The **analytic rank** of an abelian variety is the analytic rank of its L-function $L(A,s)$. The analytic rank of a curve is the analytic rank of its Jacobian. The weak form of the BSD conjecture implies that the analytic rank is equal to the rank of the Mordell-Weil group of the abelian variety.

Analytic ranks are always computed under the assumption that $L(A,s)$ satisfies the Hasse-Weil conjecture (they are not necessarily well-defined otherwise). When $A$ is defined over $\Q$, the parity of the analytic rank is always compatible with the sign of the functional equation.

In general, analytic ranks stored in the LMFDB are only upper bounds on the true analytic rank (they could be incorrect if $L(A,s)$ has a zero very close to but not on the central point). For abelian varieties over $\Q$ of analytic rank less than 2 this upper bound is necessarily tight, due to parity.

**Knowl status:**

- Review status: reviewed
- Last edited by John Voight on 2020-01-28 12:53:30

**Referred to by:**

- g2c.587.a.587.1.top
- g2c.analytic_sha
- g2c.bsd_invariants
- g2c.conditional_mw_group
- rcs.rigor.g2c
- lmfdb/genus2_curves/main.py (line 403)
- lmfdb/genus2_curves/main.py (line 587)
- lmfdb/genus2_curves/templates/g2c_curve.html (line 177)
- lmfdb/genus2_curves/templates/g2c_isogeny_class.html (lines 33-35)
- lmfdb/genus2_curves/templates/g2c_search_results.html (line 18)

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

- 2020-01-28 12:53:30 by John Voight (Reviewed)
- 2020-01-17 12:13:11 by Andrew Sutherland
- 2020-01-17 12:07:56 by Andrew Sutherland
- 2020-01-17 11:56:28 by Andrew Sutherland
- 2020-01-07 12:19:06 by John Voight (Reviewed)
- 2020-01-07 12:18:31 by John Voight
- 2020-01-07 12:16:58 by John Voight
- 2020-01-07 12:16:35 by John Voight
- 2020-01-07 06:11:54 by Andrew Sutherland
- 2019-09-05 19:41:12 by Kiran S. Kedlaya
- 2019-05-18 09:02:54 by Andrew Sutherland (Reviewed)
- 2019-05-18 08:51:16 by Andrew Sutherland
- 2018-05-23 16:28:30 by John Cremona (Reviewed)

**Differences**(show/hide)