• Corpus ID: 244714692

# Division in modules and Kummer theory

@inproceedings{Tronto2021DivisionIM,
title={Division in modules and Kummer theory},
author={Sebastiano Tronto},
year={2021}
}
• S. Tronto
• Published 29 November 2021
• Mathematics
In this work we generalize the concept of injective module and develop a theory of divisibility for modules over a general ring, which provides a general and unified framework to study Kummer-like field extensions arising from commutative algebraic groups. With these tools we provide an effective bound for the degree of the field extensions arising from division points of elliptic curves, extending previous results of Javan Peykar for CM curves and of Lombardo and the author for the non-CM case…

## References

SHOWING 1-10 OF 23 REFERENCES
SOME UNIFORM BOUNDS FOR ELLIPTIC CURVES OVER Q
We give explicit uniform bounds for several quantities relevant to the study of Galois representations attached to elliptic curves E/Q. We consider in particular the subgroup of scalars in the image
Galois theory of iterated endomorphisms
• Mathematics
• 2010
Given an abelian algebraic group A over a global field F, α ∈ A(F), and a prime ℓ, the set of all preimages of α under some iterate of [ℓ] generates an extension of F that contains all ℓ‐power
Kummer theory for number fields and the reductions of algebraic numbers
• Mathematics
International Journal of Number Theory
• 2019
For all number fields the failure of maximality for the Kummer extensions is bounded in a very strong sense. We give a direct proof (without relying on the Bashmakov–Ribet method) of the fact that if
Complex Multiplication Structure of Elliptic Curves
Abstract Letkbe a finite field and letEbe an elliptic curve overk. In this paper we describe, for each finite extensionlofk, the structure of the groupE(l) of points ofEoverlas a module over the
Galois representations attached to abelian varieties of CM type
Let $K$ be a number field, $A/K$ be an absolutely simple abelian variety of CM type, and $\ell$ be a prime number. We give explicit bounds on the degree over $K$ of the division fields
• Mathematics, Computer Science
• 2004
This thesis develops a method for determining entanglement, which describes unexpected additive relations between radicals, and shows how to use these to compute field degrees of radical extensions over the field of rationals.
A New Construction of the Injective Hull
The definition of injectivity, and the proof that every module has an injective extension which is a subextension of every other injective extension, are due to R. Baer [B]. An independent proof
Effective Kummer Theory for Elliptic Curves
• Mathematics
International Mathematics Research Notices
• 2021
Let $E$ be an elliptic curve defined over a number field $K$, let $\alpha \in E(K)$ be a point of infinite order, and let $N^{-1}\alpha$ be the set of $N$-division points of $\alpha$ in
Abelian groups that are direct summands of every containing abelian group
It is a well known theorem that an abelian group G satisfying G = nG for every positive integer n is a direct summand of every abelian group H which contains G as a subgroup. It is the object of this
Radical entanglement for elliptic curves
Let $G$ be a commutative connected algebraic group over a number field $K$, let $A$ be a finitely generated and torsion-free subgroup of $G(K)$ of rank $r>0$ and, for $n>1$, let $K(n^{-1}A)$ be the