RESEARCH
Seminars on Dwork Theory-4: Newton Slopes in $\mathbb{Z}_p$-Towers of Curves

2021-01-05　10:00 — 11:15

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James Upton

University of California, San Diego

Jiyou Li

Zoom No.： 922 327 63848 Passcode: 121212

Let $X/\mathbb{F}_q$ be a smooth affine curve over a finite field of characteristic $p > 2$. In this talk we discuss the $p$-adic variation of zeta functions $Z(X_n,s)$ in a pro-covering $X_\infty:\cdots \to X_1 \to X_0 = X$ with total Galois group $\mathbb{Z}_p$. For certain monodromy stable'' coverings over an ordinary curve $X$, we prove that the $q$-adic Newton slopes of $Z(X_n,s)/Z(X,s)$ approach a uniform distribution in the interval $[0,1]$, confirming a conjecture of Daqing Wan.