Counterexamples related to the Sato-Tate conjecture

Abstract

Let $E_{/\mathbf{Q}}$ be an elliptic curve. The Sato--Tate conjecture, now a theorem, tells us that the angles $\theta_p =\cos^{-1}\left(\frac{a_p}{2\sqrt p}\right)$ are equidistributed in $[0,\pi]$ with respect to the measure $\frac{2}{\pi}\sin^2\theta\, d\theta$ if $E$ is non-CM (resp.~$\frac{1}{2\pi} d \theta + \frac 1 2 \delta_{\pi/2}$ if $E$ is CM). In the non-CM case, Akiyama and Tanigawa conjecture that the discrepancy \[ D_N = \sup_{x\in [0,\pi]} \left| \frac{1}{\pi(N)} \sum_{p\leqN} 1_{[0,x]}(\theta_p) - \int_0^x \frac{2}{\pi}\sin^2\theta\, d\theta\right| \] asymptotically decays like $N^{-\frac 1 2+\epsilon}$, as is suggested by computational evidence and certain reasonable heuristics on the Kolmogorov--Smirnov statistic. This conjecture implies the Riemann hypothesis for all $L$-functions associated with $E$. It is natural to assume that the converse (``generalized Riemann hypothesis implies discrepancy estimate'') holds, as is suggested by analogy with Artin $L$-functions. We construct, for compact real tori, ``fake Satake parameters'' yielding $L$-functions which satisfy the generalized Riemann hypothesis, but for which the discrepancy decays like $N^{-\epsilon}$ for any fixed $\epsilon>0$. This provides evidence that for CM abelian varieties, the converse to ``Akiyama--Tanigawa conjecture implies generalized Riemann hypothesis'' does not follow in a straightforward way from the standard analytic methods. We also show that there are Galois representations $\rho\colon Gal(\overline{\mathbf{Q}} /\mathbf{Q}) \to GL_2(\mathbf{Z}_l)$, ramified at an arbitrarily thin (but still infinite) set of primes, whose Satake parameters can be made to converge at any specified rate to any fixed measure $\mu$ on $[0,\pi]$ for which $\cos_\ast\mu$ is absolutely continuous with bounded derivative.