For an elliptic curve $E$ defined over a field $K$, the **Mordell-Weil group** of $E/K$ is the group of $K$-rational points of $E$. It is denoted $E(K)$.

This is a special case of the Mordell-Weil group of an abelian variety.

The Mordell-Weil Theorem, first proved by Mordell for elliptic curves defined over $\Q$ and later generalized by Weil to abelian varieties $A$ over general number fields $K$, states that, if $K$ is a number field, then $A(K)$ is a finitely generated abelian group. Its rank is called the **Mordell-Weil rank** of $A$ over $K$.

The Mordell-Weil theorem implies in particular that the torsion subgroup of $E(K)$ is finite, and thus that the torsion order of $E$, one of the BSD invariants, is finite.

**Knowl status:**

- Review status: reviewed
- Last edited by John Cremona on 2020-10-08 06:11:03

**Referred to by:**

- dq.ec.reliability
- ec.canonical_height
- ec.congruent_number_curve
- ec.mordell_weil_theorem
- ec.mw_generators
- ec.q.11.a1.bottom
- ec.q.65.a1.bottom
- ec.q.88.a1.bottom
- ec.q.analytic_rank
- ec.q.regulator
- ec.rank
- ec.regulator
- ec.torsion_order
- ec.torsion_subgroup
- rcs.rigor.ec.q
- lmfdb/ecnf/templates/ecnf-curve.html (line 168)
- lmfdb/elliptic_curves/elliptic_curve.py (line 486)
- lmfdb/elliptic_curves/templates/ec-curve.html (line 89)

**History:**(expand/hide all)

- 2020-10-08 06:11:03 by John Cremona (Reviewed)
- 2019-04-08 06:41:09 by David Roe (Reviewed)
- 2019-02-08 11:18:48 by John Cremona (Reviewed)

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