TY - JOUR

T1 - Quantum modular invariant and Hilbert class fields of real quadratic global function fields

AU - Demangos, L.

AU - Gendron, T. M.

N1 - Funding Information:
We thank the referee for taking the time to carefully read this paper and making suggestions for its improvement. We also thank Federico Pellarin and Dinesh Thakur, who each responded to questions which came up during the writing up of this work. We would like to express our gratitude to the Instituto de Matemáticas (Unidad Cuernavaca) of the Universidad Nacional Autónoma de México, as well as the University of Stellenbosch (and particularly Florian Breuer) for their generous support of L. Demangos during his stays at each institution.
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG part of Springer Nature.

PY - 2021/2

Y1 - 2021/2

N2 - This is the first of a series of two papers in which we present a solution to Manin’s Real Multiplication program (Manin in: Laudal and Piene (eds) The Legacy of Niels Henrik Abel, Springer, Berlin, 2004) —an approach to Hilbert’s 12th problem for real quadratic extensions of Q—in positive characteristic, using quantum analogs of the modular invariant and the exponential function. In this first paper, we treat the problem of Hilbert class field generation. If k= Fq(T) and k∞ is the analytic completion of k, we introduce the quantum modular invariant jqt:k∞⊸k∞as a multivalued, discontinuous modular invariant function. Then if K= k(f) ⊂ k∞ is a real quadratic extension of k and f is a fundamental unit, we show that the Hilbert class field HOK (associated to OK= integral closure of Fq[T] in K) is generated over K by the product of the multivalues of jqt(f).

AB - This is the first of a series of two papers in which we present a solution to Manin’s Real Multiplication program (Manin in: Laudal and Piene (eds) The Legacy of Niels Henrik Abel, Springer, Berlin, 2004) —an approach to Hilbert’s 12th problem for real quadratic extensions of Q—in positive characteristic, using quantum analogs of the modular invariant and the exponential function. In this first paper, we treat the problem of Hilbert class field generation. If k= Fq(T) and k∞ is the analytic completion of k, we introduce the quantum modular invariant jqt:k∞⊸k∞as a multivalued, discontinuous modular invariant function. Then if K= k(f) ⊂ k∞ is a real quadratic extension of k and f is a fundamental unit, we show that the Hilbert class field HOK (associated to OK= integral closure of Fq[T] in K) is generated over K by the product of the multivalues of jqt(f).

KW - Function field arithmetic

KW - Hilbert class field

KW - Quantum j-invariant

UR - http://www.scopus.com/inward/record.url?scp=85101885368&partnerID=8YFLogxK

U2 - 10.1007/s00029-021-00619-4

DO - 10.1007/s00029-021-00619-4

M3 - Article

AN - SCOPUS:85101885368

SN - 1022-1824

VL - 27

JO - Selecta Mathematica, New Series

JF - Selecta Mathematica, New Series

IS - 1

M1 - 13

ER -