Timeline for P-adic L functions
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
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| Oct 28, 2010 at 8:03 | comment | added | Charles Matthews | I know they are measures, and so on. My first task as a grad student was to translate a paper from Russian by Manin about this (only to find that that someone else in DPMMS had already done it). I do think concentrating on the issue raised is important for a student, because Iwasawa theory, from the point of exposition, has in my considerable experience left much to be desired. The "main conjectures" - rather a promotional name, really - can be read in either direction, but are often treated too much at arm's length. | |
| Oct 28, 2010 at 0:43 | comment | added | Alex B. | Also, I don't know what merits the name "function" in your view, but $p$-adic L-functions are measures and they can be written down. In fact, explicitly writing down how to integrate against a $p$-adic L-function (say the Kubota-Leopoldt one) can hardly be avoided in a good course on this topic. | |
| Oct 27, 2010 at 20:51 | comment | added | Chris Wuthrich | The algebraic counterparts are characteristic power-series that describe a Selmer group (or class group). They are defined up to a unit power-series. So they are essentially a polynomial, indeed. They are sometimes called "algebraic $p$-adic L-functions". By a main conjecture they should be the same, up to a unit. | |
| Oct 27, 2010 at 20:51 | comment | added | Chris Wuthrich | Analytic $p$-adic L-function are NOT defined up to a unit. E.g. given an elliptic curve and a prime $p$, say good ordinary, there is one and exactly one $p$-adic L-function. They are good honest functions in one $p$-adic variable $s$. Similar for $p$-adic L-function of Dirichlet character etc. | |
| Oct 27, 2010 at 20:32 | comment | added | Arijit | Yeah I heard about the Iwasawa theory viewpoint. You want the p-adic L-function to be the annihilator of some Galois module. | |
| Oct 27, 2010 at 20:21 | history | answered | Charles Matthews | CC BY-SA 2.5 |