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My question is how should one think of p-adic L functions? I know they have been constructed classically by interpolating values of complex L-functions. Recently I have seen people think about them in terms of Euler systems. But we know only a few Euler systems and there are lot of p-adic L functions. In case of elliptic curves(at least over $\mathbb{Q}$) complex L-functions give information about the Galois representations. Should the p-adic L-function give some information about some p-adic Galois representation? It seems to be the case in case of cyclotomic fields where we think of the cyclotomic character as a 1-dimensional representation. I apologize in advance if my questions are vague. I am just starting to learn about the subject.

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I know you're looking for something more conceptual, but at an elementary level, you can think of the claim "a p-adic L-function exists corresponding to a given family of L-functions" as another way of saying "lots of interesting congruences hold between various (algebraic) special values of the L-functions in this family." Even if we didn't know any main conjectures, though, the first phrasing would still be better than the second in that it is less "Archimedio-centric." –  Hunter Brooks Oct 27 '10 at 20:33
    
Closely related question: mathoverflow.net/questions/37374/… –  Daniel Moskovich Oct 27 '10 at 20:39
    
@ Hunter I understand what you are trying to say. But that is something the p-adic L function satisfies rather than be its definition. As for the main conjectures, that is the Iwasawa point of view. A related but different question have people formulated the main conjecture for elliptic curves with supersingular reduction? –  Arijit Oct 27 '10 at 20:51
    
Thanks a lot Daniel. But I didnt get anything out of that discussion. –  Arijit Oct 27 '10 at 20:58
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Yes, there are at least three formulations of the main conjecture for elliptic curves with supersingular reduction at $p$. By Perrin-Riou, by Kato and then by Kobayashi. They are all equivalent. Probably Kobayashi's approach using his $\pm$ Selmer groups linking them to $\pm$ $p$-adic $L$-functions by Pollack is probably the best accessible one. –  Chris Wuthrich Oct 27 '10 at 22:29

2 Answers 2

up vote 20 down vote accepted

There are three way to obtain $p$-adic L-functions. The big dream is that one can do all of them for a large class of $p$-adic Galois representations $V$. To study them one starts best to look at the cases $\mathbb{Q}(1)$ for the classical Kubota-Leopoldt $p$-adic $L$-functions or the Tate-module of an elliptic curve etc.

Let $K_{\infty}=\mathbb{Q}(\mu_{p^{\infty}})$ be the union of all cyclotomic fields of roots of unity of $p$-power order. Let $G$ be its Galois group, which is isomorphic to $\mathbb{Z}_p^{\times}$.

  • Attached to $V$ there is a complex $L$-function and there are conjectures saying that certain values are algebraic and satisfy to certain congruences modulo powers of $p$, e.g. Kummer congruences. So in some cases, one can show the algebraicity and the congruences. So the values fit together to a $p$-adic analytic function. But the better way of presenting the $p$-adic $L$-function is by constructing a measure on the Galois group $G$ with values in $\mathbb{C}_p$. One can then evaluate the $p$-adic $L$-function on characters of the group $G$. This way the $p$-adic $L$-function resembles a lot its complex counterpart as they are described in Tate's thesis. See Lang's Cyclotomic Fields or Washington or Mazur-Tate-Teitelbaum for instance.

  • On the algebraic side, we have a Selmer group or a class group that we watch growing in the tower $K_{\infty}/\mathbb{Q}$. The characteristic series of the dual of this Selmer group as a $\Lambda$-module is a sort of a generating function for this growth. Like zeta-functions for varieties over finite fields. These characteristic series are in fact power-series, but they are defined up to a unit (as they are generators of some ideal). Greenberg's paper give a good introduction to this side.

  • The Euler system (if we are lucky to be in one of the few cases where we have one) is a system of norm-compatible cohomology classes. In particular they give an element in $H^1(K_n, V)$ for each intermediate field $K_n$. But there should be an element over sufficiently many abelian extensions. The norm-compatibility is involves a factor that looks like an Euler factor of the complex $L$-function. There is a general map, called the Coleman map or the logarithme élargi or whatever, from the inverse limit of the $H^1(K_{n,p}, V)$ to a ring of power-series. The image of the Euler system under this map should be the analytically defined $p$-adic $L$-function. Typically one shows that they satisfy the same interpolation property.

In some sense the Euler system is the bridge between the analytic and the algebraic world. Under the Coleman map it links to the analytic side. In the other direction, one can form derivative classes out of the cohomology classes. These derived classes can be analysed locally and they can be used to bound the Selmer group and hence the characteristic series. That is how one can prove the main conjecture in some cases in one direction. Probably a good place to start is Coates-Sujatha.

The $p$-adic $L$-function of an elliptic curve is conjectured to satisfy a $p$-adic Birch and Swinnerton-Dyer formula. (Mazur-Tate-Teitelbaum and Bernardi-Perrin-Riou in the supersingular case). On the algebraic side instead, we know almost that the characteristic series satisfies this formula. The order of vanishing is known to be at least as large as the rank and if they agree then the leading term has the desired shape involving the Tate-Shafarevich group; of course only up to a $p$-adic unit.

In the geometric case, say an elliptic curve over a function field $K$ of a curve over a finite field $k$, the complex and the $p$-adic function are the same ($p\neq\text{char}(k)$), since they are both just a polynomial with integer coefficients. Tate's Bourbaki talk on BSD shows how one can use the tower $K_{\infty} = \bar{k} \cdot K$ to prove a good deal about BSD. Iwasawa theory tries to mimic this.

So I believe that $p$-adic $L$-functions are just as nice and interesting as their complex counterparts. Even if they seem more mysterious and the definition is less straight forward, we sometimes know more about them. Now I stop otherwise I am going to write a book about it here...

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Thanks Chris. Your answer was certainly very helpful. So I am happy to think of a p-adic L function as a measure(or rather a pseudo-measure). But in Tate's thesis the zeta functions arose as eigenvectors of some 1-dimensional complex representations. So do we expect the same thing here as well like p-adic L-function arising as an eigenvector of some p-adic representation. Also when you think about p-adic measures there is no Haar measure. –  Arijit Oct 27 '10 at 23:46

It's a sensible question, for this reason: the (classical, complex) L-functions are defined in such a way that you can write them down, as Dirichlet series, at least in a right half-plane. What corresponds for p-adic L-functions? Essentially there isn't anything that matches. You can sit in many lectures on p-adic L-functions without seeing anything that merits the name "function", in the sense of function theory.

What lies behind this? It is not that p-adic numbers are less "explicit" than complex numbers at all. It is quite a different reason: the module-theoretic meaning of a p-adic L-function defines it up to a unit in a certain ring, with the implication that these functions are essentially polynomials. And it is through the modules that Iwasawa theory gains traction in number theory.

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Yeah I heard about the Iwasawa theory viewpoint. You want the p-adic L-function to be the annihilator of some Galois module. –  Arijit Oct 27 '10 at 20:32
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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. –  Chris Wuthrich Oct 27 '10 at 20:51
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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. –  Chris Wuthrich Oct 27 '10 at 20:51
    
Thanks a lot Chris. –  Arijit Oct 27 '10 at 20:57
    
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. –  Alex B. Oct 28 '10 at 0:43

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