The **Mordell-Weil group** of an abelian variety $A$ over a number field $K$ is its group of $K$-rational points $A(K)$.

Weil, building on Mordell's theorem for elliptic curves over $\Q$, proved that the abelian group $A(K)$ is finitely generated. Thus
\[
A(K)\simeq \Z^r \oplus T,
\]
where $r$ is a nonnegative integer called the **Mordell-Weil rank** of $A$, and $T$ is a finite abelian group called the **torsion subgroup**.

The torsion subgroup $T$ is the product of at most $2g$ cyclic groups, where $g$ is the dimension of $A$.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by Bjorn Poonen on 2022-03-24 16:49:09

**Referred to by:**

- ag.canonical_height
- ag.regulator
- ag.selmer_group
- dq.ec.reliability
- ec.mordell_weil_group
- ec.q.analytic_rank
- g2c.analytic_rank
- g2c.analytic_sha
- g2c.bsd_invariants
- g2c.conditional_mw_group
- g2c.jacobian
- g2c.mw_generator
- g2c.mw_generator_order
- g2c.simple_equation
- g2c.two_selmer_rank
- g2c.two_torsion_field
- modcurve.genus_minus_rank
- modcurve.rank
- rcs.rigor.ec.q
- rcs.rigor.g2c
- rcs.source.g2c
- lmfdb/genus2_curves/templates/g2c_curve.html (lines 154-157)

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

- 2022-03-24 16:49:09 by Bjorn Poonen (Reviewed)
- 2020-01-08 09:07:31 by Andrew Sutherland (Reviewed)
- 2020-01-06 14:20:07 by Andrew Sutherland (Reviewed)
- 2015-08-03 16:01:33 by Andrew Sutherland (Reviewed)

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