Seminar 2 Cartier Points on Curves
Abstract: Let $N$ be a prime number at least 23, and embed the modular curve $X_0(N)$ in its Jacobian $J_0(N)$ by sending a point $P$ to the divisor class of $(P)-(\infty)$, where $\infty$ is the cusp at infinity. A conjecture of Coleman, Kaskel, and Ribet gives a precise guess as to which complex points on $X_0(N)$ map to torsion points in this embedding. In these talks we give a proof of this conjecture.
In the first talk we will give a proof of the conjecture that holds for all $N$ outside of some explicit finite set. The main idea of the proof is that it is much easier to work with the curve $X_0^+(N)$, which is the quotient of $X_0(N)$ by its Atkin-Lehner involution.
In the second talk we explain how to deal with the exceptional cases. The main topic will be a subtle connection (discovered by Robert Coleman) between torsion points on curves and "Cartier points". A Cartier point on a curve $Y$ over an algebraically closed field $k$ of characteristic $p$ is a point $P$ in $Y(k)$ such that the space of regular differentials vanishing at $P$ is fixed by the Cartier operator. In addition to explaining the connection with torsion points on $X_0(N)$, we discuss a theorem about Cartier points on curves which generalizes the following theorem of Ekedahl: If $Y/k$ is superspecial of genus $g$, then $g \leq p(p-1)/2$. [A curve is said to be superspecial if the Cartier operator annihilates all regular differentials, in other words if the Hasse-Witt matrix is zero.]