## Local gamma factors and p-adic L-functions

(Usual pedantry: Fix a prime ${p}$, ${\overline{\mathbf{Q}_{p}}}$, ${\iota:\overline{\mathbf{Q}_{p}}\overset{\sim}{\rightarrow}\mathbf{C}}$, etc. For any ${K\subset\overline{\mathbf{Q}_{p}}}$ finite over ${\mathbf{Q}_{p}}$ let ${K_{0}}$ denote the maximal unramified subfield, and write ${G_{K}=\mathrm{Gal}(\overline{\mathbf{Q}_{p}}/K)}$, ${H_{K}=\mathrm{Gal}(\overline{\mathbf{Q}_{p}}/K(\zeta_{p^{\infty}}))}$, ${\Gamma_{K}=G_{K}/H_{K}}$ as usual. Set ${\mathcal{R}_{K}=(\mathbf{B}_{\mathrm{rig}}^{\dagger})^{H_{K}}}$ for any ${K}$, with its natural actions of ${\varphi}$ and ${\Gamma_{K}}$.)

Let ${F}$ be a number field with absolute Galois group ${G_{F}}$, and let ${V}$ be a geometric representation of ${G_{F}}$ on an ${L}$-vector space for some ${L\subset\overline{\mathbf{Q}_{p}}}$ finite over ${\mathbf{Q}_{p}}$. Suppose the L-function ${L(s,\iota V)}$ is Deligne-critical for some ${s\in\mathbf{Z}}$. What extra data do we need to specify a p-adic L-function associated with ${V}$, 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 ${V}$ is motivic, and good ordinary at all primes dividing ${p}$, Coates formulated a remarkably precise conjecture on the existence of an ordinary p-adic L-function ${L_{p}^{\mathrm{ord}}(V)}$; in particular, Coates gave an exact description of the notorious “local correction factors at ${p}$” appearing in the interpolation property of ${L_{p}^{\mathrm{ord}}(V)}$. 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 ${\Gamma_{F}=\mathrm{Gal}(F^{(p)}/F)}$ be the Galois group of the maximal abelian extension of ${F}$ unramified away from all primes dividing ${p}$. By class field theory we have

$\displaystyle \Gamma_{F}\cong F^{\times}\backslash\mathbf{A}_{F}^{\times}/\overline{(F_{\infty}\widehat{\mathcal{O}_{F}^{(p)}})^{\times}}$

and a short exact sequence

$\displaystyle 1\rightarrow(\mathcal{O}_{F}\otimes_{\mathbf{Z}}\mathbf{Z}_{p})^{\times}/\overline{\mathcal{O}_{F}^{\times}}\rightarrow\Gamma_{F}\rightarrow\mathrm{Cl}_{F}\rightarrow1.$

In particular, ${\Gamma_{F}}$ is a p-adic analytic group of dimension ${1+r_{2}+\delta_{F,p}}$ where ${\delta_{F,p}}$ is the Leopoldt defect for ${F}$ at ${p}$. Let ${\mathfrak{X}_{F}}$ denote the rigid generic fiber of the formal scheme ${\mathrm{Spf}(\mathbf{Z}_{p}[[\Gamma_{F}]])}$, so ${\mathfrak{X}_{F}(\overline{\mathbf{Q}_{p}})=\mathrm{Hom}_{\mathrm{cts}}(\Gamma_{F},\overline{\mathbf{Q}_{p}}^{\times})}$; we make the latter identification without further comment. Let ${\psi}$ denote a character ${\Gamma_{F}\rightarrow\overline{\mathbf{Q}_{p}}^{\times}}$, so by the convention of the previous sentence the value ${f(\psi)\in\overline{\mathbf{Q}_{p}}}$ is well-defined for any ${f\in\mathcal{O}(\mathfrak{X}_{F})}$. A p-adic L-function associated with ${V}$ should be an element ${L_{p}(V)\in\mathcal{O}(\mathfrak{X}_{F})\otimes L}$ such that for suitable locally algebraic ${\psi\in\mathfrak{X}_{F}(\overline{\mathbf{Q}_{p}})}$, ${L_{p}(V)(\psi)}$ is algebraic over ${\mathbf{Q}}$ and ${\iota L_{p}(V)(\psi)}$ is a multiple of ${L^{\mathrm{alg}}(0,\iota(V\otimes\psi))}$.

Definition. An oriented Galois representation is a pair ${\mathbf{V}=(V,D^{+})}$ where ${V}$ is as above and ${D^{+}=\{D_{v}^{+}\}_{v|p}}$ is a choice, for each place ${v|p}$, of a saturated ${(\varphi,\Gamma_{F_{v}})}$-submodule ${D_{v}^{+}\subset D_{v}=\mathbf{D}_{\mathrm{rig}}^{\dagger}(V|G_{F_{v}})}$. The data ${D^{+}}$ is an orientation of ${V}$.

An oriented Galois representation is balanced if

$\displaystyle \sum_{v|p}[F_{v}:\mathbf{Q}_{p}]\mathrm{rank}_{\mathcal{R}_{F_{v}}\otimes L}D_{v}^{+}=\sum_{v|\infty}\mathrm{rank}_{L}V^{c_{v}=1}.$

A character ${\psi\in\mathfrak{X}_{F}(\overline{\mathbf{Q}_{p}})}$ is critical for ${\mathbf{V}}$ if ${\psi}$ is an algebraic Hecke character, ${L(s,\iota(V\otimes\psi))}$ is Deligne-critical at ${s=0}$, and ${D_{v}^{+}\otimes\psi_{v}}$ (resp. ${(D_{v}/D_{v}^{+})\otimes\psi_{v}}$) has only negative (resp. nonnegative) Hodge-Tate weights for all ${v|p}$. ${\mathbf{V}}$ is critical if it has at least one critical character.

If ${\mathbf{V}=(V,D^{+})}$ is an oriented Galois representation, the Cartier dual is defined as ${\mathbf{V}^{\ast}(1)=\left(V^{\ast}(1),\{(D_{v}/D_{v}^{+})^{\ast}(1)\}_{v|p}\right)}$. One checks that ${\mathbf{V}}$ is balanced, resp. critical, if and only if ${\mathbf{V}^{\ast}(1)}$ 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 ${\mathbf{V}=(V,D^{+})}$ is balanced and critical, there exists a function* ${L_{p}(\mathbf{V})\in\mathcal{O}(\mathfrak{X}_{F})\otimes L}$ such that

$\displaystyle L_{p}(\mathbf{V})(\psi)=E_{\infty}(\mathbf{V},\psi)E_{p}(\mathbf{V},\psi)\frac{L^{(p\infty)}(0,V\otimes\psi)}{c_{\infty}(V)}\in\overline{\mathbf{Q}}$

for all critical characters ${\psi}$, where ${c_{\infty}}$ is a suitable Deligne period and ${E_{\infty}}$ and ${E_{p}}$ are certain local factors. The function ${L_{p}(\mathbf{V})}$ has growth specified in terms of the rational numbers ${v_{p}(\varphi^{f_{v}}|\det D_{v}^{+})}$, satisfies a functional equation

$\displaystyle L_{p}(\mathbf{V})(\psi)=\psi(\mathfrak{n})\varepsilon^{(p)}(V)L_{p}(\mathbf{V}^{\ast}(1))(\psi^{-1})$

for a certain constant ${\varepsilon^{(p)}(V)}$ and prime-to-${p}$ ideal ${\mathfrak{n}}$, and is canonically defined up to multiplication by an element of ${L^{\times}}$.

The local factor ${E_{\infty}}$ 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 ${E_{p}}$, with no restrictions on the behavior of ${V|G_{F_{v}}}$ for places ${v|p}$.

OK so let’s turn to the local picture for a minute. Let ${V}$ be a de Rham representation of ${G_{K}}$ on a finite-dimensional ${L}$-vector space for some ${L\subset\overline{\mathbf{Q}_{p}}}$. By a theorem of Berger, ${V}$ is potentially semistable, and then by a recipe of Fontaine we may associate a Weil-Deligne representation ${\mathrm{WD}(V)=(W,r,N)}$ with ${V}$ as follows: Choose some finite Galois extension ${K'/K}$ over which ${V}$ becomes semistable. Now

$\displaystyle \mathbf{D}_{\mathrm{st},K'}(V)=(V\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{st}})^{G_{K'}}$

is a finite free ${L\otimes_{\mathbf{Q}_{p}}K_{0}'}$-module with commuting actions of ${\varphi}$ and ${N}$ and of ${\mathrm{Gal}(K'/K)}$. The ${\varphi}$– and Galois actions are not ${K_{0}'}$-linear, but we may combine them into a linear action of the Weil group ${W_K}$ by defining

$\displaystyle \mathbf{r}(g)=g\otimes g\varphi^{v_{K}(g)[k:\mathbf{F}_{p}]}\in\mathrm{End}(\mathbf{D}_{\mathrm{st},K'}(V))$

for ${g\in W_{K}}$. Here ${k}$ is the residue field of ${K}$ and ${v_{K}(g)\in\mathbf{Z}}$ is the integer ${m}$ such that ${g}$ projects to ${\mathrm{Frob}_{K}^{m}}$ in ${W_{K}/I_{K}}$. Now we choose any embedding ${\tau:K_{0}'\rightarrow\overline{\mathbf{Q}_{p}}}$ and set ${W=\mathbf{D}_{\mathrm{st},K'}(V)\otimes_{L\otimes K_{0}',1\otimes\tau}\overline{\mathbf{Q}_{p}}}$, ${r=\mathbf{r}}$, ${N=1\otimes N}$. 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 ${W^{+}\subset\mathrm{WD}(V)}$ and saturated sub-${(\varphi,\Gamma_{K})}$-modules

$\displaystyle D^{+}\subset\mathbf{D}_{\mathrm{rig}}^{\dagger}(V)=\left(V\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{rig}}^{\dagger}\right)^{H_{K}}$

over ${\mathcal{R}_{K}\otimes L}$.

Proof. By a theorem of Berger, we may adjoin a formal element ${\ell_{X}}$ to ${\mathcal{R}_{\mathbf{Q}_{p}}}$ such that

$\displaystyle \begin{array}{rcl} \mathbf{D}_{\mathrm{st},K'}(V)\otimes_{K_{0}'}\mathcal{R}_{K'}[\tfrac{1}{t},\ell_{X}] & \cong & \mathbf{D}_{\mathrm{rig},K'}^{\dagger}(V)[\tfrac{1}{t},\ell_{X}]\\ & \cong & \mathbf{D}_{\mathrm{rig}}^{\dagger}(V)\otimes_{\mathcal{R}_{K}}\mathcal{R}_{K'}[\tfrac{1}{t},\ell_{X}]\end{array}$

compatibly with all structures. In particular, since ${(\mathcal{R}_{K}[\frac{1}{t},\ell_{X}])^{\Gamma_{K}}=K_{0}}$ this lets us recover ${\mathbf{D}_{\mathrm{st}}}$ functorially from the right-hand side. Now, given ${D^{+}}$ as in the lemma, the subspace ${(D^{+}\otimes_{\mathcal{R}_{K}}\mathcal{R}_{K'}[\frac{1}{t},\ell_{X}])^{\Gamma_{K'}}\subset\mathbf{D}_{\mathrm{st},K'}(V)}$ is stable under all the relevant actions and gives a sub-Weil-Deligne representation by the above recipe. In the other direction, given a ${\overline{\mathbf{Q}_{p}}}$-subspace ${W^{+}\subset W}$ stable under the Weil group and monodromy actions, there’s a unique maximal ${\varphi}$-, ${N}$-, ${\mathrm{Gal}(K'/K)}$-invariant subspace ${\mathbf{D}^{+}\subset\mathbf{D}_{\mathrm{st},K'}(V)}$ free of the correct rank over ${L\otimes K_{0}'}$ such that ${\mathbf{D}^{+}\otimes_{L\otimes K_{0}',1\otimes\tau}\overline{\mathbf{Q}_{p}}=W^{+}}$ (form the preimage of ${W^{+}}$ under ${1\otimes\tau}$ and then intersect all of its ${\varphi^{\mathbf{Z}}}$-pullbacks), and we set

$\displaystyle D^{+}=\mathbf{D}_{\mathrm{rig}}^{\dagger}(V)\cap\left(\mathbf{D}^{+}\otimes_{K_{0}'}\mathcal{R}_{K'}[\frac{1}{t},\ell_{X}]\right).$

It’s easy to see these functors are mutually inverse. ${\square}$

Given an oriented representation ${\mathbf{V}}$, the recipe above produces a sub-Weil-Deligne representation ${W_{v}^{+}\subset W_{v}=\mathrm{WD}(V|G_{F_{v}})}$ from the data of the orientation. Now we define

$\displaystyle E_{p}(\mathbf{V},\psi)=\prod_{v|p}\frac{L(0,W_{v}^{+}\otimes\psi_{v})}{\varepsilon_{v}(W_{v}^{+}\otimes\psi_{v})L(0,(W_{v}^{+})^{\ast}(1)\otimes\psi_{v}^{-1})}.$

Here ${\varepsilon_{v}(W)=\varepsilon_{v}(W,\nu,dx)}$ is the local epsilon factor defined by some choice of Haar measure ${dx}$ on ${F_{v}}$ and additive character ${\nu:F\rightarrow\mathbf{C}^{\times}}$, cf. section 4.1 of Tate’s Corvallis article. So this isn’t an unfamiliar sort of expression – it’s a product of local ${\gamma}$-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 ${p}$ is unramified in ${F_{v}}$, in which case we have canonical choices for ${\nu}$ and ${dx}$. Then ${\varepsilon_{v}}$ enjoys the following properties, among others:

• If ${W'}$ is unramified and ${W}$ has trivial monodromy, then ${\varepsilon_{v}(W\otimes W')=\varepsilon_{v}(W)^{\dim W'}\cdot(\det W')(\varpi_{v}^{f(W)})}$ where ${f(W)}$ is the conductor of ${W}$.
• If ${W=\chi}$ is a character of ${F_{v}^{\times}}$, then

$\displaystyle \varepsilon_{v}(\chi)=\chi(c)\sum_{u\in\mathcal{O}_{F_{v}}^{\times}/(1+\mathfrak{m}_{v}^{f(\chi)})}\chi^{-1}(u)\nu(c^{-1}u),$

where ${c\in F_{v}}$ is any element of conductor ${f(\chi)}$.

Now we return to the global setup above. It’s time for some examples.

1. Take ${F=\mathbf{Q}}$, so ${\Gamma_{F}=\mathrm{Gal}(\mathbf{Q}(\zeta_{p^{\infty}})^{+}/\mathbf{Q})}$. Take ${V=L(\chi)}$, with ${\chi}$ an odd Dirichlet character. Then a balanced refinement corresponds to choosing ${D^{+}}$ empty, and so ${E_{p}=1}$. An algebraic Hecke character ${\chi_{\mathrm{cyc}}^{j}\eta\in\mathfrak{X}_{\mathbf{Q}}(\overline{\mathbf{Q}_{p}})}$ is critical for ${\mathbf{V}=(V,\emptyset)}$ iff ${j\leq0}$ (and ${\eta(-1)=(-1)^{j}}$ by evenness), and the sought-after ${L_{p}(\mathbf{V})}$ is of course the Kubota-Leopoldt p-adic L-function of ${\chi}$.

A similar analysis holds for an even Dirichlet character, although now ${D^{+}=D}$ and all the factors ${E_{\infty}}$, ${E_{p}}$ and ${c(V)}$ will contribute nontrivially.

2. Take ${F=\mathbf{Q}}$ and ${V=V_{f}}$ the (geometrically normalized) Galois representation associated with some cuspidal eigenform ${f\in S_{k}^{\mathrm{new}}(\Gamma_{1}(N))}$ for ${p\nmid N}$. Suppose ${V_{f}|G_{\mathbf{Q}_{p}}}$ is indecomposable. Then choosing a balanced refinement of ${V_{f}}$ amounts to choosing one of the two eigenvalues of ${\varphi}$ on ${\mathbf{D}_{\mathrm{crys}}(V_{f}|G_{\mathbf{Q}_{p}})}$ (there are conjecturally always two!), which coincide with the roots of ${X^{2}-a_{f}(p)X+p^{k-1}\epsilon(p)}$. Choose such a root ${\alpha}$, which corresponds to choosing ${D_{\alpha}^{+}}$ such that

$\displaystyle \begin{array}{rcl} \mathbf{D}_{\mathrm{crys}}(D_{\alpha}^{+}) & = & \mathbf{D}_{\mathrm{crys}}(D)^{\varphi=\alpha}\\ & \subset & \mathbf{D}_{\mathrm{crys}}(D)\\ & = & \mathbf{D}_{\mathrm{crys}}(V_{f}|G_{\mathbf{Q}_{p}}).\end{array}$

With this choice, ${W^{+}}$ is the one-dimensional representation associated with the unramified character ${x\mapsto\alpha^{v_{p}(x)}}$ of ${\mathbf{Q}_{p}^{\times}}$. Now the refined form ${f_{\alpha}=f(q)-\beta f(q^{p})}$ (with ${\beta=\frac{p^{k-1}\epsilon(p)}{\alpha}}$ the complementary root) is not ${\theta}$-critical, and the canonical p-adic L-function ${L_{p}(f,\alpha)}$ (as constructed by Mazur/Swinnerton-Dyer, Amice-Velu, Vishik, Pollack-Stevens and Bellaiche) satisfies the interpolation property

$\displaystyle \begin{array}{rcl} L_{p}(f,\alpha)(\chi_{\mathrm{cyc}}^{j}\eta) & = & (1-\beta\overline{\eta}(p)p^{-j})(1-\alpha^{-1}\eta(p)p^{j-1})\left(\frac{p^{j}}{\alpha}\right)^{n}\frac{(j-1)!L(j,f\otimes\eta^{-1})}{\tau(\eta^{-1})(-2\pi i)^{j}\Omega_{f}^{+}}\\ & = & L_{(p)}(j,f\otimes\eta^{-1})(1-\beta\overline{\eta}(p)p^{-j})(1-\alpha^{-1}\eta(p)p^{j-1})\left(\frac{p^{j}}{\alpha}\right)^{n}\frac{(j-1)!L^{(p\infty)}(j,f\otimes\eta^{-1})}{\tau(\eta^{-1})(-2\pi i)^{j}\Omega_{f}^{+}}\end{array}$

for all ${1\leq j\leq k-1}$, where ${p^{n}}$ is the conductor of ${\eta}$. (Here we follow the convention that ${\eta(p)=0}$ for ${\eta}$ nontrivial, and ${L_{(p)}}$ denotes the local L-factor at ${p}$, not to be confused with ${L_{p}}$!) Now one finds that:

• The factor ${E_{\infty}}$ is exactly ${(j-1)!/(-2\pi i)^{j}}$.
• The factor ${E_{p}}$ is exactly

$\displaystyle L_{(p)}(j,f\otimes\eta^{-1})(1-\beta\overline{\eta}(p)p^{-j})(1-\alpha^{-1}\eta(p)p^{j-1})\left(\frac{p^{j}}{\alpha}\right)^{n}\tau(\eta^{-1})^{-1}.$

The ${\varepsilon}$-factor contributes ${(p^{j}/\alpha)^{n}/\tau(\eta^{-1})}$, and the ratio of L-factors contributes the rest.

• ${\Omega_{f}^{+}}$ is the relevant period.

This all still goes through if ${f}$ is Steinberg at ${p}$, and the factor ${L(0,(W_{p}^{+})^{\ast}(1)\otimes\psi_{p}^{-1})}$ contributes the trivial zero.

3. Keep ${f}$ and ${D_{\alpha}^{+}}$ as in the previous example, and choose some ${g\in S_{k'}^{\mathrm{new}}(\Gamma_{1}(M))}$ with ${k'. Take ${V=V_{f}\otimes V_{g}}$ and

$\displaystyle \begin{array}{rcl} D^{+} & = & D_{\alpha}^{+}\otimes\mathbf{D}_{\mathrm{rig}}^{\dagger}(V_{g})\\ & \subset & \mathbf{D}_{\mathrm{rig}}^{\dagger}(V_{f})\otimes\mathbf{D}_{\mathrm{rig}}^{\dagger}(V_{g})\\ & \cong & \mathbf{D}_{\mathrm{rig}}^{\dagger}(V_{f}\otimes V_{g}).\end{array}$

The critical characters for ${\mathbf{V}=(V,D^{+})}$ are exactly the ${\chi_{\mathrm{cyc}}^{j}\eta}$ with ${k'\leq j. 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 ${g}$ at p! From the point of view of ${\varepsilon}$-factors, we are “beyond the realm of Gauss sums” if ${g}$ is supercuspidal at ${p}$. And still everything works out – ${W^{+}}$ is an unramified twist of ${\mathrm{WD}(V_{g}|G_{\mathbf{Q}_{p}})}$, so (up to this twist) ${\varepsilon_{p}(W^{+}\otimes\eta)}$ is essentially the p-part of the root number of ${g\otimes\eta}$, and this exactly matches the term “${W_{p}(g)}$” and its twisty variants appearing in part ii. of Theorem 5.1b of Hida.

*Strictly speaking, ${L_{p}(\mathbf{V})}$ might not lie in ${\mathcal{O}(\mathfrak{X}_{F})\otimes L}$; this function could have poles at the points ${\psi}$ where ${\mathbf{Q}_p(1)}$ is a subquotient of ${V \otimes \psi}$.