12 problems

Let $p$ be a prime and $k\ge 1$ an integer. Let $\chi\colon \mathbb{F}_q^\times\to\mathbb{C}^\times$ be a non-trivial character of order $o=\operatorname{ord}(\chi)$, where $o=p^k$…

Let $p$ be prime, let $n$ be an integer, and for a Dirichlet character $\chi\bmod p$ define the generalized quadratic Gauss sum … where $e(y)=e^{2\pi i y}$. Let $\chi_0$ denote the…

Let $e(x):=e^{2\pi i x}$ for $x\in\mathbb{R}$ and $\check{e}(z):=e(z+\overline{z})$ for $z\in\mathbb{C}$. Let $\mathbb{Q}(i)$ be the Gaussian quadratic field with ring of integers…

Let $\Psi_6(X)$ be the primitive Dirichlet characters of order $6$ with conductor at most $X$, and let $\tau(\chi)$ be the Gauss sum of $\chi$. Sextic Gauss-sum equidistribution co…

For odd integers $k$, consider primitive quartic Dirichlet characters $\chi$ of odd conductor $q$, and let $\tau(\chi)$ denote their Gauss sums. Odd-moment equidistribution conject…

Let $p$ be a sufficiently large prime, let $m$ be a positive integer, and let $n$ be an integer satisfying $(n,p)=1$. Let $\chi_0$ be the principal character modulo $p$, let…

Let $p$ be a sufficiently large prime, let $m$ be a positive integer, let $n$ be an integer, and let $\chi$ range over Dirichlet characters modulo $p$. Set $e(y)=e^{2\pi iy}$. Bag,…

Let $n$ be prime and let $\omega$ be a character of $\mathbb F_{p^n}^{\times}$; write $S(\chi)$ for the corresponding Gauss sum. Let $\hat k=k(p^n-1)/(p-1)$. Prime-characteristic G…

Let $n>m$. Let $\pi$ and $\tau$ be irreducible cuspidal representations of $\operatorname{GL}_n(\mathbb F_q)$ and $\operatorname{GL}_m(\mathbb F_q)$, corresponding to regular chara…

Conjectural Gauss-sum formula. There exists a Boolean function $h_{a,b}(\omega)$ such that

Let $\mathfrak{a}\subseteq\mathfrak{o}$ be a $G$-stable ideal. Write $E_{\mathfrak{a}}$ for the relevant ray-unit group,…

Asymptotic-growth conjecture. There is a constant $c\in(1,3)$ such that