We study the elliptic curves in Cremona’s tables that are predicted by the Birch–SwinnertonDyer conjecture to have elements of order 7 in their Tate–Shafarevich group. We show that in many cases these elements are visible in an abelian surface or abelian 3-fold.

Wood anatomy of Betula uber (Ashe) Fernald is described and compared with woods of other birches belonging to series Humiles and series Costatae. Anatomically, wood of B. uber is typical of birches in general. On the basis of pore size and frequency, fiber characteristics, axial xylem parenchyma distribution, and absence of aggregate rays, it is argued that B. uber is properly classified in ser...

A new species of Eulophinae, Pnigalio eriocraniae Li & Yang, sp. n., is described and illustrated. This new species is a larval ectoparasitoid of Eriocrania semipurpurella alpina Xu (Lepidoptera, Eriocraniidae), a leaf miner in birch trees, Betula spp. (Betulaceae), in Qinghai Province, northwest China. The biology of the new species and a key to the known species from China are provided.

Let N ≡ 1(mod 4) be a positive integer and let be the single even primitive quadratic Dirichlet character on (Z/NZ)×. Let f ∈ S2(Γ0(N), ) be a newform with nebentypus . By the Shimura construction, f corresponds to an abelian variety Af defined over Q whose dimension is [Kf : Q] where Kf is the number field associated with f . When dimAf = 2, the Fricke involution wN acts on Af and is defined o...

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 Takayuki Oda on his 60th birthday.