RESEARCH
Seminars on Dwork Theory-3:$p$-adic estimates of exponential sums on curves

2021-01-05　09:00 — 10:00

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Joe Kramer-Miller

University of California, Irvine

Jiyou Li

Zoom No.： 922 327 63848 Passcode: 121212

Let $X$ be a smooth proper curve over a finite field
$\mathbb{F}_q$ of characteristic $p\geq 3$ and let $V \subset X$ be an
affine curve. For a regular function $\overline{f}$ on $V$, we may form
the $L$-function $L(\overline{f},V,s)$ associated to the exponential
sums of $\overline{f}$. We prove a lower estimate on the Newton polygon
of $L(\overline{f},V,s)$. This confirms a conjecture of Deligne that the
"irregular Hodge filtration forces bounds on $p$-adic valuations of
Frobenius eigenvalues. We will explain the main strategy underlying the
proof. This strategy was subsequently applied in joint work with James
Upton to study $\mathbb{Z}_p$-towers over ordinary curves.