On Morphisms Between Connected Commutative Algebraic Groups over a Field of Characteristic 0.

Let be a field of characteristic 0 and let and be connected commutative algebraic groups over . Let Mor(,) denote the set of morphisms of algebraic varieties → that map the neutral element to the neutral element. We construct a natural retraction from Mor(,) to Hom(,) (for arbitrary and ) which commutes with the composition and addition of morphisms.

On the frequency of height values.

We count algebraic numbers of fixed degree and fixed (absolute multiplicative Weil) height with precisely conjugates that lie inside the open unit disk. We also count the number of values up to that the height assumes on algebraic numbers of degree with precisely conjugates that lie inside the open unit disk. For both counts, we do not obtain an asymptotic, but only a rough order of growth, which arises from an asymptotic for the logarithm of the counting function; for the first count, even this rough order of growth exists only if or .