Arithmetic of CM Center Polynomials for the Riemann Xi Kernel

Abstract

This preprint studies the arithmetic structure of the even derivatives at the symmetry center of the standard Jacobi-theta kernel associated with the Riemann Xi function. Using Romik’s centered expansion of the Jacobi theta constant, the paper constructs a family of rational polynomials that exactly encode all even center derivatives of the kernel. Their degrees and leading coefficients are determined explicitly. Nesterenko’s algebraic-independence theorem is then used to prove transcendence and nonvanishing results for the center derivatives and the associated derivative Laguerre quantities. The paper further introduces an integral normalization of these center polynomials and studies their local arithmetic. Exact Newton polygon results are obtained at the primes 2 and 5, while a uniform initial Newton polygon edge is established at the prime 3. These results imply the existence of a quadratic Eisenstein factor at both 3 and 5 for every nontrivial member of the family, and show that 3 and 5 are precisely the odd primes with this universal property. The results concern the arithmetic of a derived polynomial family associated with the Xi kernel and are distinct from the previously studied congruence properties of the underlying Taylor coefficients at complex multiplication points.