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ON MOD 2 ARITHMETIC DIJKGRAAF–WITTEN INVARIANTS FOR CERTAIN REAL QUADRATIC NUMBER FIELDS

Hirano, Hikaru 大阪大学 DOI:10.18910/93066

2023.10

概要

theory for number fields as an arithmetic analogue of the Dijkgraaf–Witten theory for 3manifolds [7], based on the analogies between number rings and 3-manifolds, and primes
and knots in arithmetic topology [13]. We note that Dijkgraaf–Witten theory may be regarded as a 3-dimensional Chern–Simons gauge theory with finite gauge groups. Kim’s
theory is concerned with totally imaginary number fields, since it employs some results on
e´ tale cohomology groups of the ring of integers of totally imaginary number fields, which
no longer hold for number fields with real primes. Therefore, it is desirable to extend Kim’s
theory for number fields with real primes.
In this paper, we extend Kim’s theory for number fields with real primes, by using the
modified e´ tale cohomology groups and the modified e´ tale fundamental groups which take
real primes into account. We then explicitly calculate the mod 2 arithmetic Dijkgraaf–Witten
√
invariants for real quadratic fields Q( p1 p2 · · · pr ), where pi ’s are distinct prime numbers
congruent to 1 mod 4, in terms of the Legendre symbols of pi ’s.
Let us outline the construction of arithmetic Chern–Simons invariants and arithmetic
Dijkgraaf–Witten invariants in the following. Let n be a positive integer and let K be a finite
algebraic number field containing n-th roots of unity. Note that if K has a real prime, then n
must be 2. We choose a primitive n-th root of unity ζn in K, which induces an isomorphism
Z/nZ  μn . Let K denote the ring of integers of K and let X = Spec K denote the prime
spectrum of K . Let X∞ denote the set of infinite primes of K and set X = X  X∞ . Following [3] and [1], we may introduce a Grothendieck topology (site) X e´ t called the Artin–Verdier
site, the topos Sh(X e´ t ) of abelian sheaves on X e´ t , and the modified e´ tale cohomology groups
Hi (X, F) for F ∈ Sh(X e´ t ) and i ≥ 0. These cohomology groups admit the 3-dimensional
2020 Mathematics Subject Classification. Primary 11R37, 81T45; Secondary 11R80, 14F20, 57K31. ...

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Filelds, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978.

Faculty of Mathematics, Kyushu University

744, Motooka, Nishi-ku

Fukuoka, 819–0395

Japan

e-mail: simentos1026@gmail.com

...

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