For modular forms, it is known that you can construct p-adic L-functions by interpolating (p-power conductor) twists of their associated L-functions at special values. Similarly, Kubota-Leopoldt's p-adic L-function can be constructed by interpolating Dirichlet L-functions at negative integers. My question is: how general is this method? For example, for which (L-functions of) automorphic representations have p-adic L-functions been constructed?
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1$\begingroup$ For future reference, it's really not necessary to TeX individual letters and things like that in titles. They don't render on the front page and lag when they do render. $\endgroup$– Qiaochu YuanCommented Apr 23, 2010 at 5:34
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2 Answers
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You should distinguish between the method (of which there is not one, but several) and the result (namely, the existence of $p$-adic $L$-functions).
It is expected that $p$-adic $L$-function exist in great generality (say for algebraic automorphic representations, i.e. automorphic representations that are expected to have a connection with Galois representations). But they have not yet been constructed in that level of generality.
For Grossencharacters of totally real fields, they were constructed by Serre, and studied further by Deligne and Ribet, and others. For Grossencharacters on CM fields, they were constructed by Katz. They are also constructed for anticyclotomic twists of (base-changes to a quad. imag. field) of classical modular forms, and I think for symmetric squares of classical modular forms. I have omitted many other cases that are known, and I'm not going to attempt to make additional attributions, since I'll likely get them wrong.
In general, the construction and study of $p$-adic $L$-functions is an active and ongoing research area.
Just to give one idea of the current state of play, let me mention that there is an ongoing project of Eischen, Harris, Li, Skinner and myself which will construct them for twists of certain conjugate self-dual cuspforms on $GL_n$ over a CM field. (This generalizes Katz's result, which is the $n = 1$ case, and the anti-cyclotomic $p$-adic $L$-functions for modular forms, which is the $n = 2$ case.) In this construction, as in many of the other constructions mentioned above, there will be technical restrictions and caveats; I won't try to describe them here.
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$\begingroup$ @Emerton: let me ask a related question. If G is connected reductive over a global field, and pi is an automorphic representation for G, then it's not pi that has an L-function, but the pair pi,rho, where rho is a representation of the L-group of G. If we believe functoriality for a second, this seems to reduce us to the case of GL_n (because rho_*(pi) will be an auto rep of GL_n). Can you see how anticyclotomic p-adic L-functions for ell curves fit into this story? Is it somehow one part of a conjectural 2-variable p-adic L-function? $\endgroup$ Commented Apr 23, 2010 at 6:45
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$\begingroup$ Comment on your answer: for Grossencharacters of any field, one can consider the job done now, because for silly reasons they vanish identically if the field isn't totally real or CM. Presumably there is a conjectural analogue of this statement for GL_n, so, in particular, just sticking to GL_n over a CM field isn't as restrictive as it might sound perhaps? $\endgroup$ Commented Apr 23, 2010 at 6:47
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1$\begingroup$ See Olivier's answer to my question [mathoverflow.net/questions/18884/… here]. The verdict there seems to be that there are examples where interpolating values of classical L-functions gives you zero, but if you assume a whole constellation of conjectures you might still hope to be able to interpolate leading terms of classical L-functions. $\endgroup$ Commented Apr 23, 2010 at 7:23
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$\begingroup$ "Comment on your answer: for Grossencharacters of any field, one can consider the job done now, because for silly reasons they vanish identically if the field isn't totally real or CM." Is your terminology like mine? Grossencharacters for totally imaginary but not CM fields exist, but factor through the norm down to the CM-subfield. With an example, the Deligne conjecture was proved by Blasius for CM fields and Harder for totally imaginary. The paper of Harder and Schappacher discusses this, page 36 and on to 43. dx.doi.org/10.1007/BFb0084583 $\endgroup$– JunkieCommented Apr 23, 2010 at 12:35
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$\begingroup$ Katz's result, as well as what the first Harris–Li–Skinner paper states, restricts to the "ordinary" case (for Katz, this is that every prime above $p$ splits in the CM field). Has there been any work generalizing to the non-ordinary case (for Größencharaktere/automorphic forms on unitary groups)? The symmmetric square $p$-adic $L$-functions have been constructed for non-ordinary newforms, for example (Dabrowski–Delbourgo (ams.org/mathscinet-getitem?mr=1434442)). $\endgroup$ Commented Apr 23, 2010 at 16:45
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The following is more a long comment than an answer per se.
One thing to keep in mind when discussing $p$-adic $L$-functions is that to a given algebraic automorphic representation $\pi$ or Galois representation $\rho$ is potentially attached several objects which could reasonably called the $p$-adic $L$-function of $\rho/\pi$. Largely for historical reasons, when one speaks of the $p$-adic $L$-function of $\rho$ without further comment, one generally speaks of the $p$-adic $L$-function coming from the cyclotomic $\mathbb Z_{p}$-extension, as I assume you do in your question. The most natural object from a strictly mathematical point of view seems to me to be the $p$-adic $L$-function attached to the universal deformation ring of $\bar{\rho}$ (at least when this universal deformation ring exists).
Even restricting yourself to the simplest case of the cyclotomic $p$-adic $L$-function, the case of $GL_{n}$ over $\mathbb Q$ has not been done (that I know of) and I doubt (euphemism) that it will follow from the work of Eischen, Emerton, Harris, Li and Skinner (Emerton claims nothing of the sort). Unless I am very much mistaken, the cyclotomic case for $GL_{n}$ over $\mathbb Q$ would be an extremely impressive progress. Somehow, the case of the anticyclotomic $\mathbb Z_{p}$-extension of a CM field is sometimes easier because one can use the Rankin-Selberg method to prove that special values are algebraic and the Rankin-Selberg method is quite amenable to $p$-adic methods. I imagine that this is an ingredient in the work of EEHLS (but I know nothing about it, so please M.Emerton correct me if I'm wrong).
Leaving the real world for a second: conjecturally, cyclotomic $p$-adic $L$-functions are now constructed for any motive over $\mathbb Q$ (though you will have a really hard time finding this in the literature, as one has to combine an impressive series of very involved papers). Of course, the conjectural construction would not tell you much in way of an actual construction (the conjectural construction gives you an element in some local cohomology group and you will have somehow to identify it as a global element), even though I admit I have been more than mildly impressed by an answer of Idoneal to a question here on MO about $p$-adic $L$-functions here which seems to indicate that analytic argument allows you to do just that in the case of modular forms.
Kevin Buzzard, sure the anticyclotomic $p$-adic $L$-function of an elliptic curve is part (technically, a specialization) of a two-variable $p$-adic $L$-function. In this setting and at least in the ordinary case, this has been known for more than 25 years (it was done in his thesis by S.Haran and later widely expanded by H.Hida in his Invent. Math. 79 paper). And further, this two-variable $p$-adic $L$-function is a specialization of a three-variable $p$-adic $L$-function taking into account variation of the weight in the Hida family passing through this elliptic curve. Even in the finite slope non-ordinary case, I think this three-variable $p$-adic $L$-function is known to exist by the work of A.Panciskin.
