Related papers: Algebraic functions and class number formulas

Let $\ell$ be an odd prime, $q$ an odd prime power such that $q \not\equiv 0 \pmod \ell$, and $m$ the order of $q$ in $\F_\ell^\times$. We propose an explicit $L$-polynomial of hyperelliptic function field $K:=\F_q(T,…

Let $K$ be an imaginary quadratic field and $\mathcal{O}$ be an order in $K$. We construct class fields associated with form class groups which are isomorphic to certain $\mathcal{O}$-ideal class groups in terms of the theory of canonical…

Consider the real numbers $$ \ell_{n,k} = \ln\left( \tfrac{3}{2}\,k+\sqrt{\left(\tfrac{3}{2}\,k \right)^2 + 3\,n} \right) $$ and the intervals $\mathcal{L}_{n,k} = \left]\ell_{n,k}-\ln 3,\ell_{n,k}\right]$. For all $n \geq 1$, define $$…

The main result is to show that if $K \ncong \mathbb Q(\sqrt{-15})$ is an imaginary quadratic field and $E$ is an elliptic curve over $K$ with a torsion point of order 16, then the class number of $K$ is divisible by 10. This gives an…

For any odd prime $p,$ we construct an infinite family of pairs of imaginary quadratic fields $\mathbb{Q}(\sqrt{d}),\mathbb{Q}(\sqrt{d+1})$ whose class numbers are both divisible by $p.$ One of our theorems settles Iizuka's conjecture for…

Weinberger in 1972, proved that the ring of integers of a number field with unit rank at least $1$ is a principal ideal domain if and only if it is a Euclidean domain, provided the generalised Riemann hypothesis holds. Lenstra extended the…

Let $D<0$ be a fundamental discriminant and denote by $E(D)$ the exponent of the ideal class group $\text{Cl}(D)$ of $K={\mathbb Q}(\sqrt{D})$. Under the assumption that no Siegel zeros exist we compute all such $D$ with $E(D)$ is a divisor…

Following the natural instinct that when a group operates on a number field then every term in the class number formula should factorize `compatibly' according to the representation theory (both complex and modular) of the group, we are led…

Let $n>1$ be an odd integer. We prove that there are infinitely many imaginary quadratic fields of the form $\mathbb{Q}(\sqrt{x^2-2y^n})$ whose ideal class group has an element of order $n$. This family gives a counter example to a…

Let $\mathbb{A}[n]$ be the group of $n$-torsion points of a commutative algebraic group $\mathbb{A}$ defined over a number field $F$. For a prime ideal $\mathfrak{p}$, we let $N_{\mathfrak{p}}(\mathbb{A}[n])$ be the number of…

Here we constructively classify quadratic $d$-numbers: algebraic integers in quadratic number fields generating Galois-invariant ideals. We prove the subset thereof maximal among their Galois conjugates in absolute value is discrete in…

It is shown that every Leavitt path algebra L of an arbitrary directed graph E over a field K is an arithmetical ring, that is, the two-sided ideals of L form a distributive lattice. It is also shown that L is a multiplication ring, that…

Ordinary binary multiplication of natural numbers can be generalized in a non-trivial way to a ternary operation by considering discrete volumes of lattice hexagons. With this operation, a natural notion of `3-primality' -- primality with…

An integral quadratic lattice is called indefinite $k$-universal if it represents all integral quadratic lattices of rank $k$ for a given positive integer $k$. For $k\geq 3$, we prove that the indefinite $k$-universal property satisfies the…

For any integer $k\ge 1$, we show that there are infinitely many complex quadratic fields whose 2-class groups are cyclic of order $2^k$. The proof combines the circle method with an algebraic criterion for a complex quadratic ideal class…

Let $p$ be an odd prime and $d = p^{\tau}(p-1)$. In the spirit of Aritn's conjecture, consider the system of two diagonal forms of degree $d$ in $s$ variables given by \begin{equation*}\begin{split} a_1x_1^d + \cdots + a_sx_s^d = 0\\…

This paper contains an account of arbitrary cubic function fields of characteristic three. We define a standard form for an arbitrary cubic curve and consider its function field. By considering an integral basis for the maximal order of…

We construct, for imaginary quadratic number fields with class number 1, an arithmetic site of Connes-Consani type. The main difficulty here is that the constructions of Connes and Consani and part of their results strongly rely on the…

We give an algebraic proof of a class number formula for dihedral extensions of number fields of degree $2q$, where $q$ is any odd integer. Our formula expresses the ratio of class numbers as a ratio of orders of cohomology groups of units…

In this article, we investigate factorization problems for twisted triple product Galois representations over real quadratic fields, arising from families of Hilbert cusp forms. Specifically, we address the factorization in two distinct…