p-adic L functions from Selmer groups - how canonical are they?

Question

For this question, I am going to be very concrete but very much appreciate broader viewpoints. Let $F$ be a number field and define $F_n = F(\mu_{p^n})$ and let us suppose for simplicity that $\mu_p \subset F$ but $\mu_{p^2} \not\subset F$ so that $Gal(F_n/F) = \mathbb Z/p^n$. In thfis case, one can consider the inverse limit $X_\infty = \lim_n Cl(F_n)[p^\infty]$ with an action of $\Gamma \cong \mathbb Z_p = \lim_n Gal(F_n/F)$.

Iwasawa showed that $X_\infty$ is essentially a torsion module over $\mathbb Z_p[[\Gamma]] \cong \mathbb Z_p[[T]]$ so that one can define a well defined characteristic ideal. In some cases, people have shown that this ideal is generated by the p-adic L-function of $F$.

Questions:

1. Is the expectation in general that the characteristic ideal should be generated by the p-adic L function? I am not even sure if we have a definition of a p-adic L function for arbitrary $F$.
2. To what extent can we use $X_\infty$ to pin down the p-adic L functions exactly (and not just upto units)? For instance, do people expect there to be an element $\gamma \in \mathbb Z_p[[\Gamma]]$ so that the power series $f(\gamma)$ that annihilates $X_\infty$ is exactly the p-adic L function of $F$?
3. Related to the previous point, in direct analogy with the characteristic $p$ case, one might expect that picking $\gamma$ to be the element sending $\zeta \to \zeta^\ell$ for $\ell \neq p$ might define something close to the p-adic L function. Is anything known in this direction?
4. Even if we can't capture the p-adic L function entirely, can we extract some information out of it (such as some special values) by picking appropriate $\gamma$?

Answer 1

All of your questions are undermined by the same fundamental issue: you cannot talk about "the" p-adic $L$-function in this generality, because there is no sensible definition of what a $p$-adic $L$-function should be.

It's common to state the Iwasawa main conjecture (either the version proved by Wiles for class groups of cyclotomic extensions over totally-real fields, or its analogues in fancier settings) as an equality

"algebraic $p$-adic $L$-function = analytic p-adic L-function"

but this is perhaps a tiny bit misleading, because the two sides are not quite on the same footing.

• the algebraic $p$-adic $L$-function is an element of $\Lambda(\Gamma)$ unique up to units, defined as a generator of the characteristic ideal of the Selmer group;
• the analytic $p$-adic $L$-function is an element of $\Lambda(\Gamma)$ which is uniquely determined (on the nose), defined as the unique thing which interpolates $L$-values.

So there is more information on one side than the other: we should more accurately state the conjecture as saying "the algebraic $p$-adic $L$-function is the image in $\Lambda(\Gamma) / {\{\text{units}\}}$ of the analytic p-adic $L$-function". [*]

If you try to formulate main conjectures over non-totally-real fields, you find that the definition of the algebraic $p$-adic $L$-function still makes sense, but the definition of the analytic one doesn't – the fundamental problem is that if your ground field is not totally real, then $L$-functions of ray class characters don't have any critical values, so there is nothing to interpolate. [**] So there is a fundamental obstacle to extending Iwasawa main conjectures to this setting.

(A long time ago, I wondered about whether one can use Stark's conjecture, which gives an arithmetic interpretation of leading terms of complex $L$-functions at non-critical values, to define some kind of $p$-adic $L$-function. Here is the link: Stark's conjecture and p-adic L-functions. But this is still $\pi$-in-the-sky stuff, even 13 years on from the original question: Stark's conjecture in this setting is just too hard.[***])

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[*] This is, in fact, more or less the formulation that is used in the noncommutative main conjectures that were very fashionable around 2005-2010. There you have some relative $K$-group which generalises $\Lambda(\Gamma) / \{units\}$, in which there is an element arising from Selmer groups; and some other relative $K$-group which generalises $\Lambda(\Gamma)$, in which there is an element which interpolates $L$-values; and the conjecture says that the former element is the image of the latter under some boundary map.

[**] Over CM-fields you can do something: in this setting there exist "interesting" Hecke Grössencharacters, not of finite order, whose $L$-functions do have critical values. But if your field isn't CM and doesn't have a CM subfield, then there are no Grössencharacters whose $L$-functions have critical values. More subtly, even in the CM case, the "correct" Selmer group to match these $L$-functions turns out to be rather different from Iwasawa's $X_\infty$, and it is not obvious if it has any relation to limits of class groups.

[***] Before you ask, the recent breakthroughs on Stark's conjecture by Dasgupta et al don't help here, since they are specific to totally real / CM settings.