Glossary of Arithmetic and Diophantine Geometry - I

I

Igusa zeta-function

An Igusa zeta-function, named for Jun-ichi Igusa, is a generating function counting numbers of points on an algebraic variety modulo high powers pn of a fixed prime number p. General rationality theorems are now known, drawing on methods of mathematical logic.

Infinite descent

Infinite descent was Pierre de Fermat's classical method for Diophantine equations. It became one half of the standard proof of the Mordell–Weil theorem, with the other being an argument with height functions (q.v.). Descent is something like division by two in a group of principal homogeneous spaces (often called 'descents', when written out by equations); in more modern terms in a Galois cohomology group which is to be proved finite. See Selmer group.

Iwasawa theory

Iwasawa theory builds up from the analytic number theory and Stickelberger's theorem as a theory of ideal class groups as Galois modules and p-adic L-functions (with roots in Kummer congruence on Bernoulli numbers). In its early days in the late 1960s it was called Iwasawa's analogue of the Jacobian. The analogy was with the Jacobian variety J of a curve C over a finite field F (qua Picard variety), where the finite field has roots of unity added to make finite field extensions F′ The local zeta-function (q.v.) of C can be recovered from the points J(F′) as Galois module. In the same way, Iwasawa added pn-power roots of unity for fixed p and with n → ∞, for his analogue, to a number field K, and considered the inverse limit of class groups, finding a p-adic L-function earlier introduced by Kubota and Leopoldt.