School of Mathematics and Statistics   Number Theory Seminar

# Ben Smith

## Thursday , 3:05-4:00PM, Carslaw 275

### Explicit real multiplication from cyclotomic coverings of curves

 If $C \to X$ is a covering of curves, then the Jacobian $J_X$ of $X$ is isogenous to an abelian subvariety of the Jacobian $J_C$ of $C$, so the endomorphisms (and particularly, the automorphisms) of $J_C$ induce endomorphisms of $J_X$. In this talk, we will give a simple construction of a one-parameter family of coverings of hyperelliptic curves $C_t \to X_t$, and give explicit, efficiently computable forms for the real multiplications on $X_t$ induced by the automorphisms of $C_t$. These explicit endomorphisms be used to significantly speed up scalar multiplications on $J_{X_t}$.