Timeline for p-adic L functions from Selmer groups - how canonical are they?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Nov 20, 2023 at 20:34 | comment | added | Chris Wuthrich | Don't know. Maybe classical Stickelberger elements are a good choice of an annihilator of the class group and their definition can be viewed as algebraic. They fit together to a generator of the characteristic ideal of the class group. | |
| Nov 20, 2023 at 20:26 | comment | added | David Loeffler | "In a few instances one can give an algebraic characterisation of the generator of the characteristic ideal" — which cases do you mean, and how? | |
| Nov 20, 2023 at 19:57 | comment | added | Chris Wuthrich | I didn't mean that all Euler systems are algebraic. Not sure how vigorously I would claim that cyclotomic units can be viewed as purely algebraic. In any case, what I meant should be that in a few instances one can give an algebraic characterisation of the generator of the characteristic ideal without reference to L-values. It is interesting to note that in all other cases we cannot do this at the moment; we have to rely on automorphic methods. But I don't see a reason why it is theoretically impossible. I understood this as part of the question; hence my comment. | |
| Nov 20, 2023 at 10:52 | comment | added | David Loeffler | In your example, I would argue that cyclotomic units aren't a purely algebraic concept: they rely on the fact that we can generate the abelian extensions of $\mathbb{Q}$ using values of a transcendental function (the exponential map). | |
| Nov 20, 2023 at 10:49 | comment | added | David Loeffler | I would disagree vigorously with your implicit suggestion that Euler systems are "purely algebraic". In all cases where we have a reasonably canonical construction of an Euler system (not just a proof that some module of Euler systems is nonempty), this construction involves automorphic methods. | |
| Nov 20, 2023 at 9:00 | comment | added | Chris Wuthrich | As expected, your answer is way better and clearer than the questions. One underlying question sort of remains: Is there an purely algebraic characterisation of the generator of the characteristic ideal that ends up being the unique p-adic L-function? In the presence of an Euler system, like the (algebraic) cyclotomic units, one could answer this with yes. | |
| Nov 20, 2023 at 6:37 | comment | added | David Loeffler | @Wojowu I've never found this viewpoint terribly convincing. Yes $Spec O_k$ behaves like a curve with some points removed; but the machinery that purports to "put them back in" is itself treating the Archimedean places in a radically different way from the non-Archimedean ones, so the claim that the resulting object restores the symmetry between arch and non-arch places seems highly suspect to me. | |
| Nov 19, 2023 at 20:08 | comment | added | Wojowu | @DavidLoeffler In the context of the unit theorem, the different places can actually be treated on essentially the same footing - $O_K^\times$ is the group of invertible sections on $Spec\, O_K$, which is a "compact curve" (the set of all places) with $r_1+r_2-1$ places removed. If you remove more places, for a total of $k$ places excluded, you will get a group of rank $k-1$, analogously to a statement for functions on punctured curves. Of course putting those places back in is harder, but I believe there may be some analogous "Arakelov-type" statement for units bounded by $1$ at some places. | |
| Nov 19, 2023 at 10:46 | comment | added | David Loeffler | That's an important principle but it does have limitations – one sees this much earlier on in the theory, with Dirichlet's unit theorem, where the Archimedean signature determines the rank of the unit group in an essential way. | |
| Nov 19, 2023 at 10:25 | comment | added | Asvin | Thank you, that's helpful! It feels strange that we can only meaningfully define p adic L functions (even conjecturally) for those fields with some archimedean condition. Feels very much against the spirit of all the places being on an equal footing. | |
| Nov 19, 2023 at 9:01 | history | answered | David Loeffler | CC BY-SA 4.0 |