Chebotarev-Sato-Tate distribution for abelian surfaces potentially of \(\mathrm{GL}_2\)-type
The Sato-Tate conjecture in the case of abelian varieties can be extended to a conjecture predicting the equidistribution with respect to a certain probability measure of Frobenius conjugacy classes in the Chebotarev-Sato-Tate group associated to an abelian variety and a finite Galois extension of the underlying algebraic number field. The main result of this paper is the proof of the conjecture under certain technical assumptions in certain cases of abelian surfaces potentially of \(\mathrm{GL}_2\)-type over a totally real number field. In order to explain the result more precisely, some notation is required. Let \(A\) be a polarized abelian variety of dimension \(g\) over an algebraic number field \(F\). Given a rational prime \(\ell\), the \(\ell\)-adic rational Tate module \(V_\ell(A)\) is defined as the vector space \[ V_\ell(A)=T_\ell(A)\otimes_{\mathbb{Z}}\mathbb{Q} \] of dimension \(2g\) over \(\mathbb{Q}_\ell\), where \(T_\ell(A)\) is the usual \(\ell\)-adic Tate module. The symplectic basis of this vector space with respect to the Weil pairing arises from a fixed symplectic basis on the singular homology \(H_1(A(\mathbb{C}),\mathbb{Q})\). Then, the action of the absolute Galois group \(G_F\) of the underlying number field \(F\), gives the representation \[ \rho_{A,\ell}:G_F\to \mathrm{GSp}_{2g}(\mathbb{Q}_\ell). \] Let \(G_\ell\) denote its image, \(G_\ell^{\mathrm{Zar}}\) its Zariski closure, and \(G_\ell^{1,\mathrm{Zar}}\) the kernel of the similitude character of \(G_\ell^{\mathrm{Zar}}\). For simplicity, one may fix an isomorphism of the algebraic closure of \(\mathbb{Q}_\ell\) and \(\mathbb{C}\), and view these groups as subgroups of the complex group \(\mathrm{GSp}_{2g}(\mathbb{C})\). The Sato-Tate group \(ST_A\) of the abelian variety \(A\) is defined as the maximal compact subgroup of the complex Lie group \(G_\ell^{1,\mathrm{Zar}}(\mathbb{C})\), which lies inside \(\mathrm{USp}(2g)\), i.e., the intersection of the unitary group and the symplectic group. Let \(K/F\) be a finite Galois extension of \(F\). The Chebotarev-Sato-Tate group \(ST\) is then defined as \[ ST=ST_A\times\mathrm{Gal}(K/F), \] where \(\mathrm{Gal}(K/F)\) is the Galois group of the extension \(K/F\). Let \(S\) be the set of primes of \(F\) consisting of all primes that lie over \(\ell\), all primes at which \(A\) has bad reduction, and all primes at which the extension \(K/F\) ramifies. Given a prime \(\mathfrak{p}\) of \(F\) such that \(\mathfrak{p}\not\in S\), let \(g_\mathfrak{p}\) be the image under \(\rho_{A,\ell}\) of the arithmetic Frobenius at \(\mathfrak{p}\) in \(G_F\). According to [\textit{J. Tate}, Invent. Math. 2, 134--144 (1966; Zbl 0147.20303)], it follows that the element \(g_\mathfrak{p}q_\mathfrak{p}^{-1/2}\), where \(q_\mathfrak{p}\) is the order of the residue field at \(\mathfrak{p}\), is conjugate to an element of \(ST_A\). Let \(x_{\mathfrak{p}}\) denote the conjugacy class in the Chebotarev-Sato-Tate group \(ST\) of the element \[ g_\mathfrak{p}q_\mathfrak{p}^{-1/2}\times \sigma_\mathfrak{p}, \] where \(\sigma_\mathfrak{p}=\left(\frac{K/F}{\mathfrak{p}}\right)\) is the Artin symbol of \(K/F\). The Haar measure \(\mu\) on \(ST\) is defined as the product of the Haar measure on \(ST_A\) and the discrete measure on \(\mathrm{Gal}(K/F)\). Let \(E/F\) be the Galois extension of \(F\) such that the quotient \(ST_A / ST_A^0\) of the Sato-Tate group by its connected component of the identity is isomorphic to the Galois group \(\mathrm{Gal}(E/F)\). In the notation as above, the conjecture considered in this paper predicts the following. Suppose that the field \(K\) is linearly disjoint from the field \(E\) over \(F\). Then, the sequence \(\{x_\mathfrak{p}\}_{\mathfrak{p}\not\in S}\), ordered according to the norm of primes \(\mathfrak{p}\), is equidistributed on the set \(X\) of all conjugacy classes in \(ST\) with respect to the measure on \(X\) arising from the Haar measure \(\mu\) on \(ST\). The main result of the paper, providing evidence for the conjecture, says that the conjecture holds in certain cases. Let \(A\) be an abelian surface over a totally real number field \(F\) potentially of \(\mathrm{GL}_2\)-type. Let \(K/F\) be a finite Galois extension with the Galois group Gal\((K/F)\cong G_1\times G_2\) such that \(G_1\) is abelian and the extension \(K^{G_1}/F\) is totally real, where \(K^{G_1}\) is the fixed field of \(G_1\). Suppose that \(K\) is linearly disjoint from \(E\) over \(F\). Then, the conjecture above holds for \(A\) and \(K/F\).
scientific article; zbMATH DE number 7753451
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