A new generating function for calculating the Igusa local zeta function

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• 作者：Raemeon A. Cowan ; Daniel J. Katz ; ^{daniel.katz@csun.edu" class="auth_mail" title="E-mail the corresponding author} ; Lauren M. White
• 关键词：Igusa local zeta function ; Generating function ; Quadratic form ; p-Adic field
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：2 January 2017
• 年：2017
• 卷：304
• 期：Complete
• 页码：355-420
• 全文大小：979 K

文摘

A new method is devised for calculating the Igusa local zeta function 311549&_mathId=si1.gif&_user=111111111&_pii=S0001870816311549&_rdoc=1&_issn=00018708&md5=5e5d6971527cab894d8d7d2e145127d7" title="Click to view the MathML source">Z_f$Z_f$ of a polynomial 311549&_mathId=si2.gif&_user=111111111&_pii=S0001870816311549&_rdoc=1&_issn=00018708&md5=0113f7dab6775f1733284096c00a0faa" title="Click to view the MathML source">f(x₁,…,x_n)$f(x_1,\dots ,x_n)$ over a p  -adic field. This involves a new kind of generating function 311549&_mathId=si3.gif&_user=111111111&_pii=S0001870816311549&_rdoc=1&_issn=00018708&md5=8158e3abcbe8edfad71fd320fed79990" title="Click to view the MathML source">G_f$G_f$ that is the projective limit of a family of generating functions, and contains more data than 311549&_mathId=si1.gif&_user=111111111&_pii=S0001870816311549&_rdoc=1&_issn=00018708&md5=5e5d6971527cab894d8d7d2e145127d7" title="Click to view the MathML source">Z_f$Z_f$. This 311549&_mathId=si3.gif&_user=111111111&_pii=S0001870816311549&_rdoc=1&_issn=00018708&md5=8158e3abcbe8edfad71fd320fed79990" title="Click to view the MathML source">G_f$G_f$ 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 are presented, as well as for polynomials over unramified 2-adic fields of the form 311549&_mathId=si4.gif&_user=111111111&_pii=S0001870816311549&_rdoc=1&_issn=00018708&md5=e33c28fd13bf65c5c9dcad433e5e4cc5" title="Click to view the MathML source">Q+L$Q+L$ where Q is a quadratic form and L is a linear form such that Q and L have disjoint variables. For a quadratic form over an arbitrary p-adic field with odd p, this new technique makes clear precisely which of the three candidate poles are actual poles.