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Register a new userOn abelian $2$-ramification torsion modules of quadratic fields
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For a number field $F$ and a prime number $p$, the $mathbb{Z}_p$-torsion module of the Galois group of the maximal abelian pro-$p$ extension of $F$ unramified outside $p$ over $F$, denoted as $mathcal{T}_p(F)$, is an important subject in abelian $p$-ramification theory. In this paper we study the group $mathcal{T}_2(F)=mathcal{T}_2(m)$ of the quadratic field $F=mathbb{Q}(sqrt{ m})$. Firstly, assuming $m>0$, we prove an explicit $4$-rank formula for $mathcal{T}_2(-m)$. Furthermore, applying this formula and exploring the connection of $mathcal{T}_2(-m)$ to the ideal class group of $mathbb{Q}(sqrt{-m})$ and the tame kernel of $mathbb{Q}(sqrt{m})$, we obtain the $4$-rank density of $mathcal{T}_2$-groups of imaginary quadratic fields. Secondly, for $l$ an odd prime, we obtain results about the $2$-divisibility of orders of $mathcal{T}_2(pm l)$ and $mathcal{T}_2(pm 2l)$. In particular we find that $#mathcal{T}_2(l)equiv 2# mathcal{T}_2(2l)equiv h_2(-2l)bmod{16}$ if $lequiv 7bmod{8}$ where $h_2(-2l)$ is the $2$-class number of $mathbb{Q}(sqrt{-2l})$. We then obtain density results for $mathcal{T}_2(pm l)$ and $mathcal{T}_2(pm 2l)$. Finally, based on our density results and numerical data, we propose distribution conjectures about $mathcal{T}_p(F)$ when $F$ varies over real or imaginary quadratic fields for any prime $p$, and about $mathcal{T}_2(pm l)$ and $mathcal{T}_2(pm 2 l)$ when $l$ varies, in the spirit of Cohen-Lenstra heuristics. Our conjecture in the $mathcal{T}_2(l)$ case is closely connected to Shanks-Sime-Washingtons speculation on the distributions of the zeros of $2$-adic $L$-functions and to the distributions of the fundamental units.
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Given a natural number n and a number field K, we show the existence of an integer ell_0 such that for any prime number ellgeq ell_0, there exists a finite extension F/K, unramified in all places above ell, together with a principally polarized abelian variety A of dimension n over F such that the resulting ell-torsion representation rho_{A,ell} from G_F to GSp(A[ell](bar{F})) is surjective and everywhere tamely ramified. In particular, we realize GSp_{2n}(mathbb{F}_ell) as the Galois group of a finite tame extension of number fields F/F such that F is unramified above ell.
Let E/F be a CM field split above a finite place v of F, let l be a rational prime number which is prime to v, and let S be the set of places of E dividing lv. If E_S denotes a maximal algebraic extension of E unramified outside S, and if u is a place of E dividing v, we show that any field embedding E_S to bar{E_u} has a dense image. The unramified outside S number fields we use are cut out from the l-adic cohomology of the simple Shimura varieties studied by Kottwitz and Harris-Taylor. The main ingredients of the proof are then the local Langlands correspondence for GL_n, the main global theorem of Harris-Taylor, and the construction of automorphic representations with prescribed local behaviours. We explain how stronger results would follow from the knowledge of some expected properties of Siegel modular forms, and we discuss the case of the Galois group of a maximal algebraic extension of Q unramified outside a single prime p and infinity.
The Mordell-Weil groups $E(mathbb{Q})$ of elliptic curves influence the structures of their quadratic twists $E_{-D}(mathbb{Q})$ and the ideal class groups $mathrm{CL}(-D)$ of imaginary quadratic fields. For appropriate $(u,v) in mathbb{Z}^2$, we define a family of homomorphisms $Phi_{u,v}: E(mathbb{Q}) rightarrow mathrm{CL}(-D)$ for particular negative fundamental discriminants $-D:=-D_E(u,v)$, which we use to simultaneously address questions related to lower bounds for class numbers, the structures of class groups, and ranks of quadratic twists. Specifically, given an elliptic curve $E$ of rank $r$, let $Psi_E$ be the set of suitable fundamental discriminants $-D<0$ satisfying the following three conditions: the quadratic twist $E_{-D}$ has rank at least 1; $E_{text{tor}}(mathbb{Q})$ is a subgroup of $mathrm{CL}(-D)$; and $h(-D)$ satisfies an effective lower bound which grows asymptotically like $c(E) log (D)^{frac{r}{2}}$ as $D to infty$. Then for any $varepsilon > 0$, we show that as $X to infty$, we have $$#, left{-X < -D < 0: -D in Psi_E right } , gg_{varepsilon} X^{frac{1}{2}-varepsilon}.$$ In particular, if $ell in {3,5,7}$ and $ell mid |E_{mathrm{tor}}(mathbb{Q})|$, then the number of such discriminants $-D$ for which $ell mid h(-D)$ is $gg_{varepsilon} X^{frac{1}{2}-varepsilon}.$ Moreover, assuming the Parity Conjecture, our results hold with the additional condition that the quadratic twist $E_{-D}$ has rank at least 2.
Given a finite endomorphism $varphi$ of a variety $X$ defined over the field of fractions $K$ of a Dedekind domain, we study the extension $K(varphi^{-infty}(alpha)) : = bigcup_{n geq 1} K(varphi^{-n}(alpha))$ generated by the preimages of $alpha$ under all iterates of $varphi$. In particular when $varphi$ is post-critically finite, i.e., there exists a non-empty, Zariski-open $W subseteq X$ such that $varphi^{-1}(W) subseteq W$ and $varphi : W to X$ is etale, we prove that $K(varphi^{-infty}(alpha))$ is ramified over only finitely many primes of $K$. This provides a large supply of infinite extensions with restricted ramification, and generalizes results of Aitken-Hajir-Maire in the case $X = mathbb{A}^1$ and Cullinan-Hajir, Jones-Manes in the case $X = mathbb{P}^1$. Moreover, we conjecture that this finite ramification condition characterizes post-critically finite morphisms, and we give an entirely new result showing this for $X = mathbb{P}^1$. The proof relies on Faltings theorem and a local argument.
Let $K$ be a number field and $S$ a finite set of places of $K$. We study the kernels $Sha_S$ of maps $H^2(G_S,fq_p) rightarrow oplus_{vin S} H^2(G_v,fq_p)$. There is a natural injection $Sha_S hookrightarrow CyB_S$, into the dual $CyB_S$ of a certain readily computable Kummer group $V_S$, which is always an isomorphism in the wild case. The tame case is much more mysterious. Our main result is that given a finite $X$ coprime to $p$, there exists a finite set of places $S$ coprime to $p$ such that $Sha_{Scup X} stackrel{simeq}{hookrightarrow} CyB_{Scup X} stackrel{simeq}{twoheadleftarrow} CyB_X hookleftarrow Sha_X$. In particular, we show that in the tame case $Sha_Y$ can {it increase} with increasing $Y$. This is in contrast with the wild case where $Sha_Y$ is nonincreasing in size with increasing $Y$.
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