Algebraic Dependence of Arithmetic Functions

Arithmetic Algebraic Geometry

[3] , Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank zero, Math. Finiteness results for modular curves of genus at least 2, Amer.

Arithmetic Genera of Algebraic Varieties.

AS e 5m_2(S), AE e O5m-2(E) and (As)r = (AE)r; thus A -* AS + AE iS a linear mapping of Om -2(9ON) into (M. It follows from the proposition (II) that, for every element a, + a2 of 5, there exists one and only one element A Of O5m-2() with AE = a2. This element A satisfies (a, As)r= a2r (AE)r = 0, while r = ES is a hyperplane section of S. Hence, by the proposition (I), we get AS = a,. Thus we s...

Arithmetic upon an Algebraic Surface

The title of my lecture is, I am afraid, probably misleading and certainly too ambitious. For, on the one hand, the connection between arithmetic and geometry suggested by it is not the modern development in divisors theory, but an application of algebraic geometry for arithmetical purposes. On the other hand, I shall confine the subject of this lecture to cubic surfaces in ordinary space, cons...

Arithmetic complexity in algebraic extensions

Given a polynomial f with coefficients from a field F, is it easier to compute f over an extension R of F than over F? We address this question, and show the following. For every polynomial f , there is a noncommutative extension R so that over R, f has polynomial size formulas. On the other hand, if F is algebraically closed, no commutative extension R can decrease formula or circuit complexit...

Algebraic Groups and Arithmetic Groups