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Register a new userThe arithmetic of arithmetic Coxeter groups
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In the 1990s, J.H. Conway published a combinatorial-geometric method for analyzing integer-valued binary quadratic forms (BQFs). Using a visualization he named the topograph, Conway revisited the reduction of BQFs and the solution of quadratic Diophantine equations such as Pells equation. It appears that the crux of his method is the coincidence between the arithmetic group $PGL_2({mathbb Z})$ and the Coxeter group of type $(3,infty)$. There are many arithmetic Coxeter groups, and each may have unforeseen applications to arithmetic. We introduce Conways topograph, and generalizations to other arithmetic Coxeter groups. This includes a study of arithmetic flags and variants of binary quadratic forms.
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Motivated by a recent result of Prasad, we consider three stronger notions of arithmetic equivalence: local integral equivalence, integral equivalence, and solvable equivalence. In addition to having the same Dedekind zeta function (the usual notion of arithmetic equivalence), number fields that are equivalent in any of these stronger senses must have the same class number, and solvable equivalence forces an isomorphism of adele rings. Until recently the only nontrivial example of integral and solvable equivalence arose from a group-theoretic construction of Scott that was exploited by Prasad. Here we provide infinitely many distinct examples of solvable equivalence, including a family that contains Scotts construction as well as an explicit example of degree 96. We also construct examples that address questions of Scott, and of Guralnick and Weiss, and shed some light on a question of Prasad.
We prove that there are only finitely many conjugacy classes of arithmetic maximal hyperbolic reflection groups.
We formulate a general question regarding the size of the iterated Galois groups associated to an algebraic dynamical system and then we discuss some special cases of our question.
Given an abelian group $G$, it is natural to ask whether there exists a permutation $pi$ of $G$ that destroys all nontrivial 3-term arithmetic progressions (APs), in the sense that $pi(b) - pi(a) eq pi(c) - pi(b)$ for every ordered triple $(a,b,c) in G^3$ satisfying $b-a = c-b eq 0$. This question was resolved for infinite groups $G$ by Hegarty, who showed that there exists an AP-destroying permutation of $G$ if and only if $G/Omega_2(G)$ has the same cardinality as $G$, where $Omega_2(G)$ denotes the subgroup of all elements in $G$ whose order divides $2$. In the case when $G$ is finite, however, only partial results have been obtained thus far. Hegarty has conjectured that an AP-destroying permutation of $G$ exists if $G = mathbb{Z}/nmathbb{Z}$ for all $n eq 2,3,5,7$, and together with Martinsson, he has proven the conjecture for all $n > 1.4 times 10^{14}$. In this paper, we show that if $p$ is a prime and $k$ is a positive integer, then there is an AP-destroying permutation of the elementary $p$-group $(mathbb{Z}/pmathbb{Z})^k$ if and only if $p$ is odd and $(p,k) otin {(3,1),(5,1), (7,1)}$.
The arithmetic complexity $c(mathscr{A}_{theta})$ of a noncommutative torus $mathscr{A}_{theta}$ measures the rank $r$ of a rational elliptic curve $mathscr{E}(K)cong mathbf{Z}^r oplus mathscr{E}_{tors}$ via the formula $r= c(mathscr{A}_{theta})-1$. The number $c(mathscr{A}_{theta})$ is equal to the dimension of a connected component $V_{N,k}^0$ of the Brock-Elkies-Jordan variety associated to a periodic continued fraction $theta=[b_1,dots, b_N, overline{a_1,dots,a_k}]$ of the period $(a_1,dots, a_k)$. We prove that the component $V_{N,k}^0$ is a fiber bundle over the Fermat-Pell conic $mathscr{Q}$ with the structure group $mathscr{E}_{tors}$ and the fiber an $r$-dimensional affine space. As an application, we evaluate the Tate-Shafarevich group $W (mathscr{E}(K))$ of elliptic curve $mathscr{E}(K)$ in terms of the group $W (mathscr{Q})$ calculated by Lemmermeyer.
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