What is the motivation for defining the conductor of an abelian variety?

Question

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?

Answer 1

The reference to Serre is good, but for a somewhat more elementary introduction (for elliptic curves), you could look at Chapter IV Section 10 of my book Advanced Topics in the Arithmetic of Elliptic Curves. It includes an explanation of why the tame part of the conductor can be read off from the reduction type (good, multiplicative, additive) of the elliptic curve, and why the wild part of the conductor is 0 for primes $p\ge5$. As noted in one of the comments, the wild part of the conductor is defined in terms of the action of inertia (and the higher inertia groups) on the $\ell$-torsion $E[\ell]$, so one can use the definition that you gave with $T_\ell(E)$ replaced by the finite Galois module $E[\ell]$. There's also a proof that the exponent of the conductor $f(E/K)$ over a local field $K$ with normalized valuation $v_K$ satisfies $$ f(E/K)\le 2+3v_K(3)+8v_K(2). $$ (This was generalized to abelian varieties by Lockhart, Rosen, and me [1], and then Brumer and Kramer [2] gave the best possible upper bounds.)

Again for elliptic curves, there is a also the beautiful formula of Ogg [3] and Saito [4] relating the minimal discriminant $\text{Disc}_{E/K}$, the exponent of the conductor $f(E/K)$, and the number of components on the special fiber of the Neron model $m(E/K)$: $$ v_K(\text{Disc}_{E/K}) = f(E/K) + m(E/K) - 1. $$ Ogg proved every case except char($K)=0$ and residue characteristic 2, while Saito gave a more conceptual proof that covers all cases

[1] Lockhart, P., Rosen, M. and Silverman, J., An upper bound for the conductor of an abelian variety J. Alg. Geo. 2 (1993), 569-601.

[2] Brumer, A. and Kramer, K., The conductor of an abelian variety, Compositio Math. (1994).

[3] Ogg, A., Elliptic curves and wild ramification, Amer. J. Math. 89 (1967), 1-21.

[4] Saito, T., Conductor, discriminant, and the Noether formula of arithmetic surfaces, Duke Math. J. 57 (1988), 151-173.