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Triple product p-adic L-functions for balanced weights

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Abstract

We construct p-adic triple product L-functions that interpolate (square roots of) central critical L-values in the balanced region. Thus, our construction complements that of Harris and Tilouine. There are four central critical regions for the triple product L-functions and two opposite settings, according to the sign of the functional equation. In the first case, three of these regions are of interpolation, having positive sign; they are called the unbalanced regions and one gets three p-adic L-functions, one for each region of interpolation (this is the Harris-Tilouine setting). In the other setting there is only one region of interpolation, called the balanced region: we produce the corresponding p-adic L-function. Our triple product p-adic L-function arises as p-adic period integrals interpolating normalizations of the local archimedean period integrals. The latter encode information about classical representation theoretic branching laws. The main step in our construction of p-adic period integrals is showing that these branching laws vary in a p-adic analytic fashion. This relies crucially on the Ash-Stevens theory of highest weight representations over affinoid algebras.

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Notes

  1. The fact that the central characters are the inverse of the usual ones is due to the fact that the \(L^{2}\)-automorphic forms on B that appear in Sect. 3.2 enjoy the equivariance property \(f\left( xg\right) =f\left( x\right) \), as opposite to the usual convention \(f\left( gx\right) =f\left( x\right) \). Thus we consider right \(\mathbf {B}^{\times }\left( \mathbb {A} _{f}\right) \) action \(\left( fu\right) \left( x\right) :=f\left( ux\right) \) on them, rather than the usual left action \(\left( uf\right) \left( x\right) :=f\left( xu^{-1}\right) \). The rule \(f^{*}\left( x\right) :=f\left( x^{-1}\right) \), which satisfies \(\left( fu\right) ^{*}=uf^{*}\), exchange the two spaces, but the central characters of the corresponding spaces are reversed.

  2. Indeed note that \(K_{1}\pi K_{2}\) is compact, being the image of \( K_{1}\times K_{2}\) by means of the continuous map given by \(\left( x,y\right) \mapsto x\pi y\). Since \(K_{1}\) is open, \(K_{1}\pi K_{2} =\bigsqcup \nolimits _{i}K_{1}\pi _{i}\) is an open covering which, by compactness, admits a finite refinement.

  3. In order to determine \(\lambda \), note that

    $$\begin{aligned} \lambda \left\langle P,\Delta _{k_{1},k_{2},k_{3}}\right\rangle _{k_{1},k_{2},k_{3}}=\left\langle P,\delta _{3}^{*}\left( \Delta _{k_{1}+1,k_{2}+1,k_{3}}\right) \right\rangle _{k_{1}+1,k_{2}+1,k_{3}}=\left\langle \delta _{3}P,\Delta _{k_{1}+1,k_{2}+1,k_{3}}\right\rangle _{k_{1}+1,k_{2}+1,k_{3}}. \end{aligned}$$

    A good choice is to take \(P=Y_{1}^{k_{1}}\otimes X_{2}^{\underline{k} _{3}^{*}}Y_{2}^{\underline{k}_{1}^{*}}\otimes X_{3}^{k_{3}}\).

  4. We write \(V\otimes _{\iota }W\) (resp. \(V\otimes W\)) to denote \(V\otimes W\) with the inductive (resp. projective) tensor topology.

  5. The trilinear form \(t_{\underline{k}}\) satisfies the invariance formula

    $$\begin{aligned} t_{\underline{k}}\left( \varphi _{1}u,\varphi _{2}u,\varphi _{3}u\right) = \mathrm {Nrd}_{f}\left( u\right) ^{\underline{k}^{*}}t_{\underline{k} }\left( g_{1},\varphi _{2},\varphi _{3}\right) \end{aligned}$$

    and we have \(\mathrm {Nrd}_{f}\left( u\right) =1\) for \(u\in K^{\#}\) or \(u\in \omega _{p}^{-1}K^{\#}\omega _{p}\).

  6. We have \(\mathrm {Nrd}_{f}\left( \widehat{\omega }_{p}\right) =\left| \mathrm {nrd}\left( \widehat{\omega }_{p}\right) \right| _{\mathbb {A} _{f}}^{-1}=\left| p\right| _{p}^{-1}=p.\)

  7. Via the morphism \(\mathcal {O}_{\mathcal {C}_{N,\varepsilon }}\rightarrow \mathcal {O}_{\mathcal {C}_{N,\varepsilon }^{\le h}}\) which sends a Hecke operator acting on finite slope overconvergent modular forms to the Hecke operator acting on overconvergent modular forms of slope \(\le h\).

  8. By a special unramified representation, we mean the twist by an unramified character of the special representation. Of course, this is a ramified representation of conductor 1.

  9. There is a typos in [43, Proposition 4.4]: the quantity \((1-\varepsilon )\) should be \((1+\varepsilon )\), which is 2 in our case, in accordance with the Prasad’s results.

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Authors and Affiliations

  1. Universita degli Studi di Milano, Milan, Italy

    Marco Adamo Seveso

  2. University of Calgary, Calgary, Canada

    Matthew Greenberg

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  1. Matthew Greenberg
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Correspondence to Marco Adamo Seveso.

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Greenberg, M., Seveso, M.A. Triple product p-adic L-functions for balanced weights. Math. Ann. 376, 103–176 (2020). https://doi.org/10.1007/s00208-019-01865-w

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