Abstract. Answering problems of Manin, we use the critical L-values of even weight k ≥ 4 newforms f ∈ S k (Γ 0 (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) keep track of 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.

In this paper, we prove the Iwasawa main conjecture for elliptic curves at an odd supersingular prime p. Some consequences are the p-parts of the leading term formulas in the Birch and Swinnerton-Dyer conjectures for analytic rank 0 or 1.

The point of this paper is to give an explicit p-adic analytic construction of two Iwasawa functions L ♯ p (f, T ) and L ♭ p (f, T ) for a weight two modular form a n q n and a good prime p. This generalizes work of Pollack who worked in the supersingular case and also assumed a p = 0. The Iwasawa functions work in tandem to shed some light on the Birch and Swinnerton-Dyer conjectures in the cyclotomic direction: We bound the rank and estimate the growth of the Tate-Shafarevich group in the cyclotomic direction analytically, encountering a new phenomenon for small slopes.

Let E be an elliptic curve with good reduction at a fixed odd prime p and K an imaginary quadratic field where p splits. We give a growth estimate for the Mordell-Weil rank of E over finite extensions inside the Z 2 pextension of K.