We study function fields in one variable over the field with one element $\mathbf{F}_1$. It is proved that the Galois extensions of $\mathbf{Q}$ are isomorphic to the curves over $\mathbf{F}_1$ being understood as the Deitmar schemes. Specifically, one gets explicit formulas linking the genus and the number of cusps of an algebraic curve over the extension $\mathbf{F}_{1^m}$ of order $m\ge 1$ of the field $\mathbf{F}_1$ and the $m$-th roots of unity of the corresponding number field. It follows that elliptic curves with one cusp over $\mathbf{F}_1$ are either the cyclotomic fields or the maximal abelian unramified extensions of the quadratic number fields. Our proof depends on representaion of the Drinfeld modules by the bounded linear operators on a Hilbert space and the crossed product structure of the Cuntz-Krieger algebras.
Let $K$ be a number field with ring of integers $\mathcal{O}_K$, and let $A,B,C \in \mathcal{O}_K \setminus\{0\}$. Denote by $S_K'$ the set of prime ideals of $\mathcal{O}_K$ dividing $2ABC$. Assuming two standard conjectures concerning the modularity of mod-$p$ Galois representations and the Eichler-Shimura correspond...
For each $r\in\{4,6\}$, we construct an explicit one-parameter family of elliptic curves over $\mathbb{Q}(i)$ containing infinitely many pairwise nonisomorphic curves genuinely defined over $\mathbb{Q}(i)$ with $j$-invariant $1728$ and rank exactly $r$. We construct explicit $\mathbb{Q}(i)$-rational points to bound the...
In this paper, we study the unramified Iwasawa module over the cyclotomic $\mathbb{Z}_2$-extension of the real quadratic field $\mathbb{Q}(\sqrt{p_1p_2p_3p_4})$, where $p_1, p_2, p_3$, and $p_4$ are distinct odd prime numbers. We give a criterion for the finiteness of an unramified Iwasawa module of a number field. The...
Researchers studying the inverse Galois problem realized 25 of the 26 sporadic finite simple groups as Galois groups over $\mathbb{Q}$ during 1984--1989. We complete this program by proving that the last remaining sporadic group, the Mathieu group $M_{23}$, occurs as a Galois group over $\mathbb{Q}$. In fact, we produc...
Xiaoyu Huang, Blake Jackson, Kyu-Hwan Lee et al.· 0 citations
We extend Gauss composition and Gauss genus theory over $\mathbf{Z}$ to $\mathbb{F}_{2^n}[T]$, a polynomial ring over a finite field $\mathbb{F}_{2^n}$ of characteristic 2. We find new invariants of binary quadratic forms over $\mathbb{F}_{2^n}[T]$ by using Arf invariant and introduce new definitions of proper equivale...
In this article we generalise a classical Kolyvagin's result on the vanishing of the $p$-part of the Shafarevich--Tate group of an elliptic curve (defined over $\mathbb{Q}$) over an imaginary quadratic field $K$ to modular abelian varieties of $\mathrm{GL}_2$-type (defined over a totally real number field $F$) and a CM...
Luca Mastella, Ahmed Matar, Francesco Zerman· 0 citations
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