Timeline for Non-vanishing of p-adic L-functions
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
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| Apr 5, 2011 at 1:23 | comment | added | monodromy | @Jupiter: I think there is some confusion in your discussion. | |
| Apr 5, 2011 at 1:15 | comment | added | Jupiter Jones | @Kevin II: corresponding weight 2 modular form and chi runs over characters of p-power conductor. So as long as Mazur knew the Riemann hypothesis, BSD, and the existence of the p-adic L-function (which he did), he could have known that his conjecture was true if f_E was replaced by a modular form of weight strictly greater than 2. In this way, weight 2 forms constitute almost a degenerate case -- there aren't enough critical values to see the whole picture. But I've never seen things cast in that kind of light. Hmm, I wonder if any Iwasawa-type people have read this far and want to chime in. | |
| Apr 5, 2011 at 1:14 | comment | added | Jupiter Jones | @Kevin: I'm not sure about any of it, but I find the whole thing pretty weird. It does indeed seem that one single p-adic L-function encodes all of those L-values. Thus the non-vanishing of L(f,1) implies the non-vanishing of the whole lot of them (except for finitely many). Back in the 70s (?) when Mazur was conjecturing things like the rank of an elliptic curve over the cyclotomic Zp-extension is finite, was this kind of consideration in his mind at all? Namely, that conjecture (by BSD) is equivalent to the non-vanishing of all but finitely many L(f_E,chi,1) where f_E is the | |
| Apr 4, 2011 at 22:39 | comment | added | Rob Harron | @Jupiter Jones: I did not ask him that, though I'd be surprised if he were surprised. | |
| Apr 4, 2011 at 22:08 | comment | added | Rob Harron | @Kevin: There is indeed one p-adic L-function for all the critical twists, see for example section I.14 of Mazur–Tate–Teitelbaum. | |
| Apr 4, 2011 at 18:14 | comment | added | Kevin Buzzard | @Jupiter Jones: are you really sure that things are as easy as you suggest? For example are you sure that there is one $p$-adic L-function which is interpolating special values of all twists at all critical points at once, in the higher weight case? I'm no expert but somehow I didn't expect things to be so easy... | |
| Apr 4, 2011 at 17:41 | vote | accept | Jupiter Jones | ||
| Apr 4, 2011 at 17:40 | comment | added | Jupiter Jones | Excellent! I didn't realize that Rohrlich was at BU. Did you ask him if he was surprised that you get this result "for free" in the ordinary weight greater than 2 case (via p-adic L-functions)? | |
| Apr 4, 2011 at 16:27 | history | answered | Rob Harron | CC BY-SA 2.5 |