Abstract
We define a family of pseudodifferential operators on the Hilbert space \(L^2({\mathbb{Q}_p})\) of complex valued square-integrable functions on the \(p\)-adic number field \({\mathbb{Q}_p}\). These generalise the Vladimirov derivative, using the Dirichlet and other suitable multiplicative characters.The Riemann zeta-function and the related Dirichlet \(L\)-functions can be expressed as a trace of these operators on a subspace of \(L^2({\mathbb{Q}_p})\). We also extend this to the \(L\)-functions associated with modular (cusp) forms. Wavelets on \({\mathbb{Q}_p}\) are common sets of eigenfunctions of all these operators.
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Note added
Recently a paper (arXiv:2001.01721 [hep-th]) that discussed the construction of pseudodifferential operators involving multiplicative characters appeared on the arXiv.
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Data sharing is not applicable to this article as no new data were created or analysed in this study.
Notes
The indicator function \(\Omega^{(p)}(\xi) = 1\) for \(|\xi|_p \le 1\), and 0 otherwise.
These operators, together with \(\log_p D = \displaystyle{\lim_{\alpha\to 0}}\frac{D^\alpha - 1}{\alpha\ln p}\), (formally) generate a large symmetry of the wavelets [9].
The extension we need is really quite minimal. The original Dirichlet character is a map \(\chi: \mathbb{Z} \rightarrow \mathbb{C}\). This of course makes sense as a map \(\chi\big|_{(p)} : p^{\mathbb{N}} \rightarrow \mathbb{C}\). We need to extend this restricted form to \(\mathfrak{x} : p^{\mathbb{Z}} \rightarrow \mathbb{C}\). Moreover, only the non-zero values of the characters are precisely defined, the rest was by extension. Therefore, our definition should also be thought of as an extension in the same spirit.
It should also be noted that the extended character we need is really a combination of the ‘norm function’ on \({\mathbb{Q}_p}\) (\(| \cdot|_p : {\mathbb{Q}_p} \rightarrow \mathbb{R}\)) and an extended character on rational numbers. It may be possible to define an extended character on \({\mathbb{Q}_p}\) itself, however, the present construction serves our purpose.
More precisely, the relevant groups are the projective special linear groups PSL(2,\(\mathbb{R}\)) = SL(2,\(\mathbb{R}/\{\pm\}\)) and PSL(2,\(\mathbb{Z}\)) = SL(2,\(\mathbb{Z})/\{\pm\}\).
The cusp forms of a more general modular group \(\Gamma\) vanish as \(z\) approaches certain rational points on \(\mathbb{R} = \partial\mathbb{H}\).
The Hecke operators \(T(m)\), \(m\in\mathbb{N}\), are a set of commuting operators whose action on the modular form is to return the coefficients in the \(q\)-expansion as eigenvalues \(T(m) f(z) = a(m) f(z)\). In other words a modular form is an eigenvector of the Hecke operators with the eigenvalues as the coefficients in its \(q\)-expansion [11, 12, 13, 14].
Similarly, the operators in Eqs. (2.7) and (3.6) may be conjugated to get \(\mathrm{Tr}_{\mathcal{H}_-} \left(a_+^\ell D^{-s}_{\mathfrak{x}} a_-^\ell\right) = \left(\chi(p) p^{-s}\right)^\ell L_p(s,\chi)\), i.e, the \(\ell\)-th coefficient as a prefactor, in the case of the Dirichlet \(L\)-functions.
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Acknowledgments
One of us (DG) presented the results in this paper at the VII-th International Conference on \(p\)-Adic Mathematical Physics & its Applications held at the Universidade Beira Interior, Covilhã, Portugal during Sep 30 – Oct 4, 2019, as well as in the National String Meeting 2019 held at IISER Bhopal, India during Dec 22–27, 2019. We thank the participants of these meetings, especially P. Bradley and W. Zuñiga-Galindo for their comments. We also thank V. Patankar for discussions and the anonymous referee for a critical evaluation and valuable suggestions to improve the manuscript.
Funding
DG is supported in part by the MATRICS grant no. MTR/2020/000481 of the SERB, Department of Science & Technology, Government of India. Note added: Recently a paper (arXiv:2001.01721 [hep-th]) that discussed the construction of pseudodifferential operators involving multiplicative characters appeared on the arXiv. Data availability statement: Data sharing is not applicable to this article as no new data were created or analysed in this study.
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Dutta, P., Ghoshal, D. Pseudodifferential Operators on \({\mathbb{Q}_p}\) and \(L\)-Series. P-Adic Num Ultrametr Anal Appl 13, 280–290 (2021). https://doi.org/10.1134/S2070046621040038
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DOI: https://doi.org/10.1134/S2070046621040038