A new generating function for calculating the Igusa local zeta function

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• 作者：Raemeon A. Cowan ; Daniel J. Katz<sup> ; sup><sup>daniel.katz@csun.edu" class="auth_mail" title="E-mail the corresponding authorsup> ; 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 <span id="mmlsi1" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816311549&_mathId=si1.gif&_user=111111111&_pii=S0001870816311549&_rdoc=1&_issn=00018708&md5=5e5d6971527cab894d8d7d2e145127d7" title="Click to view the MathML source">Z<sub>fsub>span><span class="mathContainer hidden"><span class="mathCode">scroll">sub>Zfsub>span>span>span> of a polynomial <span id="mmlsi2" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816311549&_mathId=si2.gif&_user=111111111&_pii=S0001870816311549&_rdoc=1&_issn=00018708&md5=0113f7dab6775f1733284096c00a0faa" title="Click to view the MathML source">f(x<sub>1sub>,…,x<sub>nsub>)span><span class="mathContainer hidden"><span class="mathCode">scroll">fstretchy="false">(sub>x1sub>,…,sub>xnsub>stretchy="false">)span>span>span> over a p  -adic field. This involves a new kind of generating function <span id="mmlsi3" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816311549&_mathId=si3.gif&_user=111111111&_pii=S0001870816311549&_rdoc=1&_issn=00018708&md5=8158e3abcbe8edfad71fd320fed79990" title="Click to view the MathML source">G<sub>fsub>span><span class="mathContainer hidden"><span class="mathCode">scroll">sub>Gfsub>span>span>span> that is the projective limit of a family of generating functions, and contains more data than <span id="mmlsi1" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816311549&_mathId=si1.gif&_user=111111111&_pii=S0001870816311549&_rdoc=1&_issn=00018708&md5=5e5d6971527cab894d8d7d2e145127d7" title="Click to view the MathML source">Z<sub>fsub>span><span class="mathContainer hidden"><span class="mathCode">scroll">sub>Zfsub>span>span>span>. This <span id="mmlsi3" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816311549&_mathId=si3.gif&_user=111111111&_pii=S0001870816311549&_rdoc=1&_issn=00018708&md5=8158e3abcbe8edfad71fd320fed79990" title="Click to view the MathML source">G<sub>fsub>span><span class="mathContainer hidden"><span class="mathCode">scroll">sub>Gfsub>span>span>span> 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 <span id="mmlsi4" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816311549&_mathId=si4.gif&_user=111111111&_pii=S0001870816311549&_rdoc=1&_issn=00018708&md5=e33c28fd13bf65c5c9dcad433e5e4cc5" title="Click to view the MathML source">Q+Lspan><span class="mathContainer hidden"><span class="mathCode">scroll">Q+Lspan>span>span> 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.