Daniel Miller

First Position
Data Scientist, MicrosoftDissertation
Counterexamples related to the Sato–Tate conjectureAdvisor
Research Area
Abstract
Let E/Q be an elliptic curve. The Sato--Tate conjecture, now a theorem, tells us that the angles θp=cos−1(ap2√p) are equidistributed in [0,π] with respect to the measure 2πsin2θdθ if E is non-CM (resp.~12πdθ+12δπ/2 if E is CM). In the non-CM case, Akiyama and Tanigawa conjecture that the discrepancy DN=sup 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.