Abstract
We establish an analogue of the classical Polya–Vinogradov inequality for \(GL(2, {\mathbbm {F}}_p)\), where p is a prime. In the process, we compute the ‘singular’ Gauss sums for \(GL(2, {\mathbbm {F}}_p)\). As an application, we show that the collection of elements in \(GL(2,{\mathbbm {Z}})\) whose reduction modulo p are of maximal order in \(GL(2, {\mathbbm {F}}_p)\) and whose matrix entries are bounded by x has the expected size as soon as \(x\gg p^{1/2+\varepsilon }\) for any \(\varepsilon >0\).
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Acknowledgements
This work was started when the second author visited ISI, Kolkata in March, 2016. Both the authors thank ISI and TIFR, Mumbai where much of the work was carried out for excellent working condition. The second author acknowledge the support of the Department of Atomic Energy, Government of India under project no. 12-RD-TFR-RT14001, while being a member of TIFR. The second author thanks MPIM, Bonn for two visits during May of 2018 and 2019, for an excellent working environment allowing the authors to make progress on these questions. It is a pleasure to acknowledge J.-M. Deshouillers, É. Fouvry, E. Ghate, H. Iwaniec, F. Jouve, D. Prasad, O. Ramaré, D.S. Ramana, S. Sen, S. Varma for their interest, suggestions and encouragement. Finally, the authors thank the anonymous reviewer for a careful reading of the manuscript and for several correction and suggestions that led to a better exposition of the work done in this article.
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The second author acknowledge the support of the Department of Atomic Energy, Government of India under project no. 12-RD-TFR-RT14001.
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Communicated by Kannan Soundararajan.
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Ganguly, S., Rajan, C.S. Singular Gauss sums, Polya–Vinogradov inequality for GL(2) and growth of primitive elements. Math. Ann. 386, 943–985 (2023). https://doi.org/10.1007/s00208-022-02413-9
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DOI: https://doi.org/10.1007/s00208-022-02413-9