Let $K$ be a $p$-adic field, and let $A$ be an abelian variety over $K$. The conductor of the abelian variety is often defined as $2u+t+\delta$, where $u$, $t$ and $\delta$ are invariants related to the special fiber of the neron model of $A$ over the ring of integers of $K$. (See for example the wikipedia page: https://en.wikipedia.org/wiki/Conductor_of_an_abelian_variety). While I have seen this definition used in several texts, it has never been made clear to me where this definition came from, and why it is helpful.

What is the motivation for this odd definition? In what sense is it related to the conductor in number theory? In particular, is the following assertion correct?

### Guess pertaining to motivation

Let $T_l$ be the Tate module of $A$, and let $L$ be the unique minimal field over $K$ such that $Gal(L)$ acts trivially on $T_l$. Let $G=Gal(L/K)$, and let the $G_i$'s be the lower numbering of the ramification of $L/K$. Then is the conductor of $A$ defined above the same as $\sum_{i=0}^{\infty} \frac{|G_i|}{|G|}dim(T_l/(T_l^{G_i}))$? If so, where is this proven?

globalfield as a product of local factors. The global conductor is invariant under isogeny (so the local conductor has this property too), and also enters into the functional equation for the $L$-function of the variety. $\endgroup$5more comments