(Usual pedantry: Fix a prime , , , etc. For any finite over let denote the maximal unramified subfield, and write , , as usual. Set for any , with its natural actions of and .)
Let be a number field with absolute Galois group , and let be a geometric representation of on an -vector space for some finite over . Suppose the L-function is Deligne-critical for some . What extra data do we need to specify a p-adic L-function associated with , and what should the resulting function look like? Now Perrin-Riou has formulated some remarkable conjectural answers to this question, but her approach is so beautifully motivic and equivariant and canonical that her conjectures are rather inaccessible, even in the simplest cases (for example, they assume the truth of the Beilinson conjectures). In the situation when is motivic, and good ordinary at all primes dividing , Coates formulated a remarkably precise conjecture on the existence of an ordinary p-adic L-function ; in particular, Coates gave an exact description of the notorious “local correction factors at ” appearing in the interpolation property of . In this post, I want to point out that, thanks to some advances in p-adic Hodge theory over the years, Coates’s formula for these correction factors now makes sense in total generality.
OK, so let be the Galois group of the maximal abelian extension of unramified away from all primes dividing . By class field theory we have
and a short exact sequence
In particular, is a p-adic analytic group of dimension where is the Leopoldt defect for at . Let denote the rigid generic fiber of the formal scheme , so ; we make the latter identification without further comment. Let denote a character , so by the convention of the previous sentence the value is well-defined for any . A p-adic L-function associated with should be an element such that for suitable locally algebraic , is algebraic over and is a multiple of .
Definition. An oriented Galois representation is a pair where is as above and is a choice, for each place , of a saturated -submodule . The data is an orientation of .
An oriented Galois representation is balanced if
A character is critical for if is an algebraic Hecke character, is Deligne-critical at , and (resp. ) has only negative (resp. nonnegative) Hodge-Tate weights for all . is critical if it has at least one critical character.
If is an oriented Galois representation, the Cartier dual is defined as . One checks that is balanced, resp. critical, if and only if is balanced, resp. critical. These conditions are motivated partially by experience, and partially by considerations on the algebraic side of Iwasawa theory: they’re necessary in order for the numerology behind a putative main conjecture to work out correctly. But anyway, in my humble opinion, the data of a balanced critical orientation is exactly the “extra data” necessary to specify a p-adic L-function, at least in the Deligne-critical setting. More precisely, putting ourselves in this setup, something like the following should be true:
Optimistic Conjecture. If is balanced and critical, there exists a function* such that
for all critical characters , where is a suitable Deligne period and and are certain local factors. The function has growth specified in terms of the rational numbers , satisfies a functional equation
for a certain constant and prime-to- ideal , and is canonically defined up to multiplication by an element of .
The local factor should be as specified in Coates’s article, and I won’t say anything about it here. What I really want to talk about is a uniform expression for , with no restrictions on the behavior of for places .
OK so let’s turn to the local picture for a minute. Let be a de Rham representation of on a finite-dimensional -vector space for some . By a theorem of Berger, is potentially semistable, and then by a recipe of Fontaine we may associate a Weil-Deligne representation with as follows: Choose some finite Galois extension over which becomes semistable. Now
is a finite free -module with commuting actions of and and of . The – and Galois actions are not -linear, but we may combine them into a linear action of the Weil group by defining
for . Here is the residue field of and is the integer such that projects to in . Now we choose any embedding and set , , . It’s easy to check that this really is a Weil-Deligne representation, independent (up to isomorphism) of all choices.
Lemma. There is a bijection between sub-Weil-Deligne representations and saturated sub--modules
Proof. By a theorem of Berger, we may adjoin a formal element to such that
compatibly with all structures. In particular, since this lets us recover functorially from the right-hand side. Now, given as in the lemma, the subspace is stable under all the relevant actions and gives a sub-Weil-Deligne representation by the above recipe. In the other direction, given a -subspace stable under the Weil group and monodromy actions, there’s a unique maximal -, -, -invariant subspace free of the correct rank over such that (form the preimage of under and then intersect all of its -pullbacks), and we set
It’s easy to see these functors are mutually inverse.
Given an oriented representation , the recipe above produces a sub-Weil-Deligne representation from the data of the orientation. Now we define
Here is the local epsilon factor defined by some choice of Haar measure on and additive character , cf. section 4.1 of Tate’s Corvallis article. So this isn’t an unfamiliar sort of expression – it’s a product of local -factors! In the ordinary case, an expression of this form already appears in Coates’s article (on p. 51, halfway between equations (34) and (35)). Let us assume that is unramified in , in which case we have canonical choices for and . Then enjoys the following properties, among others:
- If is unramified and has trivial monodromy, then where is the conductor of .
- If is a character of , then
where is any element of conductor .
Now we return to the global setup above. It’s time for some examples.
1. Take , so . Take , with an odd Dirichlet character. Then a balanced refinement corresponds to choosing empty, and so . An algebraic Hecke character is critical for iff (and by evenness), and the sought-after is of course the Kubota-Leopoldt p-adic L-function of .
A similar analysis holds for an even Dirichlet character, although now and all the factors , and will contribute nontrivially.
2. Take and the (geometrically normalized) Galois representation associated with some cuspidal eigenform for . Suppose is indecomposable. Then choosing a balanced refinement of amounts to choosing one of the two eigenvalues of on (there are conjecturally always two!), which coincide with the roots of . Choose such a root , which corresponds to choosing such that
With this choice, is the one-dimensional representation associated with the unramified character of . Now the refined form (with the complementary root) is not -critical, and the canonical p-adic L-function (as constructed by Mazur/Swinnerton-Dyer, Amice-Velu, Vishik, Pollack-Stevens and Bellaiche) satisfies the interpolation property
for all , where is the conductor of . (Here we follow the convention that for nontrivial, and denotes the local L-factor at , not to be confused with !) Now one finds that:
- The factor is exactly .
- The factor is exactly
The -factor contributes , and the ratio of L-factors contributes the rest.
- is the relevant period.
This all still goes through if is Steinberg at , and the factor contributes the trivial zero.
3. Keep and as in the previous example, and choose some with . Take and
The critical characters for are exactly the with . In this case a p-adic L-function has been constructed by Hida and Urban. Things are exciting now because we are making no assumption on the local behavior of at p! From the point of view of -factors, we are “beyond the realm of Gauss sums” if is supercuspidal at . And still everything works out – is an unramified twist of , so (up to this twist) is essentially the p-part of the root number of , and this exactly matches the term “” and its twisty variants appearing in part ii. of Theorem 5.1b of Hida.
*Strictly speaking, might not lie in ; this function could have poles at the points where is a subquotient of .