Related papers: Hecke reciprocity and class groups

Motivated by the work of Liu, we study certain canonical quotients of $G_{\emptyset}^T(K)$ -- the Galois group of the maximal unramified extension of a global field $K$ that is split completely at a finite nonempty set of places in $T$ --…

Let $A$ be an abelian variety over a number field $F$, and suppose that $\mathbb Z[\zeta_n]$ embeds in $\mathrm{End}_{\bar F} A$, for some root of unity $\zeta_n$ of order $n = 3^m$. Assuming that the Galois action on the finite group…

We report on the computation of torsion in certain homology theories of congruence subgroups of SL(4,Z). Among these are the usual group cohomology, the Tate-Farrell cohomology, and the homology of the sharbly complex. All of these theories…

Given a random real quadratic field from $\{ \mathbb{Q}(\sqrt{p}\,) ~|~ p \text{ primes} \}$, the conjectural probability $\mathbb{P}(h=q)$ that it has class number $q$ is given for all positive odd integers $q$. Some related conjectures of…

The purpose of this paper is to give a reciprocity between $U_q(h)$ and $\Cal H_{n,r}$, the Hecke algebra of $(\Bbb Z / r\Bbb Z)\wr \frak S_n$ introduced by Ariki and Koike. Let $K=\Bbb Q(q,u_1,\dots ,u_r)$ be the field of rational…

We extend the unramified class field theory for arithmetic schemes of K. Kato and S. Saito to the tame case. Let $X$ be a regular proper arithmetic scheme and let $D$ be a divisor on $X$ whose vertical irreducible components are normal…

Given an abelian algebraic group $A$ over a global field $F$, $\alpha \in A(F)$, and a prime $\ell$, the set of all preimages of $\alpha$ under some iterate of $[\ell]$ generates an extension of $F$ that contains all $\ell$-power torsion…

Cohen and Lenstra have given a heuristic which, for a fixed odd prime $p$, leads to many interesting predictions about the distribution of $p$-class groups of imaginary quadratic fields. We extend the Cohen-Lenstra heuristic to a…

Let $F$ be a number field, $f$ an algebraic automorphic newform on $\mathrm{GL}(2)$ over $F$, $p$ an odd prime does not divide the class number of $F$ and the level of $f$. We prove that $f$ is determined by its $L$-values twisted by Galois…

The action of the absolute Galois group $\text{Gal}(K^{\text{ksep}}/K)$ of a global field $K$ on a tree $T(\phi, \alpha)$ of iterated preimages of $\alpha \in \mathbb{P}^1(K)$ under $\phi \in K(x)$ with $\text{deg}(\phi) \geq 2$ induces a…

We calculate the murmuration density for the family of Hecke $L$-functions of imaginary quadratic fields associated to non-trivial characters. This density exhibits a universality property like Zubrilina's density for the murmurations of…

The {\em Prym} of a cyclic covering of smooth projective curves is the ``new'' part of the Jacobian: the quotient of the Jacobian of the covering curve by the Jacobians of the intermediate covers. Given a family of such coverings, the…

Let $K = \mathbb Q(\sqrt{-q})$, where $q$ is a prime congruent to $3$ modulo $4$. Let $A = A(q)$ denote the Gross curve over the Hilbert class field $H$ of $K$. In this note we use Magma to calculate the values $L(E/H, 1)$ for all such…

We study cyclotomic quiver Hecke algebras $R^{\Lambda_0}(\beta)$ in type $A^{(2)}_{2\ell}$, where $\Lambda_0$ is the fundamental weight. The algebras are natural $A^{(2)}_{2\ell}$-type analogue of Iwahori-Hecke algebras associated with the…

Let $\mathbb{F}_q$ denote a finite field of odd cardinality $q$, $\mathbb{A}=\mathbb{F}_q[T]$ the polynomial ring over $\mathbb{F}_q$ and $k=\mathbb{F}_q(T)$ the rational function field over $\mathbb{F}_q$. In this paper, we compute the…

Let $G$ be a Frobenius group with an abelian Frobenius kernel $F$ and let $k$ be a finite extension of $\mathbb{Q}$. We obtain an upper bound for the number of degree $|F|$ algebraic extensions $K/k$ with Galois group $G$ with the norm of…

Let $\mathrm{k}=\mathbb{Q}(\sqrt[3]{d},\zeta_3)$, with $d$ a cube-free positive integer. Let $C_{\mathrm{k},3}$ be the $3$-component of the class group of $\mathrm{k}$. By the aid of genus theory, arithmetic proprieties of the pure cubic…

We compute the algebraic K-theory of the Hecke algebra of a reductive p-adic group G using the fact that the Farrell-Jones Conjecture is known in this context. The main tool will be the properties of the associated Bruhat-Tits building and…

Let $p$ be an odd prime. For a number field $K$, we let $K_\infty$ be the maximal unramified pro-$p$ extension of $K$; we call the group $\mathrm{Gal}(K_\infty/K)$ the $p$-class tower group of $K$. In a previous work, as a non-abelian…

In this paper, we use the theory of genus fields to study the Euclidean ideals of certain real biquadratic fields $K.$ Comparing with the previous works, our methods yield a new larger family of real biquadratic fields $K$ having Euclidean…