show · st_group.degree all knowls · up · search:

The degree $d$ of a Sato-Tate group is the degree of the characteristic polynomials of its elements, equivalently, the dimension of the $d\times d$ matrices it contains.

For an abelian variety $A$ over a number field, the degree $d$ of its Sato-Tate group is twice its dimension $g$ as an abelian variety (if $A=\mathrm{Jac}(C)$ is the Jacobian of a curve $C$, then $g$ is also the genus of $C$). The degree $d=2g$ is then also the degree of the characteristic polynomials of the Frobenius endomorphism of the reductions of $A$ modulo good primes.

Authors:
Knowl status:
• Review status: reviewed
• Last edited by Andrew Sutherland on 2021-01-01 15:19:50
Referred to by:
History:
Differences