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 by .
Theorem 4.1. Let be a complete local discrete valuation ring with quotient field and residue field . Set , and let be an Abelian scheme. If is invertible on , the reduction map gives us an injection . If is the characteristic of , one has an injection .
The closed immersion induces a reduction map . As is invertible on , is a finite étale group scheme. Since an étale morphism is formally étale and is complete, one has a bijection . Thus, one has an injection
the first equality by the valuative criterion of properness. Passing to the colimit over all invertible on gives us an injection
Now assume to be the valuation ring of the completion of a global field . Then is finite, and choosing such that extends to an Abelian scheme over an open subscheme of the model of . Choose two different primes of . Then the previous discussion gives us an injection .
Thus, we have a (finite) upper bound for . Checking for roots of division polynomials gives all torsion points.
In fact, there is a constant such that for all number fields of degree and all elliptic curves (Theorem of Merel). If , one can take (Theorem of Mazur).
(This is example VII.3.3.2 from [Silverman].) Let . Then , so has good reduction for . One sees that and , so . Since , this must be a point of infinite order.
The torsion subgroup of
Example 4.5 (congruent numbers).
A congruent number is a positive integer which is the area of a right triangle with rational side lengths. One can assume square-free. A natural number is congruent iff has a point with iff it has a point of infinite order (see Example 4.6), and if this is the case, there are infinitely many such triangles.
For an Abelian scheme and a non-archimedean place of good reduction set the reduction of the Néron model at , the residue field.
Let and consider the elliptic curve . It is uniquely determined by , so we can assume to be fourth-power-free. The elliptic curve has discriminant and thus good reduction outside the primes dividing . Let be a prime of good reduction. By [Silverman], V.4.1 (a), the reduction is supersingular iff the coefficient of in is zero. Thus for .
If is a prime of good reduction, the above discussion gives us an injection for . Thus for all not dividing , and it follows that it divides . Since one has a -rational -torsion point , the torsion subgroup is , or .
The torsion subgroup of is the full -torsion subgroup , so all points with , namely .
Thus, if 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 .)
Similarly, one finds for , sixth-power-free:
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