Zeta-polynomials for modular form periods

Answering problems of Manin, we use the critical L-values of even weight k≥4 newforms f∈S_k(Γ₀(N)) to define zeta-polynomials Z_f(s) which satisfy the functional equation Z_f(s)=±Z_f(1−s), and which obey the Riemann Hypothesis: if Z_f(ρ)=0, then Re(ρ)=1/2. The zeros of the Z_f(s) on the critical line in t-aspect are distributed in a manner which is somewhat analogous to those of classical zeta-functions. These polynomials are assembled using (signed) Stirling numbers and “weighted moments” of critical L-values. In analogy with Ehrhart polynomials which keep track of integer points in polytopes, the Z_f(s) encode arithmetic information. Assuming the Bloch–Kato Tamagawa Number Conjecture, they encode the arithmetic of a combinatorial arithmetic–geometric object which we call the “Bloch–Kato complex” for f. Loosely speaking, these are graded sums of weighted moments of orders of Šafarevič–Tate groups associated to the Tate twists of the modular motives.