turkmath.org

Modular forms have attracted attention for many years. Recently, one of the breakthrough and very significant results in pure mathematics is the proof of the Sato-Tate conjecture for non-CM modular eigenforms (even for Hilbert eigenforms) by Taylor, Barnet-Lamb, Geraghty and Harris. A special case of the Sato-Tate theorem states that signs of coefficients of integral weight Hecke eigenforms are equidistributed. That such should also be the case for half-integral weight forms was conjectured by Kohnen and Bruinier. We will explain how the Shimura lift and the Sato-Tate theorem can be exploited to obtain sign equidistribution for certain subsets of the coefficients of half-integral weight eigenforms. We will obtain the results in different notions of density. Furthermore, an explicit error bound for the Sato-Tate conjecture in the CM-case will be also given. Finally, if time permits, then we will conclude with the current position of the Bruinier-Kohnen sign equidistribution conjecture and giving some open problems in theory of automorphic forms. This is a joint work with G. Wiese (Luxembourg) and Sara Arias-de-Reyna (Luxembourg).