Let $A$ be a non-zero abelian variety defined over a number field $F$. Let $v$ be a finite place of $F$, and let $f_v(A)$ be the usual conductor exponent of $A$ at $v$ (defined e.g. on p.500 of the article of Serre and Tate on good reduction of abelian varieties).
Questions (a): Let $B$ be a non-zero abelian variety over $F$. Is it true that $$f_v(A\times_F B)=f_v(A)+f_v(B)?$$ If not, what is the relation of $f_v(A\times_F B)$ to $f_v(A)$ and $f_v(B)$?
Suppose $A=E$ has dimension one (i.e. $A$ is an elliptic curve). Then $f_v(E)=1$ if and only if $E$ has semi-stable bad reduction at $v$.
Question (b): For arbitrary $A$ as above, can one say something similar (probably involving $dim(A)$) about $f_v$ in the case when $A$ has semi-stable reduction at $v$?