Let \(A\) be an abelian variety defined over a number field \(k\) of dimension \(g\ge 1\), and let \(N\) be the absolute value of the conductor of \(A\). For a fixed rational prime \(\ell\), there is a well-known Galois representation on the rational \(\ell\)-adic Tate module of \(A\), and given a nonzero prime ideal \(\mathfrak p\) of the ring of integers of \(k\) not dividing \(\ell N\), the trace of the Frobenius map at \(\mathfrak p\) is denoted by \(a_{\mathfrak p}\). The absolute value of the norm of \(\mathfrak p\) is denoted by \(\mathrm{Nm}(\mathfrak p)\), and \(\overline a_{\mathfrak p}:=a_{\mathfrak p}/\sqrt{\mathrm{Nm}(\mathfrak p)}\). By Hasse-Weil bound, the normalized trace is contained in \([-2g, 2g]\).
Attached to \(A\) is a compact real Lie subgroup \(\mathrm{ST}(A)\) of the unitary symplectic group \(\mathrm{USp}(2g)\), called the Sato-Tate group of \(A\), and the Sato-Tate conjecture states that \[ \sum_{\mathrm{Nm}(\mathfrak p)\le x} \delta_I( \overline a_{\mathfrak p}) \sim \mu(I) \mathrm{Li}(x) \] where \(I\) is a nonzero subinterval of \([-2g, 2g]\), \(\delta_I\) is the characteristic function on \(I\), \(\mu\) is the pushforward measure on \([-2g, 2g]\) via the trace map of the normalized Haar meausure of \(\mathrm{ST}(A)\), and \(\mathrm{Li}(x):=\int_2^x dt/\log t\). Let \(\mathfrak g\) be the complexified Lie algebra of \(\mathrm{ST}(A)\), which can be written as \(\mathfrak s \times \mathfrak a\) where \(\mathfrak s\) is semi-simple and \(\mathfrak a\) is abelian. Let \(\epsilon_{\mathfrak g}:= \frac1{2(q+\varphi)}\) where \(q\) is the rank of \(\mathfrak g\) and \(\varphi\) is the size of the positive roots of \(\mathfrak s\).
The authors of the paper under review prove that if \(\mathrm{ST}(A)\) is connected, and if we assume the Mumford-Tate conjecture and the generalized Riemann hypothesis for \(L(\chi,s)\) for every irreducible character \(\chi\) of \(\mathrm{ST}(A)\), then \[ \sum_{\mathrm{Nm}(\mathfrak p)\le x} \delta_I( \overline a_{\mathfrak p}) = \mu(I) \mathrm{Li}(x) + O\ \frac{x^{1-\epsilon_{\mathfrak g}} \log(Nx)^{2\epsilon_{\mathfrak g}} } { \log(Nx)^{1-4\epsilon_{\mathfrak g}} } \] where the sum runs over primes not dividing \(N\). The authors also note that the implied constant of \(O\)-notation depends on \(k\) and \(g\) for \(x\ge x_0\) not on \(A\), but \(x_0\) depends on \(\mathfrak g\). This result was proved for elliptic curves with complex multiplication in [V. K. Murty, Rocky Mt. J. Math. 15, 535–551 (1985; Zbl 0587.14009)]. The authors’ theorem can be obtained by using an estimate of [V. K. Murty, Rocky Mt. J. Math. 15, 535–551 (1985; Zbl 0587.14009)], as presented in [A. Bucur and K. S. Kedlaya, Contemp. Math. 663, 45–56 (2016; Zbl 1417.11103)], on truncated sums of an irreducible character over the prime ideals of \(k\), and they use the Mumford-Tate conjecture to make the implied constant of \(O\)-notation independent of \(A\).