4 The Birch and Swinnerton-Dyer conjecture 18 4.1 Analytic rank 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.1.1 Kolyvagin’s proof . . . . . . . . . . . . . . . . . . . . . 18 4.1.2 A variant . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.1.3 Kato’s proof . . . . . . . . . . . . . . . . . . . . . . . . 20 4.2 Analytic rank 1 . . . . . . . . . . . . . . . . . . . . . . . . ...

The location of Meteor Crater is shown on nearly all highway maps of Arizona. It lies 6 mi (9.7 km) south of 1-40 between Flagstaff and Winslow in northern Arizona; the turnoff from 1-40, 34 mi (55 km) east of Flagstaff, is well marked by signs along the highway. Access to the crater is by a paved road that leads directly to a visitor center and museum on the rim of the crater. Qualified scient...

In his unpublished manuscripts (referred to by Birch [1] as Fragment V, pp. 247-249), Ramanujan [3] gave a whole list of assertions about various (transforms of) modular forms possessing naturally associated Euler products, in more or less the spirit of his extremely beautiful paper entitled "On certain arithmetical functions" (in Trans. Camb. Phil. Soc. 22 (1916)). It is simply amazing how Ram...

Based on the tree-ring growth characteristics of Erman’s birch (Betula ermanii charm.) and the relationships between it and climatic factors at elevation of 1950m, the sensitivity of tree lines in Changbai Mountain to climatic factors was assessed. The results indicated tree line forest in Changbai Mountain had an obvious sensitivity to climate factors. However, difference from other study sits...

We formulate an analogue of the conjecture of Birch and Swinnerton-Dyer for Abelian schemes with everywhere good reduction over higher dimensional bases over finite fields. We prove some conditional results for the p′-part on it, and prove the p′-part of the conjecture for constant or isoconstant Abelian schemes, in particular the p′-part for (1) relative elliptic curves with good reduction or ...

This discussion paper reveals the contribution of pollen transport conditions to the inter-annual variability of the seasonal pollen index (SPI). This contribution is quantified as a sensitivity of the pollen model predictions to meteorological variability and is shown to be a noticeable addition to the SPI variability caused by plant reproduction cycles. A specially designed SILAM model re-ana...

is the Hecke L-series attached to the eigenform f . Hecke’s theory shows that L(f, s) has an Euler product expansion identical to (2), and also that it admits an integral representation as a Mellin transform of f . This extends L(f, s) analytically to the whole complex plane and shows that it satisfies a functional equation relating its values at s and 2− s. The modularity of E thus implies tha...

Precipitation amounts and patterns at high latitude sites have been predicted to change as a result of global climatic changes. We addressed vegetation responses to three years of experimentally increased summer precipitation in two previously unaddressed tundra types: Betula nana-dominated shrub tundra (northeast Siberia) and a dry Sphagnum fuscum-dominated bog (northern Sweden). Positive resp...