TORSION POINTS OF ELLIPTIC CURVES WITH BAD REDUCTION AT SOME PRIMES II

Title & Authors
TORSION POINTS OF ELLIPTIC CURVES WITH BAD REDUCTION AT SOME PRIMES II
Yasuda, Masaya;

Abstract
Let K be a number field and fix a prime number $\small{p}$. For any set S of primes of K, we here say that an elliptic curve E over K has S-reduction if E has bad reduction only at the primes of S. There exists the set $\small{B_{K,p}}$ of primes of K satisfying that any elliptic curve over K with $\small{B_{K,p}}$-reduction has no $\small{p}$-torsion points under certain conditions. The first aim of this paper is to construct elliptic curves over K with $\small{B_{K,p}}$-reduction and a $\small{p}$-torsion point. The action of the absolute Galois group on the $\small{p}$-torsion subgroup of E gives its associated Galois representation $\small{\bar{\rho}_{E,p}}$ modulo $\small{p}$. We also study the irreducibility and surjectivity of $\small{\bar{\rho}_{E,p}}$ for semistable elliptic curves with $\small{B_{K,p}}$-reduction.
Keywords
reduction of elliptic curves;torsion points;Galois representation;
Language
English
Cited by
1.
Ramification of the Kummer extension generated from torsion points of elliptic curves, International Journal of Number Theory, 2015, 11, 06, 1725
2.
KUMMER GENERATORS AND TORSION POINTS OF ELLIPTIC CURVES WITH BAD REDUCTION AT SOME PRIMES, International Journal of Number Theory, 2013, 09, 07, 1743
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