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Results : Local Zeta Functions for Quadratic Polynomials

A new method is devised for calculating the Igusa local zeta function Zf of a polynomial f(x1, . . . , xn) over a p-adic field. This involves a new kind of generating function Gf that is the projective limit of a family of generating functions, and contains more data than Zf . This Gf resides in an algebra whose structure is naturally compatible with operations on the underlying polynomials, facilitating calculation of local zeta functions. This new technique is used to expand significantly the set of quadratic polynomials whose local zeta functions have been calculated explicitly. Local zeta functions for arbitrary quadratic polynomials over p-adic fields with p odd, and polynomials without constant term over unramified 2-adic fields are presented. For a quadratic form over an arbitrary p-adic field, this new technique renders transparent the fact that there are only three candidate poles, and when p is odd, makes clear precisely which ones are truly poles.