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

As proved by Weil, $A(K)$ is a finitely generated. This implies that the abelian group $A(K)$ has a unique decomposition of the form
\[
A(K)\simeq \Z^r \oplus T,
\]
where the finite group $T$ is the **torsion subgroup** $T$, and the nonnegative integer $r$ is the **Mordell-Weil rank** of $A$.

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

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by Andrew Sutherland on 2020-01-08 09:07:31

**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
- 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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