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

Question

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 $p$.

Answer 1

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 $p$.

If I had to pick one definition over the other, I'd pick Silverman's. This is related to how the reduction type changes when the base field of the elliptic curve is extended. There is a discussion of this on pages 180 and 181 of Silverman, where the semi-stable reduction theorem is stated and proved.