# P-adic L-functions for GL(3)

## TL;DR

This paper constructs p-adic L-functions for a specific class of automorphic representations of GL(3), proving conjectures and offering new methods for general-type representations.

## Contribution

The first construction of p-adic L-functions for general-type RACARs of GL(3) without self-duality assumptions.

## Key findings

- A bounded measure $L_p(\Pi)$ is constructed on $\mathbb{Z}_p^\times$ that interpolates critical values of $L(\Pi \times \eta, -j)$.
- The results confirm conjectures by Coates–Perrin-Riou and Panchishkin for this case.
- The method uses a 'Betti Euler system' and applies to arbitrary cohomological weights and ramification at p.

## Abstract

Let \documentclass[12pt]{minimal}
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\begin{document}$$\Pi $$\end{document}Π be a regular algebraic cuspidal automorphic representation (RACAR) of \documentclass[12pt]{minimal}
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\begin{document}$$\textrm{GL}_3(\mathbb {A}_{\mathbb {Q}})$$\end{document}GL3(AQ). When \documentclass[12pt]{minimal}
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\begin{document}$$\Pi $$\end{document}Π is p-nearly-ordinary for the maximal standard parabolic with Levi \documentclass[12pt]{minimal}
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\begin{document}$$\textrm{GL}_1 \times \textrm{GL}_2$$\end{document}GL1×GL2, we construct a p-adic L-function for \documentclass[12pt]{minimal}
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\begin{document}$$\Pi $$\end{document}Π. More precisely, we construct a (single) bounded measure \documentclass[12pt]{minimal}
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\begin{document}$$L_p(\Pi )$$\end{document}Lp(Π) on \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {Z}_p^{\times }$$\end{document}Zp× attached to \documentclass[12pt]{minimal}
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\begin{document}$$\Pi $$\end{document}Π, and show it interpolates all the critical values \documentclass[12pt]{minimal}
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\begin{document}$$L(\Pi \times \eta ,-j)$$\end{document}L(Π×η,-j) at p in the left-half of the critical strip for \documentclass[12pt]{minimal}
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\begin{document}$$\Pi $$\end{document}Π (for varying \documentclass[12pt]{minimal}
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\begin{document}$$\eta $$\end{document}η and j). This proves conjectures of Coates–Perrin-Riou and Panchishkin in this case. We also prove a corresponding result in the right half of the critical strip, assuming near-ordinarity for the other maximal standard parabolic. Our construction uses the theory of spherical varieties to build a “Betti Euler system”, a norm-compatible system of classes in the Betti cohomology of a locally symmetric space for \documentclass[12pt]{minimal}
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\begin{document}$$\textrm{GL}_3$$\end{document}GL3. We work in arbitrary cohomological weight, allow arbitrary ramification at p along the Levi factor of the standard parabolic, and make no self-duality assumption. We thus give the first constructions of p-adic L-functions for RACARs of \documentclass[12pt]{minimal}
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\begin{document}$$\textrm{GL}_n(\mathbb {A}_{\mathbb {Q}})$$\end{document}GLn(AQ) of ‘general type’ (i.e. those that do not arise as functorial lifts) for any \documentclass[12pt]{minimal}
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\begin{document}$$n >2$$\end{document}n>2.