On the Gross-Prasad conjecture with its refinement for \((\mathrm{SO}(5), \mathrm{SO}(2))\) and the generalized Böcherer conjecture (Q6622797)

In a previous work [J. Eur. Math. Soc. (JEMS) 23, No. 4, 1295--1331 (2021; Zbl 1486.11072)] the authors studied the Gross-Prasad period for \((\mathrm{SO}(2n+1),\mathrm{SO}(2))\) when the character on \(\mathrm{SO}(2)\) is trivial. In this work, the condition of trivial character on \(\mathrm{SO}(2)\) is dropped in the case of the Gross-Prasad period for \((\mathrm{SO}(5),\mathrm{SO}(2))\).\N\NThe main theorem proves the identity predicted by Ichino-Ikeda relating the Gross-Prasad period to the central \(L\)-value of an automorphic \(L\)-function. This extends the result of the authors' previous paper in the case of \((\mathrm{SO}(5),\mathrm{SO}(2))\). The proof uses theta correspondence to reduce the identity for Gross-Prasad period to that of a Whittaker period, similar to the authors' previous paper. However the use of theta correspondence is quite different here. While in the previous paper the theta correspondence relates the Gross-Prasad period to Whittaker periods on metaplectic groups (where a conditional result of Lapid-Mao can be used), here the authors use a sequence of theta correspondences along with several accidental isomorphisms to relate the Gross-Prasad period with Whittaker periods on \(\mathrm{PGL}(2)\times \mathrm{PGL}(2)\) and \(\mathrm{PGL}(4)\). They are able to get an unconditional formula as the Whittaker period formula on \(\mathrm{PGL}(n)\) is unconditional.\N\NThe result is also translated into the language of Siegel modular forms, which gives an explicit identity that generalizes Böcherer's conjecture to the non-trivial toroidal character case. This identity is expected to have a broad spectrum of interesting applications both arithmetic and analytic.