A well-known conjecture of Artin states that if $a$ is an integer not equal to $0, \pm 1$ or a perfect square, then there exist infinitely many primes $p$ such that $a$ is a primitive root $(\text{ mod } p)$. In this article, we study a generalization of the classical (abelian) large sieve inequality in non-abelian settings. Assuming the non-abelian large sieve inequality, we provide a proof of Artin's primitive root conjecture. Further, using duality techniques, we derive unconditional results towards the conjectured non-abelian large sieve inequality.
In these notes, we introduce the 50-year-old $K(\pi, 1)$ conjecture alongside Coxeter and Artin groups. Roughly speaking, the conjecture states that the complement in $\mathbb{C}^n$ of a"symmetric"configuration of hyperplanes is a $K(\pi, 1)$ space. Our end goal is to present a proof of the conjecture in the so-called...
Giovanni Paolini· Winter Braids Lecture Notes· 0 citations
We show that every faithful, transitive, and generically $(n+2)$-transitive action of a connected group $G$ on an irreducible variety $X$ of dimension $n>0$, all defined over an algebraically closed field $F$, is isomorphic to the natural action of the projective linear group $PGL_{n+1}(F)$ on the projective space $\ma...
Let $G$ be a finite group and let $p$ be a prime. If $P$ is a nonabelian Sylow $p$-subgroup of $G$ and $m(P)$ is the smallest non-linear irreducible character degree of $P$, we prove that there exists $\chi \in {\rm Irr}(G)$ in the principal $p$-block of $G$ such that $1<\chi(1)_p\le m(P)$, giving one inequality of the...
Asier Arranz, Javier G'omez-Serrano, Gabriel Navarro et al.· 1 citation
In 1988, Koblitz conjectured an asymptotic formula for the number of primes $p \le x$ for which the reduction of an elliptic curve over $\mathbb{Q}$ has prime order. Building on the work of Balog, Cojocaru, and David, who proved this conjecture on average in 2011, we extend the result to primes $p$ lying in arithmetic...
A. Güloğlu, Asimina S. Hamakiotes, Sung-Min Lee et al.· 0 citations
Based on a conjecture of Loughran and the second author, we give an explicit prediction for the leading constant in Malle's conjecture for Galois $\mathcal{H}$-extensions of $\mathbb{Q}$ ordered by discriminant, where $\mathcal{H}$ is the $3\times 3$ Heisenberg group over $\mathbb{F}_4$. The predicted leading constant...
Let $C_n$ be a cyclic group of order $n$. We prove that if $(n,6)=1$, then every minimal zero-sum sequence of length four over $C_n$ has index one, thereby resolving the length-four index conjecture. After the gcd reduction, the nonunit case follows from the theorem of Shen-Xia-Li, and the remaining unit case is solved...
Hong-Jian Li, Ping-Zhi Yuan, Shijie Yuan et al.· 0 citations
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