Invariants in Towers of Curves over Finite Fields

Abstract: A $Z_p$ tower of curves in characteristic $p$ is a sequence $C_0, C_1, C_2, ...$ of smooth projective curves over a perfect field of characteristic $p$ such that $C_n$ is a branched cover of $C_{n-1}$ and $C_n$ is a branched Galois $Z/(p^n)$-cover of $C_0$. For nice examples of $Z_p$ towers, the growth of the genus is stable: for sufficiently large $n$, the genus of $C_n$ is a quadratic polynomial in $p^n$. In characteristic $p$, there are additional curve invariants like the a-number which are poorly understood. They describe the group-scheme structure of the $p$-torsion of the Jacobian. I will discuss work in progress with Bryden Cais studying these invariants and suggesting that their growth is also stable in genus stable $Z_p$ towers. This is a new kind of Iwasawa theory for function fields.

number theory

Audience: researchers in the discipline

Comments: To join the talks on January 19th, please register here:

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POINT: New Developments in Number Theory

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