The **Hasse-Weil conjecture** predicts that the L-function $L(A,s)$ of a (positive-dimensional) abelian variety $A$ over a number field $K$ has an analytic continuation to $\C$ with no poles in the critical strip and that it satisfies its functional equation; equivalently, $L(A,s)$ lies in the Selberg class.

The Hasse-Weil conjecture is implied by modularity and is thus known to hold, for example, when $K=\Q$ and $A$ is an elliptic curve or an abelian variety of $\GL_2$-type.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by John Voight on 2020-01-28 12:56:15

**Referred to by:**

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

- 2020-01-28 12:56:15 by John Voight (Reviewed)
- 2020-01-28 12:55:49 by John Voight (Reviewed)
- 2020-01-28 12:54:51 by John Voight
- 2020-01-20 11:33:40 by David Roe
- 2020-01-17 11:52:14 by Andrew Sutherland

**Differences**(show/hide)