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 $K$ be an imaginary quadratic number field, let $p$ be an odd prime, and let $G=G_{\varnothing}(K)(p)$ 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...
Let $X/\mathbb{Q}$ be a smooth, projective, geometrically integral curve of genus $g \geq 2$ with a point $ b \in X(\mathbb{Q})$ and geometrically CM Jacobian. Fix an odd prime $p$ of good reduction, and let $U$ be the \'etale pro-unipotent $\mathbb{Q}_p$-fundamental group of $X$. We prove, for all $n \gg 0,$ the inequ...
Let $K$ be an imaginary quadratic field and let $p \ge 5$ be a prime that is unramified in $K$. Let $\mathcal{A}_f/\mathbb{Q}$ be an abelian variety of $\mathrm{GL}_2$-type associated with a weight-two modular form $f$, with good non-ordinary reduction at $p$, and suppose that $(f,K)$ satisfies the generalized Heegner...
Let $K/\Qp$ be a finite extension, $\bG$ a connected reductive group over $K$, and set $G = \bG(K)$, considered as a $p$-adic Lie group. We show first that the Bernstein center $\cC_G$ of the category of solid locally $\Qp$-analytic representations of $G$ is isomorphic to the center of the locally analytic distribution...
Daniel Le, Matthias Strauch, Zi-Chuan Wang· 0 citations
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 $p$, t...
Let $F$ be a totally real field of degree $n$ and discriminant $\Delta_F$, and let $X_1(\eta)$ be the cover of the Hilbert modular variety parametrizing abelian varieties with real multiplication by $\mathcal O_F$, together with a torsion point having annihilator $\eta$. Let $L=K_{\overline X_1(\eta)}+D$ be the log-can...
Soheil Memariansorkhabi· 1 citation
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
