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Spectral aspect subconvex bounds for \(U_{n+1} \times U_n\)

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Abstract

Let \((\pi ,\sigma )\) traverse a sequence of pairs of cuspidal automorphic representations of a unitary Gan–Gross–Prasad pair \((U_{n+1},U_n)\) over a number field, with \(U_n\) anisotropic. We assume that at some distinguished archimedean place, the pair stays away from the conductor drop** locus, while at every other place, the pair has bounded ramification and satisfies certain local conditions (in particular, temperedness). We prove that the subconvex bound

$$\begin{aligned} L(\pi \times \sigma ,1/2) \ll C(\pi \times \sigma )^{1/4 - \delta } \end{aligned}$$

holds for any fixed

$$\begin{aligned} \delta < \frac{1}{8 n^5 + 28 n^4 + 42 n^3 + 36 n^2 + 14 n}. \end{aligned}$$

Among other ingredients, the proof employs a refinement of the microlocal calculus for Lie group representations developed with A. Venkatesh and an observation of S. Marshall concerning the geometric side of the relative trace formula.

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Notes

  1. These difficulties have since been addressed in the preprint [45].

  2. This convention differs from that in [46], where \({{\,\textrm{Op}\,}}\) refers to the “unscaled” operator assignment, i.e., the case \({\text {h}}= 1\). We refer here to the latter more verbosely as \({{\,\textrm{Op}\,}}_{1}\).

  3. The space \({\underline{S}}^m\) was denoted in [46, Section 4.3] by \(S^m\). We do not adopt the latter notation here.

  4. The space \({\underline{S}}^m_\delta \) was denoted “\(S^m_{\delta }\)” in [46, Section 4.4] The latter notation is used differently here.

  5. The notation here differs from that in [46, Section 3] There, we used “\(\Psi ^m\)” in place of \({\underline{\Psi }}^m\) and “\(\Psi ^m_\delta \)” for the space of \({\text {h}}\)-dependent operators whose norms are bounded in the indicated manner. The class \(\Psi ^m_\delta \) defined here plays a similar role. The relationship between these definitions is as described in the proof of Theorem 9.5.

  6. Some readers have told us that they found the “argument” of [46, Section 17.3] insufficiently detailed, so we elaborate upon the key step here. Let F be a field of characteristic zero with algebraic closure \({\bar{F}}\). Let G be an algebraic group that acts on a variety X, with G, X and the action all defined over F. Suppose that the action of \(G({\bar{F}})\) on \(X({\bar{F}})\) is simply-transitive and that X(F) is nonempty. Then the action of G(F) on X(F) is likewise simply-transitive. In verifying this, the main step is to check that if \(g \in G({\bar{F}})\) and \(x_0 \in X(F)\) satisfy \(g x_0 \in X(F)\), then in fact \(g \in G(F)\). It is enough to check that g is fixed by every element of \({{\,\textrm{Gal}\,}}({\bar{F}}/F)\). Since the action of \(G({\bar{F}})\) on \(X({\bar{F}})\) is free and defined over F, it is enough to check the same for \(g x_0\), which follows from our hypothesis that \(g x_0 \in X(F)\).

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Acknowledgements

We would like to thank Valentin Blomer, Simon Marshall, Philippe Michel, Abhishek Saha, Peter Sarnak, Akshay Venkatesh, Liyang Yang and Wei Zhang for their helpful feedback on an earlier draft. We are also grateful to Liyang Yang for suggesting an improvement to Lemma 15.3 and to the anonymous referee for a very detailed reading and many helpful comments and corrections.

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    Paul D. Nelson

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Nelson, P.D. Spectral aspect subconvex bounds for \(U_{n+1} \times U_n\). Invent. math. 232, 1273–1438 (2023). https://doi.org/10.1007/s00222-023-01180-x

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