Math 2.0

Number theory Arithmetic geometry Theorem

Mazur's torsion theorem

Builds on Elliptic curveGroup law on an elliptic curveTorsion subgroupCyclic group

Statement

Let EE be an elliptic curve over ℚ\Q, and let E(ℚ)torsE(\Q)_{\mathrm{tors}} be the torsion subgroup of the group of rational points. Then E(ℚ)torsE(\Q)_{\mathrm{tors}} is isomorphic to one of the following 1515 groups:

  1. the cyclic group ℤ/Nℤ\Z/N\Z with 1≤N≤101 \le N \le 10 or N=12N = 12;
  2. ℤ/2ℤ×ℤ/2Nℤ\Z/2\Z \times \Z/2N\Z with 1≤N≤41 \le N \le 4.

Remarks

Each of the 1515 groups occurs for infinitely many curves with different jj-invariants; this part was known before Mazur, from explicit families. The hard part is that no other group occurs. A rational point of order NN on EE gives a non-cuspidal rational point on the modular curve X1(N)X_1(N). Mazur proved in 1977, with his study of the Eisenstein ideal, that for primes N>7N > 7 the curve X1(N)X_1(N) has no such points. Together with earlier results on the composite orders, this gives the list. Example: on y2=x3+1y^2 = x^3 + 1 the torsion subgroup is cyclic of order 66. Over larger number fields, the list of possible torsion groups is different, but it is still finite for each degree, by a theorem of Merel.

Going deeper

For a student

Together with the Nagell–Lutz theorem, it makes the torsion part of Mordell's theorem fully explicit. The method, through rational points on modular curves, is an early deep use of modular curves in arithmetic.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph
Cyclic group, concept, no evidence levelCyclic groupconceptElliptic curve, concept, no evidence levelElliptic curveconceptGroup law on an elliptic curve, concept, no evidence levelGroup law on an ellip…conceptTorsion subgroup, concept, no evidence levelTorsion subgroupconceptMazur's torsion theorem, statement, no evidence levelMazur's torsion theor…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Cyclic group concept
  2. Elliptic curve concept
  3. Group law on an elliptic curve concept
  4. Torsion subgroup concept
  5. Mazur's torsion theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Torsion_conjecture
About this page
TagRQS6YHAR8N
URIm2://RQS6YHAR8N
Kindstatement, theorem
WritingW0 stub
MSC11G05, 14H52, 11G18
StatusThis copy of the site has no build data. Run m2 status RQS6YHAR8N.
AboutElliptic curve, Torsion subgroup
UsesElliptic curve, Group law on an elliptic curve, Torsion subgroup, Cyclic group
Changed2 November 2026

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