On the indices of curves over local fields

椭圆曲线的可能adelic指数：允许有理循环同源映射的情况

The possible adelic indices for elliptic curves admitting a rational cyclic isogeny

本文证明了对于允许非平凡有理循环同源映射的非CM椭圆曲线，Zywina关于adelic指数的猜想成立，加强了Lemos对Serre统一问题的结果。

Using Indices of Points on an Elliptic Curve to Construct A Diophantine Model of $\Z$ and Define $\Z$ Using One Universal Quantifier in Very Large Subrings of Number Fields, Including $\Q$

Let $K$ be a number field and let $E$ be an elliptic curve defined and of rank one over $K$. For a set $\calW_K$ of primes of $K$, let $O_{K,\calW_K}=\{x\in K: \ord_{\pp}x \geq 0, \forall \pp \not \in \calW_K\}$. Let $P \in E(K)$ be a generator of $E(K)$ modulo the torsion subgroup. Let $(x_n(P),y_n(P))$ be the affine coordinates of $[n]P$ with respect to a fixed Weierstrass equation of $E$. We show that there exists a set $\calW_K$ of primes of $K$ of natural density one such that in $O_{K,\calW_K}$ multiplication of indices (with respect to some fixed multiple of $P$) is existentially defina…

Possible indices for the Galois image of elliptic curves over Q

For a non-CM elliptic curve $E$ over the rationals, the Galois action on its torsion points can be expressed in terms of a Galois representation $ρ_E : G \to GL_2(\hat{\mathbb{Z}})$, where $G$ is the absolute Galois group of the rationals. A well-known theorem of Serre says that the image of $ρ_E$ is open and hence has finite index in $GL_2(\hat{\mathbb{Z}})$. We will study what indices are possible assuming that we are willing to exclude a finite number of possible $j$-invariants from consideration. For example, we will show that there is a finite set $J$ of rational numbers such that if $E/\…

On Abhyankar's lemma about ramification indices

We provide a simple proof of the fact that the ramification index of the compositum of two finite extensions of local fields is equal to the least common multiple of the ramification indices when at least one of the extensions is tamely ramified.

Indices of inseparability for elementary abelian p-extensions

Let K be a local field whose residue field is a finite field of characteristic p, and let L/K be a finite totally ramified Galois extension. Fried and Heiermann defined the "indices of inseparability" of L/K, a refinement of the ramification data of L/K. We give a method for computing the indices of inseparability of the extension L/K in terms of the norm group N_{L/K}(L^*) in the case where K has characteristic p and Gal(L/K) is an elementary abelian p-group with a single ramification break. In some cases our methods lead to simple formulas for the indices of inseparability.

Extensions of local fields given by 3-term Eisenstein polynomials

Let $K$ be a local field with residue characteristic $p$ and let $L/K$ be a totally ramified extension of degree $p^k$. In this paper we show that if $L/K$ has only two distinct indices of inseparability then there exists a uniformizer $π_L$ for $L$ whose minimum polynomial over $K$ has at most three terms. This leads to an explicit classification of extensions with two indices of inseparability. Our classification extends work of Amano, who considered the case $k=1$.