Reified valuations and adic spectra

P-adic Valuations and K-regular Sequences

Abstract A sequence is said to be k-automatic if the nth term of this sequence is generated by a finite state machine with n in base k as input. Regular sequences were first defined by Allouche and Shallit as a generalization of automatic sequences. Given a prime p and a polynomial f(x) ∈ Qp[x], we consider the sequence {vp(f(n))}n=0, where vp is the p-adic valuation. We show that this sequence...

Continuous Valuations and the Adic Spectrum

Following [Hub93, §3], we introduce the spectrum of continuous valuations Cont(A) for a Huber ring A and the adic spectrum Spa(A,A) for a Huber pair (A,A). We also draw heavily from [Con14; Wed12]. These notes are from the arithmetic geometry learning seminar on adic spaces held at the University of Michigan during the Winter 2017 semester, organized by Bhargav Bhatt. See [Dat17; Ste17] for oth...

Hensel’s lemma, valuations, and p-adic numbers

The P-adic Numbers and Basic Theory of Valuations

In this paper, we aim to study valuations on finite extensions of Q. These extensions fall under a special type of field called a global field. We shall also cover the topics of the Approximation Theorems and the ring of adeles, or valuation vectors.

SOME q-CONGRUENCES RELATED TO 3-ADIC VALUATIONS

In 1992, Strauss, Shallit and Zagier proved that for any positive integer a, 3−1 ∑ k=0 (2k k ) ≡ 0 (mod 3) and furthermore 1 32a 3−1 ∑ k=0 (2k k ) ≡ 1(mod 3). Recently a q-analogue of the first congruence was conjectured by Guo and Zeng. In this paper we prove the conjecture of Guo and Zeng, and also give a q-analogue of the second congruence.