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Sato-Tate distributions and Galois endomorphism modules in genus 2. (English) Zbl 1269.11094
The paper aims at providing an analogue of the Sato-Tate conjecture for abelian surfaces defined over a number field \(k\) (the original conjecture for elliptic curves states that Euler factors are equidistributed with respect to certain measures defined over \([0,\pi]\,\)). Let \(A\) be an abelian surface defined over \(k\) and, for any rational prime \(\ell\), let \(T_\ell(A)\) be the \(\ell\)-adic Tate module of \(A\) and let \(V_{\ell}(A):=T_{\ell}(A)\otimes \mathbb{Q}\). Choosing a polarization for \(A\) and a basis for \(T_\ell(A)\) one gets a continuous homomorphism \(\rho_{A,\ell}:G_k:=\text{Gal}(\overline{k}/k) \rightarrow \text{GSp}_{2g}(\mathbb{Q}_\ell)\) providing the action of \(G_k\) on \(V_\ell(A)\) (\(g\) is the genus of \(A\)). Now let \(\mathfrak{p}\) be a prime of \(k\) of good reduction for \(A\): put \(L_{\mathfrak{p}}(A,T):=\det(1-T\text{Frob}_{\mathfrak{p}},V_{\ell}(A))\) and define the normalized \(L\)-polynomial as \(\overline{L}_{\mathfrak{p}}(A,T):=L_{\mathfrak{p}}(A,q^{-\frac{1}{2}}T)\) (where \(q\) is the cardinality of the residue field of \(\mathfrak{p}\)). Well known properties of the roots of \(\overline{L}_{\mathfrak{p}}(A,T)\) provide a correspondence with a class in \(Conj(\text{USp}(2g))\) (the conjugacy classes in the unitary symplectic group). The authors propose the following generalization for the Sato-Tate conjecture:
(C1) the \(\overline{L}_{\mathfrak{p}}(A,T)\) in \(Conj(\text{USp}(2g))\) are equidistributed with respect to the Haar measure for the Sato-Tate group \(ST_A\,\).
The group \(ST_A\) is defined as follows: let \(G^1\) be the kernel of the cyclotomic character \(\chi_\ell:G_k\rightarrow \mathbb{Q}_\ell^*\) and let \(G^{1,Zar}_\ell\) be the Zariski closure of \(\rho_{A,\ell}(G^1)\) in \(\text{GSp}_{2g}(\mathbb{Q}_\ell)\). Fix an embedding \(\iota:\mathbb{Q}_\ell \hookrightarrow \mathbb{C}\), then the Sato-Tate group for \(A\) (with respect to \(\iota\) and \(\ell\)) is the maximal compact Lie subgroup \(ST_A\) of \(G^{1,Zar}_\ell\otimes_\iota \mathbb{C}\) contained in \(\text{USp}(2g)\).
More precisely they also associate \(\mathfrak{p}\) with the conjugacy class \(s(\mathfrak{p})\in Conj(ST_A)\) of the semisimple component of the Jordan decomposition of \(q^{-\frac{1}{2}}\rho_{A,\ell}(\text{Frob}_{\mathfrak{p}})\), and refine their statement to:
(C2) the elements \(s(\mathfrak{p})\) are equidistributed with respect to the Haar measure on \(Conj(ST_A)\).
To test numerically the conjecture the authors consider the case \(g=2\) and provide a list of the admissible Sato-Tate groups for this case. There are 55 of them (modulo conjugation), the remaining ones are ruled out using conditions which a Sato-Tate group is expected to fulfill according to some related (pre-existing) conjectures (namely the Mumford-Tate conjecture and the algebraic Sato-Tate conjecture). Then the list is refined and 3 of those groups are eliminated using the Galois type associated to \(A\). Let \(K/k\) be the minimal extension of \(k\) over which all endomorphisms of \(A\) (over \(\overline{\mathbb{Q}}\)) are defined, then the Galois type associated to \(A\) is the isomorphism class of the pair (consisting of a finite group and an \(\mathbb{R}\)-algebra) \([\text{Gal}(K/k),\text{End}(A_K)\otimes \mathbb{R}]\). The authors prove that the Galois type (for \(g=2\)) uniquely determines the conjugacy class of the Sato-Tate group and vice versa, thus showing that their final list of 52 groups is minimal.
The final sections exhibit examples of Jacobians of curves of genus 2 which realize all the 52 Sato-Tate groups (or, equivalently, all the 52 Galois types) described before (it is worth mentioning that 34 of them already occur with \(k=\mathbb{Q}\)) and provide numerical evidence for the statement (C1) by checking the distributions of the first and second coefficients of the characteristic polynomial for the conjugacy classes of the \(ST_A\,\).

11M50 Relations with random matrices
11G10 Abelian varieties of dimension \(> 1\)
11G20 Curves over finite and local fields
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14K15 Arithmetic ground fields for abelian varieties
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