Bull. Korean Math. Soc. 2013; 50(1): 83-96
Printed January 31, 2013
https://doi.org/10.4134/BKMS.2013.50.1.83
Copyright © The Korean Mathematical Society.
Masaya Yasuda
Fujitsu Laboratories Ltd.
Let $K$ be a number field and fix a prime number $p$. 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 $B_{K, p}$ of primes of $K$ satisfying that any elliptic curve over $K$ with $B_{K, p}$-reduction has no $p$-torsion points under certain conditions. The first aim of this paper is to construct elliptic curves over $K$ with $B_{K, p}$-reduction and a $p$-torsion point. The action of the absolute Galois group on the $p$-torsion subgroup of $E$ gives its associated Galois representation $\REP$ modulo $p$. We also study the irreducibility and surjectivity of $\REP$ for semistable elliptic curves with $B_{K, p}$-reduction.
Keywords: reduction of elliptic curves, torsion points, Galois representation
MSC numbers: Primary 14H52; Secondary 14G05
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