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Singular Gauss sums, Polya-Vinogradov inequality for GL(2) and growth of primitive elements
[article]

Satadal Ganguly, C. S. Rajan

2021
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arXiv
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pre-print

We establish an analogue of *the* classical *Polya*-*Vinogradov* *inequality* for $GL(2, \F_p)$, where $p$ is a prime. In *the* process, we compute *the* 'singular' *Gauss* *sums* for $GL(2, \F_p)$. ...
As an application, we show that *the* collection of elements in $GL(2,\Z)$ whose reduction modulo $p$ are of maximal order in $GL(2, \F_p)$ *and* whose matrix entries are bounded by $x$ has *the* expected size ...
Main ideas behind *the* proofs *and* *the* structure of *the* paper. *The* proof of Theorem 1.4 follows *the* usual approach for proving *the* classical *Polya*-*Vinogradov* *inequality*. ...

arXiv:1912.01310v2
fatcat:h4bbkrji3bawfkf2lrf4gvsp3e