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

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?

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Your guess is not correct because $G$ will never be finite (consider the determinant, which is a power of the cyclotomic character), so neither the lower ramification groups nor their orders make sense. – Kestutis Cesnavicius Dec 22 '13 at 2:51
Ah, I see. Is it some limit of this procedure, or am I completely on the wrong track? – James D. Taylor Dec 22 '13 at 2:54
First, restrict to inertia (the conductor measures ramification afterall). Then $G$ still need not be finite (it will be finite iff the reduction is pot. good), but inertia acts through a finite quotient on the semisimplification $V$ of the $l$-adic Tate module (to see this use Grothendieck's quasi-unipotence thm., for instance) and $\delta$ in your notation (= the Swan conductor) will be the Swan conductor of $V$, which will be given by a formula of like you indicate, except you should start summing from $1$ rather than $0$ and replace $T_l$ by $V$. – Kestutis Cesnavicius Dec 22 '13 at 2:59
One motivation is to define the conductor of an abelian variety over a global field 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. – Noam D. Elkies Dec 22 '13 at 3:03
Kestutis, that sounds exactly like the kind of explanation I want. Where can I read more about this? (In particular I am not familiar with "semisimplification" and "Grothendieck's quasi-unipotence thm".) Noam, in that case what makes the definition of an abelian variety over a global field natural? – James D. Taylor Dec 22 '13 at 3:41

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.

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