Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userStronger arithmetic equivalence
125
0
0.0
(
0
)
Authors
Andrew V. Sutherland
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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.
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.
We prove that there are only finitely many conjugacy classes of arithmetic maximal hyperbolic reflection groups.
Mazur, Rubin, and Stein have recently formulated a series of conjectures about statistical properties of modular symbols in order to understand central values of twists of elliptic curve $L$-functions. Two of these conjectures relate to the asymptotic growth of the first and second moments of the modular symbols. We prove these on average by using analytic properties of Eisenstein series twisted by modular symbols. Another of their conjectures predicts the Gaussian distribution of normalized modular symbols ordered according to the size of the denominator of the cusps. We prove this conjecture in a refined version that also allows restrictions on the location of the cusps.
We study derivations and differential forms on the arithmetic jet spaces of smooth schemes, relative to several primes. As applications we give a new interpretation of arithmetic Laplacians and we discuss the de Rham cohomology of some specific arithmetic jet spaces.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
