We show that for Bruhat intervals starting from the identity in Coxeter groups the conjecture of Lusztig and Dyer holds, that is, the R-polynomials and the Kazhdan-Lusztig polynomials defined on [e, u] only depend on the isomorphism type of [e, u]. To achieve this we use the purely poset-theoretic notion of special matching. Our approach is essentially a synthesis of the explicit formula for sp...

The goal of this paper is to explain how a simple but apparently new fact of linear algebra together with the cohomological interpretation of L-functions allows one to produce many examples of L-functions over function fields vanishing to high order at the center point of their functional equation. Conjectures of Birch and Swinnerton-Dyer, Bloch, and Beilinson relate the orders of vanishing of ...

Let f be a newform of weight 2 on Γ0(N), and let Af be the corresponding optimal abelian variety quotient of J0(N). We describe an algorithm to compute the order of the component group of Af at primes p that exactly divide N . We give a table of orders of component groups for all f of level N ≤ 127 and five examples in which the component group is very large, as predicted by the Birch and Swinn...

Let E be an elliptic curve over Q and let be an odd, irreducible twodimensional Artin representation. This article proves the Birch and Swinnerton-Dyer conjecture in analytic rank zero for the Hasse-WeilArtin L-series L(E, , s), namely, the implication L(E, , 1) = 0 ⇒ (E(H)⊗ ) = 0, where H is the finite extension of Q cut out by . The proof relies on padic families of global Galois cohomology c...

Mazur [8] has introduced the concept of visible elements in the Tate-Shafarevich group of optimal modular elliptic curves. We generalized the notion to arbitrary abelian subvarieties of abelian varieties and found, based on calculations that assume the BirchSwinnerton-Dyer conjecture, that there are elements of the Tate-Shafarevich group of certain sub-abelian varieties of J0(p) and J1(p) that ...

In this paper, we introduce arithmetic Heilbronn characters that generalize the notion of the classical Heilbronn characters, and discuss several properties of these characters. This formalism has several arithmetic applications. For instance, we obtain the holomorphy of suitable quotients of L-functions attached to elliptic curves, which is predicted by the Birch–Swinnerton–Dyer conjecture, an...

To a large extent, the investigations to be brought up today arise from a curious inadequacy having to do with the arrow on the left. On the one hand, it is widely acknowledged that the theory of motives finds a strong source of inspiration in Diophantine geometry, inasmuch so many of the structures, conjectures, and results therein have as model the conjecture of Birch and Swinnerton-Dyer, whe...

We refine the techniques of our previous paper [KM1] to prove that the average order of vanishing of L-functions of primitive automorphic forms of weight 2 and prime level q satisfies ∑ f∈S2(q)∗ ords= 1 2 L(f, s) ≤ C|S2(q) ∗| with C = 6.5, for all q large enough. On the Birch and Swinnerton-Dyer conjecture, this implies rank J0(q) ≤ C dim J0(q) for q prime large enough.

Abstract The purpose of this paper is to give a formula for the leading coefficient at s = 1 {s=1} L -function one-motives over function fields in terms Weil-étale cohomology, generalizing version Birch and Swinnerton-Dyer conjecture authors’ previous work. As consequence we...

Abstract In a paper published in 1980, the author gave an adelic Tamagawa number interpretation for Birch and Swinnerton-Dyer conjecture divisors on abelian varieties. Some years later, joint work with K. Kato, more general volume zeta values of motives weights $${<}\,{-1}$$ was proposed. at hand, is generalized to deal weight $$-1$$ . As points varieties are replaced by cohomology coefficie...