The unit norm index and $p$-class group in certain degree $\ell$ extensions

The period-index problem in WC-groups I: elliptic curves

Let E/K be an elliptic curve defined over a number field, and let p be a prime number such that E(K) has full p-torsion. We show that the order of the p-part of the Shafarevich-Tate group of E/L is unbounded as L varies over degree p extensions of K. The proof uses O'Neil's period-index obstruction. We deduce the result from the fact that, under the same hypotheses, there exist infinitely many elements of the Weil-Chatelet group of E/K of period p and index p^2.

An effective open image theorem for abelian varieties

Fix an abelian variety $A$ of dimension $g\geq 1$ defined over a number field $K$. For each prime $\ell$, the Galois action on the $\ell$-power torsion points of $A$ induces a representation $ρ_{A,\ell}\colon Gal_K \to GL_{2g}(\mathbb{Z}_\ell)$. The $\ell$-adic monodromy group of $A$ is the Zariski closure $G_{A,\ell}$ of the image of $ρ_{A,\ell}$ in $GL_{2g,\mathbb{Q}_\ell}$. The image of $ρ_{A,\ell}$ is open in $G_{A,\ell}(\mathbb{Q}_\ell)$ with respect to the $\ell$-adic topology and hence the index $[G_{A,\ell}(\mathbb{Q}_\ell)\cap GL_{2g}(\mathbb{Z}_\ell): ρ_{A,\ell}(Gal_K)]$ is finite. W…

$Λ$-submodules of finite index of anticyclotomic plus and minus Selmer groups of elliptic curves

Let $p$ be an odd prime and $K$ an imaginary quadratic field where $p$ splits. Under appropriate hypotheses, Bertolini showed that the Selmer group of a $p$-ordinary elliptic curve over the anticyclotomic $\mathbb Z_p$-extension of $K$ does not admit any proper $Λ$-submodule of finite index, where $Λ$ is a suitable Iwasawa algebra. We generalize this result to the plus and minus Selmer groups (in the sense of Kobayashi) of $p$-supersingular elliptic curves. In particular, in our setting the plus/minus Selmer groups have $Λ$-corank one, so they are not $Λ$-cotorsion. As an application of our ma…

The Norm Index Theorem (An Analytic Proof)

We give an analytic proof of the norm index theorem $[I_:K^* N(I_L)] =[L:K]$ for cyclic extensions of number fields using spectral theory of the idele class group.

Explicit sections on Kuwata's elliptic surfaces

We give explicit generators for subgroups of finite index of the Mordell-Weil groups of several families of elliptic surfaces introduced by Masato Kuwata.

具有 ℓ-核指标的对称群特征标表中的零点

Zeros in the Character Tables of Symmetric Groups with an $\ell$-Core Index

本文研究对称群特征标表中以 ℓ-核分拆为指标的零元素个数，对素数 ℓ ≥ 5 证明了 Z_ℓ(n) 及双 ℓ-核情形 Z*_ℓ(n) 的渐近公式，并给出 ℓ-核指标对均为零的充分条件。