Quantum \(j\)-invariant in positive characteristic. I: Definition and convergence. (English) Zbl 1343.11068
Arch. Math. 107, No. 1, 23-35 (2016); correction ibid. 111, No. 4, 443-447 (2018).
In [Proc. Lond. Math. Soc. (3) 109, No. 4, 1014–1049 (2014; Zbl 1360.11075)] C. Castaño-Bernard and the second named author introduced a notion of modular
invariant for quantum tori. For any \(\Theta \in {\mathbb R}\) and \(\varepsilon >0\),
\(j^{\text{qt}}(\Theta)\) is defined as \(\lim_{\varepsilon \to
0}\Lambda_{\varepsilon}(\Theta)\) where \(\Lambda_{\varepsilon} (\Theta)\) is the
\(\varepsilon\)-Diophantine approximation \(\Lambda_{\varepsilon} (\Theta)=\{n\in{\mathbb
Z}\mid |n\Theta-m|<\varepsilon \text{\;for some\;} m\in{\mathbb Z}\}\). Because of the
chaotic nature of the sets \(\Lambda_{ \varepsilon}(\Theta)\), no explicit
expressions for \(j^{\text{qt}}(\Theta)\) have been obtained.
The paper under review is the first of a series of two, in which the analogue of \(j^{\text{qt}}\) for function fields over finite fields is considered. In this first paper, the authors introduce the quantum \(j\)-invariant in positive characteristic as a multi-valued, modular-invariant function of a local function field. They give basic definitions and consider questions of convergence. Let \(k= {\mathbb F}_q(T)\) and let \(k_{\infty}\) be the completion of \(k\) with respect to \(1/T\). In this case, since the absolute value is non Archimedean, the analysis simplifies considerably. For \(f\) in this setting, the set \(\Lambda_{\varepsilon}(f)\) has the structure of an \({\mathbb F}_q\)-vector space and it is possible to describe a basis for it using the sequence of denominators of best approximations of \(f\). The main result, Theorem 3, shows that if \(f\in k_{\infty}\setminus k\), then \(|j_{\varepsilon}(f)|=q^{2q-1}\) for all \(\varepsilon < 1\). It is also shown, Theorem 2, that if \(f\in k\), then for \(\varepsilon\) sufficiently small, \(j_{\varepsilon}(f)=\infty\). In particular \(j^{\text{qt}}(f)=\infty\).
With these formulas, the authors prove the multi-valued property of \(j^{\text{qt}}\) and they show that \(f\) is rational if and only if \(j_{\varepsilon}(f)=\infty\). In the second paper of the series [Arch. Math. 107, No. 2, 159–166 (2016; Zbl 06619003)], the authors study the set of values \(j^{\text{qt}}(f)\) for \(f\) quadratic.
The paper under review is the first of a series of two, in which the analogue of \(j^{\text{qt}}\) for function fields over finite fields is considered. In this first paper, the authors introduce the quantum \(j\)-invariant in positive characteristic as a multi-valued, modular-invariant function of a local function field. They give basic definitions and consider questions of convergence. Let \(k= {\mathbb F}_q(T)\) and let \(k_{\infty}\) be the completion of \(k\) with respect to \(1/T\). In this case, since the absolute value is non Archimedean, the analysis simplifies considerably. For \(f\) in this setting, the set \(\Lambda_{\varepsilon}(f)\) has the structure of an \({\mathbb F}_q\)-vector space and it is possible to describe a basis for it using the sequence of denominators of best approximations of \(f\). The main result, Theorem 3, shows that if \(f\in k_{\infty}\setminus k\), then \(|j_{\varepsilon}(f)|=q^{2q-1}\) for all \(\varepsilon < 1\). It is also shown, Theorem 2, that if \(f\in k\), then for \(\varepsilon\) sufficiently small, \(j_{\varepsilon}(f)=\infty\). In particular \(j^{\text{qt}}(f)=\infty\).
With these formulas, the authors prove the multi-valued property of \(j^{\text{qt}}\) and they show that \(f\) is rational if and only if \(j_{\varepsilon}(f)=\infty\). In the second paper of the series [Arch. Math. 107, No. 2, 159–166 (2016; Zbl 06619003)], the authors study the set of values \(j^{\text{qt}}(f)\) for \(f\) quadratic.
Reviewer: Gabriel D. Villa-Salvador (México D. F.)
MSC:
| 11K60 | Diophantine approximation in probabilistic number theory |
| 11R58 | Arithmetic theory of algebraic function fields |
| 11F03 | Modular and automorphic functions |
Citations:
Zbl 1360.11075References:
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