##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory Seminar

## James Upton

#### UC San Diego

## Newton Polygons of Abelian $L$-Functions on Curves

##### Abstract:

Let $X$ be a smooth, affine, geometrically connected curve over a finite field of characteristic $p > 2$. Let $\rho:\pi_1(X) \to \mathbb{C}^\times$ be a character of finite order $p^n$. If $\rho\neq 1$, then the Artin $L$-function $L(\rho,s)$ is a polynomial, and a theorem of Kramer-Miller states that its $p$-adic Newton polygon $\mathrm{NP}(\rho)$ is bounded below by a certain Hodge polygon $\mathrm{HP}(\rho)$ which is defined in terms of local monodromy invariants. In this talk we discuss the interaction between the polygons $\mathrm{NP}(\rho)$ and $\mathrm{HP}(\rho)$. Our main result states that if $X$ is ordinary, then $\mathrm{NP}(\rho)$ and $\mathrm{HP}(\rho)$ share a vertex if and only if there is a corresponding vertex shared by certain ``local" Newton and Hodge polygons associated to each ramified point of $\rho$. As an application, we give a local criterion that is necessary and sufficient for $\mathrm{NP}(\rho)$ and $\mathrm{HP}(\rho)$ to coincide. This is joint work with Joe Kramer-Miller.

### December 2, 2021

### 1:00 PM

### APM 6402 and Zoom; see https://www.math.ucsd.edu/$\sim$nts/

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