Calculation of the torsion subgroup of Abelian varieties over number fields
Von: rofler
Datum: So. 15. Januar 2017 17:45:24
Thema: Mathematik

4. Calculating the torsion subgroup of Abelian varieties over number fields

Wir beschreiben ein Verfahren, mit dem man die Torsionsuntergruppe Abelscher Varietäten über Zahlkörpern berechnen kann. Als Beispiel berechnen wir die Torsionsuntergruppe der elliptischen Kurven, die beim Kongruente-Zahlen-Problem auftreten.
Dies ist der dritte Teil meiner Artikelserie über rationale Punkte auf Abelschen Varietäten über globalen Körpern, siehe die Links ganz unten.
Denote the torsion subgroup of an Abelian group <math>A</math> by <math>A_{\mathrm{tors}}</math>.

Theorem 4.1. Let <math>R</math> be a complete local discrete valuation ring with quotient field <math>K</math> and residue field <math>k</math>.  Set <math>S = \mathrm{Spec}(R)</math>, and let <math>\mathscr{A}/S</math> be an Abelian scheme.  If <math>m</math> is invertible on <math>S</math>, the reduction map gives us an injection <math>\mathscr{A}[m](K) \hookrightarrow \mathscr{A}(\mathrm{Spec}(k))</math>.  If <math>p</math> is the characteristic of <math>k</math>, one has an injection <math>\mathscr{A}[p'](K) \hookrightarrow \mathscr{A}(\mathrm{Spec}(k))</math>.

The closed immersion <math>\mathrm{Spec}(k) \hookrightarrow S</math> induces a reduction map <math>\mathscr{A}(S) \to \mathscr{A}(\mathrm{Spec}(k))</math>.  As <math>m</math> is invertible on <math>S</math>, <math>\mathscr{A}[m]/S</math> is a finite étale group scheme.  Since an étale morphism is formally étale and <math>S</math> is complete, one has a bijection <math>\mathscr{A}[m](S) = \mathscr{A}[m](\mathrm{Spec}(k))</math>.  Thus, one has an injection

    <math>\mathscr{A}[m](K) = \mathscr{A}[m](S) \stackrel{\cong}{\to} \mathscr{A}[m](\mathrm{Spec}(k)) \hookrightarrow \mathscr{A}(\mathrm{Spec}(k))</math>

the first equality by the valuative criterion of properness.  Passing to the colimit over all <math>m</math> invertible on <math>S</math> gives us an injection

    <math>\mathscr{A}[p'](K) \hookrightarrow \mathscr{A}(\mathrm{Spec}(k)). \quad\quad\square</math>

Corollary 4.2.
Now assume <math>R</math> to be the valuation ring of the completion of a global field <math>K</math>.  Then <math>k = \mathrm{Spec}(\mathbf{F}_q)</math> is finite, and choosing <math>m</math> such that <math>\mathscr{A}</math> extends to an Abelian scheme over an open subscheme <math>S</math> of the model of <math>K</math>.  Choose two different primes <math>v,v'</math> of <math>S</math>. Then the previous discussion gives us an injection <math>A(K)_{\mathrm{tors}} = \mathscr{A}(S)_{\mathrm{tors}} \hookrightarrow \mathscr{A}(k_v) \times \mathscr{A}(k_{v'})</math>.

Remark 4.3.
Thus, we have a (finite) upper bound for <math>A(K)_{\mathrm{tors}}</math>.  Checking for roots of division polynomials gives all torsion points.

In fact, there is a constant <math>C(d)</math> such that <math>|E(K)_{\mathrm{tors}}| \leq C(d)</math> for all number fields <math>K/\mathbf{Q}</math> of degree <math>d</math> and all elliptic curves <math>E/K</math> (Theorem of Merel).  If <math>d = 1</math>, one can take <math>C(1) = 16</math> (Theorem of Mazur).

Example 4.4.
(This is example VII.3.3.2 from [Silverman].)  Let <math>E: y^2 = x^3 + 3</math>.  Then <math>\Delta_E = -2^43^5</math>, so <math>E</math> has good reduction for <math>p \geq 5</math>.  One sees that <math>|\mathscr{E}(\mathbf{F}_5)| = 6</math> and <math>|\mathscr{E}(\mathbf{F}_7)| = 13</math>, so <math>E(\mathbf{Q})_{\mathrm{tors}} = 0</math>.  Since <math>P = (1,2) \in E(\mathbf{Q})</math>, this must be a point of infinite order.

The torsion subgroup of <math>y^2 = x^3 + Dx</math>

Example 4.5 (congruent numbers).
A congruent number is a positive integer <math>n</math> which is the area of a right triangle with rational side lengths.  One can assume <math>n</math> square-free.  A natural number <math>n</math> is congruent iff <math>y^2 = x^3 - n^2x</math> has a point with <math>y \neq 0</math> iff it has a point of infinite order (see Example 4.6), and if this is the case, there are infinitely many such triangles.

Example 4.6.
For <math>A/K</math> an Abelian scheme and <math>v</math> a non-archimedean place of good reduction set <math>A_v := \tilde{A}_v := \mathscr{A} \times k_v</math> the reduction of the Néron model <math>\mathscr{A}</math> at <math>v</math>, <math>k_v = \mathscr{O}_{K,v}/\mathfrak{m}_v</math> the residue field.

Let <math>D \in \mathbf{Q}^\times</math> and consider the elliptic curve <math>E_D: y^2 = x^3 + Dx</math>.  It is uniquely determined by <math>D \in \mathbf{Q}^\times/(\mathbf{Q}^\times)^4</math>, so we can assume <math>D</math> to be fourth-power-free.  The elliptic curve <math>E_D</math> has discriminant <math>\Delta = -4D^3</math> and thus good reduction outside the primes dividing <math>2D</math>.  Let <math>p</math> be a prime of good reduction.  By [Silverman], V.4.1 (a), the reduction <math>\tilde{E}_D</math> is supersingular iff the coefficient of <math>x^{p-1}</math> in <math>(x^3 + Dx)^{(p-1)/2}</math> is zero.  Thus <math>|\tilde{E}_D(\mathbf{F}_p)| = p+1</math> for <math>p \equiv 3 \pmod{4}</math>.

If <math>p</math> is a prime of good reduction, the above discussion gives us an injection <math>E_D(\mathbf{Q})_{\mathrm{tors}} \hookrightarrow \tilde{E}_D(\mathbf{F}_p)</math> for <math>p > 2</math>.  Thus <math>|E_D(\mathbf{Q})_{\mathrm{tors}}| \mid (p+1)</math> for all <math>p \equiv 3 \pmod{4}</math> not dividing <math>D</math>, and it follows that it divides <math>4</math>.  Since one has a <math>\mathbf{Q}</math>-rational <math>2</math>-torsion point <math>(0,0)</math>, the torsion subgroup is <math>\mathbf{Z}/2</math>, <math>(\mathbf{Z}/2)^2</math> or <math>\mathbf{Z}/4</math>.

The torsion subgroup of <math>E_{-n^2}</math> is the full <math>2</math>-torsion subgroup <math>\mathbf{Z}/2 \times \mathbf{Z}/2</math>, so all points with <math>y = 0</math>, namely <math>(0,0), (n,0), (-n,0)</math>.

Thus, if <math>n</math> is a congruent number, one can find a point of infinite order in finite time by inspecting points ordered by their height—but if it is not, one is doomed to search ad infinitum!   (If one does not use the conjecture of Birch and Swinnerton-Dyer or the finiteness of the Tate-Shafarevich group <math>\Sha(E_D/\mathbf{Q})</math>.)

Example 4.7.
Similarly, one finds for <math>E_D: y^2 = x^3 + D</math>, <math>D \in \mathbf{Z}</math> sixth-power-free:

    <math>E_D(\mathbf{Q})_{\mathrm{tors}} = \begin{cases}
                                        \mathbf{Z}/6 & D = 1\\
                                        \mathbf{Z}/3 & D \neq 1, D \in (\mathbf{Q}^\times)^3 \text{ or } D = -432 \\
                                        \mathbf{Z}/2 & D \neq 1, D \in (\mathbf{Q}^\times)^2 \\
                                        0 & \text{otherwise}

1. Teil: "The weak Mordell-Weil theorem and descent" article.php?sid=1635
2. Teil: "Heights" article.php?sid=1643
3. Teil: "Calculating the torsion subgroup of Abelian varieties over number fields" article.php?sid=1775

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