22 problems

Let $\mathbb{Z}[i]$ be the Gaussian integers, let $\widetilde{g}_4(\pi)$ denote the quartic Gauss sum attached to a prime $\pi$, let $N(\pi)$ be its norm, and let $\tau_4(\pi)$ be…

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 $\chi_\pi$ be the cubic residue character associated with a prime element $\pi$ in the Eisenstein integers, let $g(\chi_\pi)$ be its Gauss sum, let $N(\pi)$ denote the norm, an…

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 $o=p^k$…

Loxton's conjecture. The quartic Gauss sum satisfies

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…

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 $p\equiv1\pmod8$ be a prime. Let $K=\mathbb{Q}(\zeta_{p-1},\zeta_p)$, let $\mathfrak{p}$ be a prime ideal of $\mathcal{O}_K$ above $(1-\zeta_p)\mathbb{Z}[\zeta_p]$, and let…

Let $M\in\operatorname{SL}(2,\mathbb{Z})$ be built from a finite number of factors $ST^{\#}$, where each exponent $\#$ is an arbitrary integer. The construction assigns modular mat…

Let $\mathcal{S}$ and $\mathscr{S}$ denote the two $S$-matrices appearing in the Chern–Simons and mapping-torus descriptions, and let $\widetilde{\mathscr{S}}$ be the unsymmetrized…

Let $M$ be decomposed into three factors of the form $ST^{\#}$, and let $\widetilde{\mathscr{T}}$ be the naive matrix obtained without the sign choices in the mapping-torus formula…

RT–$\hat{Z}$ correspondence. The displayed formula is true for all plumbed 3-manifolds. The formula extends the relation between WRT invariants and $\hat{Z}$ invariants from…

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 $n$ be prime, and let $\chi_1,\chi_2$ be regular characters of $\mathbb F_{q^n}^{\times}$. For each character $\eta$ of $\mathbb F_q^{\times}$, regard $\eta$ as inflated to…

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,…

Sextic square-coefficient conjecture. One has

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

Infinitely-many-minimal-values conjecture. One has $A(n)=1$ for infinitely many natural numbers $n$. Numerical evidence in the paper gives $A(19)=A(31)=1$, but the asserted infinit…