Let C/Q be the genus 3 Picard curve given by the affine model y^3=x^4-x. In this paper we compute its Sato-Tate group, show the generalized Sato-Tate conjecture for C, and compute the statistical moments for the limiting distribution of the normalized local factors of C.
According to the Birch and Swinnerton-Dyer conjectures, if A/Q is an abelian variety then its L-function must capture substantial part of the arithmetic properties of A. The smallest number field L where A has all its endomorphisms defined must also
have a role. This article deals with the relationship between these two objects in the specific case of modular abelian varieties A_f/Q associated to weight 2 newforms for the modular group Gamma_1(N). Specifically, our goal is to relate the order of L(A_f/Q,s) at s = 1 with Euler products cropped by the set of primes that split completely in L. The results we obtain for the case when f has complex multiplication are complete, while in the absence of CM, our results depend on the rate of convergence in Sato-Tate distributions.
For every normalized newform f in S_2(Gamma_1(N)) with complex multiplication, we study the modular parametrizations of elliptic curves C from the abelian variety A_f. We apply the results obtained when C is Grosss elliptic curve A(p).
