Mazur's theorem (elliptic curves)

In mathematics , Mazur's theorem describes the possible torsion subsets of elliptic curves over the rational numbers ; it goes back to Barry Mazur .

statement

Let be an elliptic curve and the group of its rational points (for the law of addition defined for points on elliptic curves see the article Elliptic curve ). Then its torsion subgroup (i.e. the group of elements of finite order) is one of the following groups: ${\displaystyle E}$ ${\displaystyle E(\mathbb {Q} )}$

• ${\displaystyle \mathbb {Z} /N\mathbb {Z} }$with or ${\displaystyle 1\leq N\leq 10}$ ${\displaystyle N=12}$
• ${\displaystyle \mathbb {Z} /2\mathbb {Z} \oplus \mathbb {Z} /2N\mathbb {Z} }$with . ${\displaystyle 1\leq N\leq 4}$

annotation

According to Mordell's theorem, is finally generated . From the main theorem about finitely generated Abelian groups it follows that they have the form ${\displaystyle E(\mathbb {Q} )}$

${\displaystyle E(\mathbb {Q} )=\mathbb {Z} ^{r}\oplus T}$

is for one of the torsion groups given above . is called the rank of the elliptic curve. This reflects the fact that emerges from Mordell's conjecture (von Faltings' theorem) that elliptic curves ( topological gender g = 1) over the rational numbers can have an infinite number of solutions (the case of one ) or finitely many (if the rational points all are in the finite torsion groups). The points in have an infinitely high order. ${\displaystyle T}$ ${\displaystyle r}$ ${\displaystyle r>0}$ ${\displaystyle \mathbb {Z} ^{r}}$

From Mazur's theorem and Lutz and Nagell's theorem (see elliptic curve ), an algorithm for determining the torsion groups results, using the fact that, according to Mazur's theorem, the points of finite order ( where , is the neutral element of the law of addition and corresponds to the point in infinity) only the order and can have. ${\displaystyle nP=O}$ ${\displaystyle O}$ ${\displaystyle n=1,2,\ldots ,10}$ ${\displaystyle 12}$

The original proof of Mazur's theorem from 1977 is considered to be extremely difficult and used many techniques of modern algebraic geometry and number theory (theory of group schemes , Néron models , theory of the pitch fields, etc.). Mazur gave a shorter proof in 1978, but it also used more advanced methods. The theorem is considered to be one of the high points of arithmetic algebraic geometry.

All 15 subgroups listed in the sentence occur in an infinite number of elliptic curves.

literature

• Joseph H. Silverman : The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.
• Joseph Silverman: Advanced Topics in the Arithmetic of Elliptic Curves. Springer-Verlag, 1994.
• Barry Mazur : Modular curves and the Eisenstein ideal. IHES Publ. Math. 47: 33-186 (1977). numdam
• Barry Mazur: Rational isogenies of prime degree. Invent. Math. 44: 129-162 (1978). On-line

Web links

• Alexander Schwartz, Elliptic Curves, Group Schemes and Mazur's Theorem, Bachelor thesis, Harvard University 2004, pdf

Individual evidence

1. ^ B. Mazur: Rational isogenies of prime degree, Invent. Math., Vol. 44, 1978, pp. 129-162.
2. ↑ General introduction to the arithmetic of elliptic curves as well as the following volume, Mazur's theorem is only mentioned in passing.