Timeline for What is the motivation for defining the conductor of an abelian variety?
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12 events
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| Jan 21, 2014 at 15:35 | answer | added | Joe Silverman | timeline score: 6 | |
| Dec 22, 2013 at 18:15 | comment | added | abz | The motivation came from elliptic curves, where there is substantial literature (and history). The definition for abelian varieties is a direct generalization. | |
| Dec 22, 2013 at 14:21 | comment | added | Kestutis Cesnavicius | I'm not sure; I don't know what's a conductor of an abelian group - conductors are typically associated to Galois representations (or close relatives of those). As far as I understand, in most situations a conductor is some sort of numerical measurement of ramification of the Galois representation at hand. | |
| Dec 22, 2013 at 4:45 | comment | added | James D. Taylor | Kestutis, is the description you gave here the motivation for defining a conductor of an abelian group? What (originally) was the purpose of this notion? | |
| Dec 22, 2013 at 4:07 | comment | added | Kestutis Cesnavicius | I don't know of an ideal reference, but you could try reading section 2 of Serre's "Facteurs locaux des fonctions zeta des varietes algebriques (definitions et conjectures)"; there the semisimplification step is done "by hand" and Grothendieck's result is referenced. To get the formula of the type you want from (10) there, express the Swan character as a sum of inductions of augmentation representations of ramification groups and use Frobenius reciprocity. | |
| Dec 22, 2013 at 4:03 | comment | added | Will Sawin | Or one can look at the mod-$l$ Galois representation, the Galois action on the $l$-torsion points. It is easy to see the Swan conductor on that representation is equivalent to the Swan conductor on the full $l$-adic representation. For proving theorems about the Swan conductor, it is often helpful to work mod $l$. | |
| Dec 22, 2013 at 3:41 | comment | added | James D. Taylor | 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? | |
| Dec 22, 2013 at 3:03 | comment | added | Noam D. Elkies | 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. | |
| Dec 22, 2013 at 2:59 | comment | added | Kestutis Cesnavicius | 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 $V$. | |
| Dec 22, 2013 at 2:54 | comment | added | James D. Taylor | Ah, I see. Is it some limit of this procedure, or am I completely on the wrong track? | |
| Dec 22, 2013 at 2:51 | comment | added | Kestutis Cesnavicius | 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. | |
| Dec 22, 2013 at 2:46 | history | asked | James D. Taylor | CC BY-SA 3.0 |