# Confusion regarding the definition of semistable reduction of an elliptic curve at a prime $p$

I am consulting the recent paper ''On the Integrality of Modular Symbols and Kato's Euler system for Elliptic Curves'' by Chris Wuthrich. But I am confused regarding the definition of semistable reduction at a prime $p$ of an elliptic curve due to the following two definitions -

1) On page 196 of ''Arithmetic of Elliptic Curves'' by Silverman, it is given that multiplicative reduction is same as semistable reduction.

2) On page 335 of the book ''Elliptic Curves'' by D Husemoeller, the author says an elliptic curve $E$ has semistable reduction at a prime $p$ if and only if $p^2$ does not divide the conductor of the curve.

So kindly suggest the correct definition of semistable reduction of an elliptic curve at a prime $p$.

• If you want to be really precise, semi-stable reduction at a prime $p$ means that the elliptic curve has multiplicative OR good reduction at $p$. May 5, 2014 at 14:05

The criteria given by Silverman and Husemoller are equivalent, and so determining the "correct definition" is a bit tricky. In particular, on page 361 of Arithmetic of Elliptic Curves, Silverman (the first edition) defines the conductor of an elliptic curve, and the exponent on $p$ in that conductor is $1$ if and only if $E$ has multiplicative reduction at $p$.