SIC-POVMs from Stark units: Prime dimensions n^2+3

We propose a recipe for constructing a SIC fiducial vector in complex Hilbert space of dimension of the form d=n²+3, focussing on prime dimensions d=p. Such structures are shown to exist in thirteen prime dimensions of this kind, the highest being p=19603. The real quadratic base field K (in the standard SIC terminology) attached to such dimensions has fundamental units u_K of norm -1. Let mathbb{Z}_K denote the ring of integers of K, then pmathbb{Z}_K splits into two ideals mathfrak{p} and mathfrak{p}'. The initial entry of the fiducial is the square ξ² of a geometric scaling factor ξ, which lies in one of the fields K\(sqrt{u_K}\). Strikingly, the other p-1 entries of the fiducial vector are each the product of ξ and the square root of a Stark unit. These Stark units are obtained via the Stark conjectures from the value at s=0 of the first derivatives of partial L functions attached to the characters of the ray class group of mathbb{Z}_K with modulus mathfrak{p}infty₁, where infty₁ is one of the real places of K.