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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 $$ST_A\,$$.

MSC:
 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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