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Pseudo-differential equations connected with p-adic forms and local zeta functions

Published online by Cambridge University Press:  17 April 2009

W. A. Zuniga-Galindo
W. A. Zuniga-Galindo
Affiliation:
Department of Mathematics and Computer Science, Barry University, 11300 N.E.Second Avenue, Miami Shores, FL 33161, United States of America, e-mail: [email protected]
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We study the asymptotics of fundamental solutions of p-adic pseudo-differential equations of type where f(∂,β) is a pseudo-differential operator with symbol , f is a form of arbitrary degree with coefficients in a p-adic field, λ ≥ 0, and g is a Schwartz-Bruhat function.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 70 , Issue 1 , August 2004 , pp. 73 - 86
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Atiyah, M.F., ‘Resolution of singularities and division of distributions’, Comm. Pure Appl. Math. 23 (1970), 145–150.CrossRefGoogle Scholar
[2]Denef, J. and Meuser, D., ‘A functional equation of Igusa's local zeta function’, Amer. J. Math. 109 (1987), 991–1008.CrossRefGoogle Scholar
[3]Gyoja, A., ‘Lectures on the functional equation satisfied by the p-adic local zeta functions of reductive prehomogeneous vector spaces’, (unpublished) Lectures at the Johns Hopkins University.Google Scholar
[4]Hironaka, H., ‘Resolution of singularities of an algebraic variety over a field of characteristic zero’, Ann. Math. 79 (1964), 109–326.CrossRefGoogle Scholar
[5]Igusa, J.-I., An introduction to the theory of local zeta functions, AMS/IP Studies in Advanced Mathematics 14 (American Mathematical Society, Providence R.I., 2000).Google Scholar
[6]Igusa, J.-I., ‘Complex powers and asymptotic expansions’, J. Reine Angew. Math. 268/269 (1974), 110–130. II, J. Reine Angew. Math., 278/279 (1975), 357–368.Google Scholar
[7]Igusa, J.-I., ‘A stationary phase formula for p-adic integrals and its applications’, in Algebraic Geometry and its Applications (Springer-Verlag, Berlin, Heidelberg, New york, 1994), pp. 175–194.CrossRefGoogle Scholar
[8]Igusa, J.-I., ‘Some results on p-adic complex powers’, Amer. J. Math. 106 (1984), 1013–1032.CrossRefGoogle Scholar
[9]Jang, Y., ‘An asymptotic expansion of the p-adic Green function’, Tohoku Math. J. 50 (1998), 229–242.CrossRefGoogle Scholar
[10]Khrennikov, A., ‘Fundamental solutions over the field of p-adic numbers’, St. Peterburgh Math. J. 4 (1993), 613–628.Google Scholar
[11]Kochubei, A.N., Pseudodifferential equations and stochastics over non-archimedean fields (Marcel Dekker, New York, 2001).Google Scholar
[12]Kochubei, A.N., ‘Fundamental solutions of pseudo-differential equations associated with p-adic quadratic forms’, Izv. Math. 62 (1998), 1169–1188.CrossRefGoogle Scholar
[13]Kochubei, A.N., ‘On p-adic Green functions’, Theoret. and Math. Phys. 96 (1993), 854–865.CrossRefGoogle Scholar
[14]Kochubei, A.N., ‘on the asymptotics of p-adic Green functions’, Proc. Steklov Inst. Math. 203 (1994/1995), 105–113.Google Scholar
[15]Sato, F., ‘On functional equations of zeta distributions’, Adv. Stud. Pure Math. 15 (1989), 465–508.CrossRefGoogle Scholar
[16]Sato, F., ‘p-adic Green functions and zeta functions’, Comment. Math. Univ. St. Pauli 51 (2002), 79–97.Google Scholar
[17]Taibleson, M.H., Fourier analysis on local fields, Mathematical Notes 15 (Princeton University Press, Princeton, N.J., 1975).Google Scholar
[18]Vladimirov, V.S., ‘On the spectrum of some pseudo-differential operators over p-adic number field’, (in Russian), Algebra i Analiz 2 (1990), 107–124.Google Scholar
[19]Vladimirov, V.S., Volovich, I.V. and Zelenov, E.I., p-adic analysis and mathematical physics (World Scientific, Singapore, 1994).CrossRefGoogle Scholar
[20]Weil, A., ‘Sur la formule de Siegel dans le théorie des groupes classiques’, Acta Math. 113 (1965), 1–87.CrossRefGoogle Scholar
[21]Zuniga-Galindo, W.A., ‘Fundamental solutions of pseudo-differential operators over p-adic fields’, Rend. Sem. Mat. Univ. Padova 109 (2003), 241–245.Google Scholar
[22]Zuniga-Galindo, W.A., ‘Igusa's local zeta functions of semiquasihomogeneous polynomials’, Trans. Amer. Math. Soc. 353 (2001), 3193–3207.CrossRefGoogle Scholar
[23]Zuniga-Galindo, W.A., ‘Local zeta functions and Newton Polyhedra’, Nagoya Math. J. 172 (2003), 31–58.CrossRefGoogle Scholar
[24]Zuniga-Galindo, W.A., ‘Local zeta function for non-degenerate homogeneous mappings’, Pacific J. Math. (to appear).Google Scholar