Special values of $p$-adic $L$-functions. - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T03:02:07Z http://mathoverflow.net/feeds/question/13287 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/13287/special-values-of-p-adic-l-functions Special values of $p$-adic $L$-functions. Kevin Buzzard 2010-01-28T21:45:31Z 2010-02-06T22:36:24Z <p>This is a very naive question really, and perhaps the answer is well-known. In other words, WARNING: a non-expert writes.</p> <p>My understanding is that nowadays there are conjectures which essentially predict (perhaps up to a sign) the value $L(M,n)$, where $M$ is a motive, $L$ is its $L$-function, and $n$ is an integer. My understanding of the history is that (excluding classical works on rank 1 motives from before the war) Deligne realised how to unify known results about $L$-functions of number fields and the B-SD conjecture, in his Corvallis paper, where he made predictions of $L(M,n)$, but only up to a rational number and only for $n$ critical. Then Beilinson extended these conjectures to predict $L(M,n)$ (or perhaps its leading term if there is a zero or pole) up to a rational number, and then Bloch and Kato went on to nail the rational number.</p> <p>Nowadays though, many motives have $p$-adic $L$-functions (the toy examples being number fields and elliptic curves over $\mathbf{Q}$, perhaps the very examples that inspired Deligne), and these $p$-adic $L$-functions typically interpolate classical $L$-functions at critical values, but the relationship between the $p$-adic and classical $L$-function is (in my mind) a lot more tenuous away from these points (although I think I have seen some formulae for $p$-adic zeta functions at $s=0$ and $s=1$ that look similar to classical formulae related arithmetic invariants of the number field).</p> <p>So of course, my question is: is there a conjecture predicting the value of $L_p(M,n)$, for $n$ an integer, and $L_p$ the $p$-adic $L$-function of a motive? Of course that question barely makes sense, so here's a more concrete one: can one make a conjecture saying what $\zeta_p(n)$ should be (perhaps up to an element of $\mathbf{Q}^\times$) for an integer $n\geq2$ and $\zeta_p(s)$ the $p$-adic $\zeta$-function? My understanding of Iwasawa theory is that it would only really tell you information about the places where $\zeta_p(s)$ vanishes, and not about actual values---Iwasawa theory is typically only concerned with the ideal generated by the function (as far as I know). Also, as far as I know, $p$-adic $L$-functions are not expected to have functional equations, so the fact that we understand $\zeta_p(s)$ for $s$ a negative integer does not, as far as I know, tell us anything about its values at positive integers.</p> http://mathoverflow.net/questions/13287/special-values-of-p-adic-l-functions/13306#13306 Answer by Rob Harron for Special values of $p$-adic $L$-functions. Rob Harron 2010-01-29T00:15:18Z 2010-01-29T00:15:18Z <p>Actually, <em>p</em>-adic <em>L</em>-functions are expected to satisfy functional equations compatible with the classical ones. For <em>M</em> an ordinary motive, Coates and Perrin-Riou conjectured the interpolation property at critical integers and the expected functional equation in some papers in the early nineties (see for example <a href="http://www.ams.org/mathscinet-getitem?mr=1097608" rel="nofollow">this</a>). In particular, the Kubota-Leopoldt <em>p</em>-adic <em>L</em>-functions interpolate <em>all</em> critical values of the classical Dirichlet <em>L</em>-functions (up to a period and a multiple). For modular forms, Mazur-Tate-Teitelbaum, in their 1986 paper) prove a <em>p</em>-adic functional equation in section 17. In fact, the two-variable <em>p</em>-adic <em>L</em>-function of an ordinary family of modular forms satisfies a two-variable functional equation interpolating the one-variable functional equation at each weight (see for example Greenberg-Stevens' inventiones paper) (I'd post more mathscinet links but it appears to be down...).</p> <p>As for the values of the <em>p</em>-adic <em>L</em>-function at non-critical integers, that's much more mysterious. Rubin has a computation outside of the critical points for a CM elliptic curve in section 3.3 of his paper in the "<em>p</em>-adic monodromy and BSD" proceedings. I think I've seen other cases, but generally it takes a lot of effort, I think.</p> <p>(Also, regarding Iwasawa theory's concern with values of <em>L</em>-functions, it is true that the Main Conjecture is only an equality of ideals in some power series ring, but one can still hope to construct <em>p</em>-adic <em>L</em>-functions on the analytic side that do a nice job at interpolating, say up to a <em>p</em>-adic unit.)</p> http://mathoverflow.net/questions/13287/special-values-of-p-adic-l-functions/13307#13307 Answer by Clark Barwick for Special values of $p$-adic $L$-functions. Clark Barwick 2010-01-29T00:18:05Z 2010-01-29T00:18:05Z <p>I'd like to write a better response, but I must be brief.</p> <p>For now, let me offer some places to read. Long story short, it is predicted that there's a relationship between special values of $p$-adic $L$-functions and syntomic regulators (which are the analogue of Beilinson's regulators in the $p$-adic world).</p> <ol> <li><p>The beautiful <a href="http://www.springerlink.com/content/vcjlmrmeeh9ktwll/" rel="nofollow">paper</a> of Manfred Kolster and Thong Nguyen Quong Do is, I think, a very readable resource.</p></li> <li><p>The best results I know in this direction are Besser's papers <a href="http://www.math.uiuc.edu/K-theory/0298/" rel="nofollow">here</a> and <a href="http://www.math.uiuc.edu/K-theory/0299/" rel="nofollow">here</a>, which use rigid syntomic cohomology.</p></li> <li><p>Besser's overview talk at the conference in Loen (notes available <a href="http://folk.uio.no/rognes/yff/alexandra.html" rel="nofollow">here</a>) was a real joy.</p></li> </ol> http://mathoverflow.net/questions/13287/special-values-of-p-adic-l-functions/13308#13308 Answer by Hunter Brooks for Special values of $p$-adic $L$-functions. Hunter Brooks 2010-01-29T00:22:05Z 2010-01-29T00:22:05Z <p>Here's a nice expository article by Colmez on Perrin-Riou's conjectures:</p> <p><a href="http://people.math.jussieu.fr/~colmez/851bourbaki.pdf" rel="nofollow">http://people.math.jussieu.fr/~colmez/851bourbaki.pdf</a></p> http://mathoverflow.net/questions/13287/special-values-of-p-adic-l-functions/14438#14438 Answer by Olivier for Special values of $p$-adic $L$-functions. Olivier 2010-02-06T22:36:24Z 2010-02-06T22:36:24Z <p>Others have hinted at it, but let me emphasize the point. At least if you are happy to assume all conjectures (and perhaps that your motive has good reduction at $p$), the conjectural landscape for $p$-adic $L$-functions is as complete as that for usual $L$-functions. Namely: there is a conjectural description of the value of the cyclotomic $p$-adic $L$-function at any integer (in fact any character $\eta\chi_{cyc}^{s}$ with $\eta$ finite).</p> <p>This can either be done in B.Perrin-Riou's style, see Fonctions $L$ $p$-adiques des représentations $p$-adiques, or in K.Kato's style, in which case it follows from the conjectures on special values of $L$-functions taking into account the action of a group algebra (the so-called equivariant conjectures). In fact, I exaggerate slightly here: at some special values, there could be an exceptional zero, in which case the leading term should incorporate an $\mathcal{L}$-invariant, and I don't think this has been (conjecturally) defined in all generality.</p> <p>Also, as Rob H. wrote, $p$-adic $L$-function are indeed expected to satisfy a functional equation. This can be seen either from Perrin-Riou's conjectural construction from motivic elements, in which case the functional equation follows from the explicit reciprocity law of Perrin-Riou (and Colmez in the de Rham case) or via Iwasawa main conjectures, in which case it follows from duality results for cohomology complexes.</p> <p>So everything you could wish for is conjectured. Not much, of course, is actually known.</p>