Abelian varieties over Q with bad reduction in one prime only

Let l be a prime. We show that there do not exist any non-zero semi-stable abelian varieties over Q with good reduction outside l if and only if l=2, 3, 5, 7 or 13. We show that any semi-stable abelian variety over Q with good reduction outside l=11 is isogenous to a power of the Jacobian variety of the modular curve X_0(11). In addition we show that there do not exist any non-zero abelian varieties over Q with good reduction outside l acquiring semi-stable reduction at l over a tamely ramified extension if and only if l \le 5

Four primality testing algorithms

In this expository paper we describe four primality tests. The first test is very efficient, but is only capable of proving that a given number is either composite or 'very probably' prime. The second test is a deterministic polynomial time algorithm to prove that a given numer is either prime or composite. The third and fourth primality tests are at present most widely used in practice. Both tests are capable of proving that a given number is prime or composite, but neither algorithm is deterministic. The third algorithm exploits the arithmetic of cyclotomic fields. Its running time is almost, but not quite polynomial time. The fourth algorithm exploits elliptic curves. Its running time is difficult to estimate, but it behaves well in practice.Comment: 21 page

On the Granular Stress-Geometry Equation

Using discrete calculus, we derive the missing stress-geometry equation for rigid granular materials in two dimensions, in the mean-field approximation. We show that (i) the equation imposes that the voids cannot carry stress, (ii) stress transmission is generically elliptic and has a quantitative relation to anisotropic elasticity, and (iii) the packing fabric plays an essential role.Comment: 6 page

The role of subtemperate slip in thermally driven ice stream margin migration

The amount of ice discharged by an ice stream depends on its width, and the widths of unconfined ice streams such as the Siple Coast ice streams in West Antarctica have been observed to evolve on decadal to centennial timescales. Thermally driven widening of ice streams provides a mechanism for this observed variability through melting of the frozen beds of adjacent ice ridges. This widening is driven by the heat dissipation in the ice stream margin, where strain rates are high, and at the bed of the ice ridge, where subtemperate sliding is possible. The inflow of cold ice from the neighboring ice ridges impedes ice stream widening. Determining the migration rate of the margin requires resolving conductive and advective heat transfer processes on very small scales in the ice stream margin, and these processes cannot be resolved by large-scale ice sheet models. Here, we exploit the thermal boundary layer structure in the ice stream margin to investigate how the migration rate depends on these different processes. We derive a parameterization of the migration rate in terms of parameters that can be estimated from observations or large-scale model outputs, including the lateral shear stress in the ice stream margin, the ice thickness of the stream, the influx of ice from the ridge, and the bed temperature of the ice ridge. This parameterization will allow the incorporation of ice stream margin migration into large-scale ice sheet models

Effectivity of Arakelov divisors and the theta divisor of a number field

We introduce the notion of an effective Arakelov divisor for a number field and the arithmetical analogue of the dimension of the space of sections of a line bundle. We study the analogue of the theta divisor for a number field.Comment: Plain Tex with 5 figures, 21 pages, revised versio