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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 as soon as $xgg p^{1/2+ep}$ for any $ep>0$. In particular, there exist elements in $GL(2,Z)$ with matrix entries that are of the order $O(p^{1/2+ep})$ whose reduction modulo $p$ are primitive elements.
By definition primitive and $2$-primitive elements of a finite field extension $mathbb{F}_{q^n}$ have order $q^n-1$ and $(q^n-1)/2$, respectively. We have already shown that, with minor reservations, there exists a primitive element and a $2$-primiti
In this paper we shall prove a subconvexity bound for $GL(2) times GL(2)$ $L$-function in $t$-aspect by using a $GL(1)$ circle method.
In the past two decades, many researchers have studied {it index $2$} Gauss sums, where the group generated by the characteristic $p$ of the underling finite field is of index $2$ in the unit group of ${mathbb Z}/m{mathbb Z}$ for the order $m$ of the
We prove that $|x-y|ge 800X^{-4}$, where $x$ and $y$ are distinct singular moduli of discriminants not exceeding $X$. We apply this result to the primitive element problem for two singular moduli. In a previous article Faye and Riffaut show that the
Let $ mathfrak{f} $ run over the space $ H_{4k} $ of primitive cusp forms of level one and weight $ 4k $, $ k in N $. We prove an explicit formula for the mixed moment of the Hecke $ L $-function $ L(mathfrak{f}, 1/2) $ and the symmetric square $L$-f