∗}(ϕ^{𝔐}), where ι : I ∖ {k} ⊆ I

If ϕ^{𝔐} ⊆ M^{0}, then ϕ is a sentence σ. If σ^{𝔐} = 1, then σ is true in 𝔐:

𝔐 ⊨ σ.

So truth is a relation between sentences and structures. Let Th(𝔐) = {σ : 𝔐 ⊨ σ}, the theory of 𝔐; then

𝔐 ≡ 𝔑 ⟺ Th(𝔐) = Th(𝔑).

4 Curves

Let K = K^{alg}. (Perhaps K = ℂ.) If K ⊆ L, let

𝔸^{n}(L) = L^{n}.

Any irreducible p in K[X, Y] determines a curve C over K:

C(L) = {(x, y) ∈ 𝔸^{2}(L) : p(x, y) = 0}.

Let (α, β) ∈ C and {α, β} ⊈ K; then (α, β) is a generic point of C over K. The field of rational functions on C over K, denoted

K(C),

is generated by

\begin{matrix}(x, y) & ⟼ x \\ (x, y) & ⟼ y \end{matrix}\} : C ⟶ 𝔸.

these are coordinates of a generic point of C; hence

K(C) ≅ K(α, β).

Say also D is a curve over K, with generic point (γ, δ), and

h : K(γ, δ) ⟶ K(α, β)

over K. Let h(γ) = f(α, β) and h(δ) = g(α, β). Then

(x, y) ⟼ (f(x, y), g(x, y)) : C −−→ D,

a dominant rational map (its image contains a generic point). Any such map ϕ induces the K-embedding ϕ^{∗} of K(D) in K(C) given by

ϕ^{∗}(f) = f ∘ ϕ.

Then

deg ϕ = [K(C) : ϕ^{∗}K(D)].

Example. K(𝔸^{1}) ≅ K(X). Then

deg(x ↦ x^{n} : 𝔸^{1} → 𝔸^{1}) = [K(X) : K(X^{n})] = n.

Example. Let C be given by x^{2} + y^{2} = 1, and let

ϕ : (x, y) ⟼ \frac{y}{1 + x} : C ⟶ 𝔸.

[figure: (−1, 0); (x, y)]

Let (α, β) be a generic point of C; then

ϕ^{∗} : f(X) ⟼ f (\frac{β}{1 + α}) : K(X) ⟶ K(α, β);

deg ϕ = [K(α, β) : K (\frac{β}{1 + α})] = 1

since ϕ^{∗} is invertible: If t = β/(1 + α), then

t^{2} = \frac{β^{2}}{(1 + α)^{2}} = \frac{1 − α^{2}}{(1 + α)^{2}} = \frac{1 − α}{1 + α};  α = \frac{1 − t^{2}}{1 + t^{2}} ;  β = \frac{2t}{1 + t^{2}} .

Each curve C has a genus g(C) in ω. (A curve over ℂ is a Riemann surface, hence an orientable surface over ℝ; its genus is the number of holes.) If ϕ : C −−→ D, dominant, then (by the Hurwitz formula)

(0) g(C) > g(D), or

(1) g(C) = g(D) ∈ {0, 1}, or

(2) h is an isomorphism. If g(C) < g(D), then every point of D(K(α, β)) has coordinates in K. Theorem.

• K is a definable subset of K(C).

• If g(C) ≠ 1 or g(D) ≠ 1, then

K(C) ≡ K(D) ⟺ K(C) ≅ K(D).

(Jean-Louis Duret proved this in case char K = 0.)

5 Function-fields (optional)

A function-field over K is K(α_{0}, …, α_{n}) (finitely generated). If L_{i} are such, then

L_{0} ≡ L_{1} ⟹ tr. deg(L_{0}/K) = tr. deg(L_{1}/K)

by the Tsen–Lang Theorem:

A quadratic form over K is a polynomial

x⃗ · A · x⃗ ^{t}

where A^{t} = A with entries from K. Then A is diagonalizable, so by change of variables, the form becomes

∑_{i<n} a_{i}x_{i}^{2}.

By Tsen and Lang, this form has a non-trivial zero from a function-field L over K if and only if

n > 2^{tr. deg(L/K)}.

Every form ax^{2} + by^{2} + cz^{2} has a non-trivial zero if and only if every nontrivial equation ax^{2} + by^{2} = 1 has a solution. So a function-field L over K is the function-field of a curve if and only if

L ⊨ ∀z ∀w ∃x ∃y (zw = 0 ∨ zx^{2} + wy^{2} = 1).

Hence in particular

K(X) ≢ K(X, Y).

6 Elliptic curves

A curve of genus 1 is an elliptic curve. A lattice is a subgroup ⟨ω_{0}, ω_{1}⟩ of ℂ, where ω_{0}ω_{1} ≠ 0 and ω_{0}/ω_{1} ∉ ℝ. Over ℂ, an elliptic curve is a torus

ℂ/Λ,

Λ a lattice. Then we may assume Λ = ⟨1, τ⟩ for some τ in ℌ. How is ℂ/Λ a curve? The Weierstraß ℘-function for Λ is given by

℘(z) = ℘(z; Λ) = \frac{1}{z^{2}} + ∑_{ω∈Λ∖\{0\}}(\frac{1}{(z − ω)^{2}} − \frac{1}{ω^{2}} ) .

℘ is doubly periodic:

℘(z + ω) = ℘(z)

if ω ∈ Λ. So ℘ is well-defined on ℂ/Λ. Now let

G_{k} = G_{k}(Λ) = ∑_{ω∈Λ∖\{0\}} \frac{1}{ω^{2k}} ,

and let E be the curve given by

y^{2} = 4x^{3} − 60G_{2}x − 140G_{3}.

Then (℘, ℘′) ∈ E, so ℂ(E) = ℂ(℘, ℘′), and there is an isomorphism

z ⟼ (℘(z), ℘′(z)) : ℂ/Λ −−→ E.

The induced group-structure of E is given by polynomials:

4(℘(a) + ℘(b) + ℘(a + b)) = λ^{2},

where

λ = \begin{cases}\frac{℘′(b) − ℘′(a)}{℘(b) − ℘(a)}, & if a ≠ b; \\ ℘′′(a), & if a = b. \end{cases}

Let E_{i} be ℂ/Λ_{i}. A non-zero homomorphism from E_{0} to E_{1} is an isogeny and corresponds to α in ℂ^{×} such that

αΛ_{0} ⊆ Λ_{1};

the degree of the isogeny is |Λ_{1}/αΛ_{0}|. Any integer induces an endomorphism of ℂ/Λ; if any other complex numbers do, then ℂ/Λ has complex multiplication.

Theorem. Let E_{i} be elliptic curves over K algebraically closed. The following are equivalent:

(0) There are two isogenies from E_{0} to E_{1} of relatively prime degrees.

(1) K(E_{0}) and K(E_{1}) agree on all sentences

∀x_{0} ∀x_{1} ⋯ ∀x_{n−1} ∃x_{n} ϕ(x⃗ ),

where ϕ is quantifier-free.

(2) K(E_{0}) and K(E_{1}) agree on all ∀∃ sentences.

If one of the E_{i} has no complex multiplication, then the following is equivalent to the foregoing:

(3) K(E_{0}) ≅ K(E_{1}).

If one of the E_{i} does have complex multiplication, and char K = 0, then the following is equivalent to (0) et al.:

(4) End(E_{0}) ≅ End(E_{1}).

(Duret proved (1) ⟺ (3) when char K = 0.)

Relevant facts:

• There are just 13 elliptic curves over ℂ that are determined by their endomorphism-rings.

• Say E = ℂ/ ⟨1, τ⟩. Then

End(E) ≅ {α ∈ ℂ : α ⟨1, τ⟩ ⊆ ⟨1, τ⟩} ⩽ ⟨1, τ⟩ .

If E has complex multiplication, then τ is quadratic (and conversely), since then

(x + Aτ)τ ∈ ⟨1, τ⟩

for some non-zero A. If |A| is minimal, then

End(E) ≅ ⟨1, Aτ̄⟩ .

Example. End(ℂ/ ⟨1, τ⟩) ≅ ⟨1, τ⟩ when τ is i or (1 + i√3)/2.

• Every isogeny α : E_{0} → E_{1} has a dual

α̂ : E_{1} → E_{0}

of the same degree d; then α̂ ∘ α = [d] (multiplication by d).

Ideas of proof:

(1) ⟹ (0). If p always divides [K(E_{0}) : ϕ^{∗}K(E_{1})], then for some E_{2},

ϕ^{∗}K(E_{1}) ⊆ ϕ^{∗}K(E_{2}) ⊆ K(E_{0});
[K(E_{0}) : K(E_{2})] = p.

Then K(E_{0}) says—but K(E_{1}) does not—that every point of E_{1} is the image of a point of some E_{2} under a map of degree p.

(0) ⟹ (2). By (0), when n > 1, some isogeny has degree prime to n. Say K(E_{0}) ⊨ ∀x⃗ ∃y⃗ ϕ(x⃗ , y⃗ ), where ϕ is quantifier-free. Let n be the factorial of the degrees of the polynomials in ϕ, and say

gcd(n, [K(E_{0}) : K(E_{1})]) = 1.

If a⃗ is from K(E_{1}), then ϕ(a⃗ , y⃗ ) must have a solution from K(E_{1}).

(0) ⟹ (4). If α_{i} : E_{0} → E_{1} and deg α_{i} = d_{i} and ∑ a_{i}d_{i} = 1, then

End(E_{1}) ⟶^{≅} End(E_{0})
β ⟼ ∑ a_{i}α̂_{i} ∘ β ∘ α_{i}.

(4) ⟹ (0). Say End(E_{0}) ≅ End(E_{1}). Then we may assume

E_{0} = ℂ/ ⟨1, τ⟩ , E_{1} = ℂ/ ⟨1, nτ⟩ ,
Aτ^{2} + Bτ + C = 0, gcd(A, B, C) = 0, n | A.

Hence

Hom(E_{0}, E_{1}) ≅ ⟨n, Aτ̄⟩ .

If α = nx + Ayτ̄, then

deg(z ↦ αz) = \frac{1}{n} |α| = nx^{2} − Bxy + \frac{AC}{n} y^{2},

gcd (n, B, \frac{AC}{n} ) = 1,

so the degree takes two relatively prime values.