Periods and distribution of cycles on Hilbert modular varieties (Q1001767)

The main result of this paper is an explicit identity between period integrals of automorphic forms and special values of automorphic \(L\)-series. Specifically, suppose that \(F\) is a totally real field of degree \(d\) and that \(k\) is a subfield of \(F\) such that \([F: k]=2\). Let \({\mathcal M}\) be the Hilbert modular variety associated to \(\mathrm{PGL}_2(\widehat{\mathcal O}_F)\) and let \({\mathcal M}^0\) be the connected component of \(\mathcal M\) corresponding to the unit element. Further let \(x \in \mathcal M^0\) be a point such that the Mumford-Tate group of \(x\) has the form \(\mathrm{Res}_{k/\mathbb Q}(D^{\times}/k^{\times})\) for some quaternion algebra \(D\) over \(k\) embedded in \(M_2(F).\) Then we denote by \(\mathcal M_x\) the minimal Shimura subvariety of Hodge type containing \(x\). Suppose now that \(\pi\) is an irreducible automorphic representation of \(\mathrm{PGL}_2(\mathbb A_F)\) which is spherical with weight \(0\) and let \(\phi\) be its newform. Then the main theorem states that, under certain conditions, \(\pi\) is a base change of an automorphic representation \(\sigma\) of \(\mathrm{GL}_2(\mathbb A_k)\) with central character associated to the extension \(F/k\). Moreover, the integral, with respect to an invariant measure \(dh\), \[ \ell_x(\phi):=\text{vol}(\mathcal M_x(\mathbb C))^{-1} \int_{\mathcal M_x(\mathbb C)} \phi(h)\, dh \] is explicitly expressed as a multiple of \(L(1, \text{Sum}^2{\sigma})\). As an application of this result, the authors prove that if \(\mathcal M_{x_i}\) is an infinite sequence of distinct compact quaternion Shimura subvarieties of \(\mathcal M\) defined as above, then \[ \lim_{i \to \infty} \text{vol}((\mathcal M_{x_i} \cap \mathcal M^0)(\mathbb C))^{-1} \int_{(\mathcal M_{x_i} \cap \mathcal M^0)(\mathbb C)}\phi(h)\,dh=\int_{\mathcal M^0(\mathbb C)} \phi(h)\, dh. \] This result is a special case of a conjecture which implies the André-Oort conjecture.