Let $K$ be an imaginary quadratic number field, let $p$ be an odd prime ($K\neq\mathbb Q(\sqrt{-3})$ if $p=3$), and let $G$ be the Galois group of the maximal everywhere unramified pro-$p$ extension of $K$. To each mod-$p$ character $x$ of $G$ we associate a linear map $D_x$ from $\mathrm{Cl}(K)[p]$ to $\mathrm{Cl}(K)/p$; a formula of Ahlqvist and Carlson expresses it through the class of a norm ideal in the unramified cyclic degree-$p$ extension attached to $x$. These maps determine all triple Massey products on $H^1(G,\mathbb F_p)$, and with them the cubic initial relations of $G$. Suppose the $p$-class rank $d$ of $K$ is at least three. The $(d-1)\times(d-1)$ minors of the family $x\mapsto D_x$ define a subscheme $\Sigma_D$ of $\mathbb P^{d-1}_{\mathbb F_p}$, the norm-degeneracy scheme of $K$. We prove: if the rank condition $\mathrm{rk}\,D_x=d-2$ holds transversally at a point of $\Sigma_D$, over some finite extension of $\mathbb F_p$, then $G$ is mild, and hence of cohomological dimension 2. For $p>3$ transversality means that $\Sigma_D$ is smooth of dimension $d-3$ at the point; at $p=3$ the kernel of the Bockstein map enters as an additional constraint. We treat every imaginary quadratic field of $p$-class rank at least three with $|D_K|<2^{30}$, for every odd prime $p$; only $p=3,5,7$ occur. We prove, unconditionally, that the $p$-class tower group is mild for 864 of the 12749 rank-three fields at $p=3$, for 203 of the 204 fields at $p=5$, and for all three fields at $p=7$; the single rank-four field remains undecided. To the best of our knowledge, these are the first number fields for which the full maximal everywhere unramified pro-$p$ Galois group is proved to be mild; in particular they provide explicit infinite $p$-class towers whose Galois groups have cohomological dimension 2.
Let $K$ be a number field containing a primitive $p$-th root of unity $\zeta_p$. Let $f(x)\in K[x]$ be a monic integral polynomial, and let $f_0$ denote its radical. Let $C/K$ be the superelliptic curve defined by $y^p=f(x)$, and let $J$ be its Jacobian variety. The variety $J$ admits multiplication by $\zeta_p$ over $K$; in particular, the endomorphism induced by $\Pi:=1-\zeta_p$ gives an isogeny of $J$ over $K$. Let $\operatorname{Sel}_\Pi(J)$ denote the Selmer group associated to $\Pi$. Under suitable hypotheses, we obtain bounds for $\operatorname{Sel}_\Pi(J)$ in terms of the $p$-torsion subgroup of the class group of $L:=K[x]/(f_0)$. Several examples illustrating the results are also discussed.
Let $p \geq 5$ be a prime, let $K$ be a finite extension of $\mathbb{Q}_p$, and let $E/K$ be an elliptic curve with good reduction. Let $F_n$ denote the $n$-division polynomial of $E$. Silverman proved that if the reduction is ordinary, then for every $P \in E(K) \setminus \hat{E}(K)$ and a suitable power $q$ of $p$, the sequences $(F_{mq^k}(P))_{k \geq 0}$ converge $p$-adically to limits that are algebraic over the field of definition of $E$. In both the ordinary and supersingular cases, we show that these sequences converge, and determine these limits explicitly in terms of the values of Mumford's algebraic sigma function attached to the Teichm\"uller lift of the prime-to-$p$ torsion component of the reduction of $P$. In particular, the limits are algebraic in the supersingular case as well. As an application, we obtain explicit $p$-adic limit formulas for nonsingular elliptic divisibility sequences.
Let $X$ be a smooth, projective, geometrically connected variety over a field $k$ containing a root of unity of order $p$. If $X$ has a Chern number prime to $p$, we show that every action of a finite $p$-group on $X$ factors through a subgroup of $\operatorname{GL}_n(k)$, where $n=\dim X$. This allows one to transfer properties of representations of finite $p$-groups to their actions on $X$. We deduce a fixed-point theorem which, unlike previously known results of this kind, is sensitive to the arithmetic of the base field. We also obtain a bound on the orders of cyclic $p$-subgroups of the Cremona groups: for instance $\operatorname{Cr}_n(\mathbb{Q})$ contains no element of order $p^2$ when $p \ge n+2$. The method is based on the following observation, of independent interest. For an affine algebraic group $G$ over a field of characteristic zero, $\operatorname{ed}_p(G) + \dim G$ is the least dimension of a smooth projective variety $Y$ with a generically free $G$-action such that the degree map $\operatorname{CH}_G(Y) \to \mathbb{F}_p$ is nonzero. A key input for our result is Karpenko and Merkurjev's computation of the essential $p$-dimension of $p$-groups.
Let $E/\Q$ be an elliptic curve and let $\widetilde K_p$ be the compositum of all $\Z_p$-extensions of a quadratic field $K$. We prove that $E(\widetilde K_p)_{\tors}=E(K)_{\tors}$ for $p\geq5$. For $p=3$ and imaginary quadratic $K\neq\Q(\sqrt{-3})$, torsion on each extension is determined by its intersections with the cyclotomic extension and the $2$-division field. Over $\Q(\sqrt{-3})$, we construct infinitely many non-CM curves with full $3$-torsion in the first anticyclotomic layer and compute the $3$-primary torsion on every slope for eight CM curves. For $p=2$, we bound the odd-primary torsion and exclude all primes greater than $7$. We also give uniform bounds for non-CM primary torsion and correct two assertions in Li's preprint about noncyclotomic $\Z_3$-extensions.
Let $E$ be an elliptic curve over $\mathbb{Q}_p$. We study the field $\mathbb{Q}_p(E[p])$ generated by the $p$-torsion points of $E$. When $E$ has good reduction, we determine not only $\mathbb{Q}_p(E[p])$ but also the field $\mathbb{Q}_p(P)$ for every point $P\in E[p]$. This classification yields criteria for the existence of $p$-torsion over unramified and Lubin--Tate extensions of $\mathbb{Q}_p$. As a global application, let $E/\mathbb{Q}$ be an elliptic curve and let $p$ be an odd prime of good reduction. Define $f_p(E)$ to be the multiplicative order of $a_p(E)$ modulo $p$ in the ordinary case, and set $f_p(E)=p^2-1$ in the supersingular case. We prove that $E(K)[p]=0$ for every number field $K/\mathbb{Q}$ with $[K:\mathbb{Q}]<f_p(E)$. When $E$ has bad reduction, we describe the ramified part of $\mathbb{Q}_p(E[p])/\mathbb{Q}_p$. For every reduction type, we determine the maximal upper ramification break of $\mathbb{Q}_p(E[p])/\mathbb{Q}_p$.
Let $K = k(X)$ be the function field of a smooth geometrically integral variety $X$ of dimension $\geq 2$ over a field $k$ of characteristic 0 and $V$ be the set of discrete valuations of $K$ associated with the prime divisors on $X$. We show that if $D$ is a $k$-defined group of multiplicative type, then the corresponding Tate-Shafarevich group $Sha(D,V) = \ker \left(H^1(K,D) \to \prod_{v \in V} H^1(K_v, D) \right)$ is finite in the following situations: (1) $k$ is finitely generated and $X(k) \neq \emptyset$; (2) $k$ is a number field. This complements previous work of Harari and Szamuely, which considered the case where $X$ is a curve.
Igor A. Rapinchuk, Avinash Roy· 0 citations
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