Zeta-polynomials for modular form periods

Answering problems of Manin, we use the critical L-values of even weight k≥4 newforms f∈Sk(Γ0(N)) to define zeta-polynomials Zf(s) which satisfy the functional equation Zf(s)=±Zf(1−s), and which obey the Riemann Hypothesis: if Zf(ρ)=0, then Re(ρ)=1/2. The zeros of the Zf(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 Zf(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 Å afarevič-Tate groups associated to the Tate twists of the modular motives.