## Tohoku Mathematical Journal

### Meromorphic continuations of local zeta functions and their applications to oscillating integrals

#### Abstract

We introduce a new method which enables us to calculate the coefficients of the poles of local zeta functions very precisely and prove some explicit formulas. Some vanishing theorems for the candidate poles of local zeta functions will be also given. Moreover we apply our method to oscillating integrals and obtain an explicit formula for the coefficients of their asymptotic expansions.

#### Article information

Source
Tohoku Math. J. (2), Volume 65, Number 2 (2013), 159-178.

Dates
First available in Project Euclid: 25 June 2013

https://projecteuclid.org/euclid.tmj/1372182720

Digital Object Identifier
doi:10.2748/tmj/1372182720

Mathematical Reviews number (MathSciNet)
MR3079283

Zentralblatt MATH identifier
1345.14010

#### Citation

Okada, Toshihisa; Takeuchi, Kiyoshi. Meromorphic continuations of local zeta functions and their applications to oscillating integrals. Tohoku Math. J. (2) 65 (2013), no. 2, 159--178. doi:10.2748/tmj/1372182720. https://projecteuclid.org/euclid.tmj/1372182720

#### References

• V. I. Arnold, S. M. Gusein-Zade and A. N. Varchenko, Singularities of differential maps, Volume II, Birkhäuser, 1988.
• D. Barlet, Développement asymptotique des fonctions obtenues par intégration sur les fibres, Invent. Math. 68 (1982), 129–174.
• J. Denef, J. Nicaise and P. Sargos, Oscillating integrals and Newton polyhedra, J. Anal. Math. 95 (2005), 147–172.
• J. Denef and P. Sargos, Polyèdre de Newton et distribution $f_+^s$ I, J. Anal. Math. 53 (1989), 201–218.
• W. Fulton, Introduction to toric varieties, Ann. of Math. Stud. 131, The William H. Roever Lectures in Geometry, Princeton University Press, Princeton, NJ, 1993.
• I. M. Gelfand and G. E. Shilov, Generalized functions–- properties and operations, Volume I, Academic Press, 1964.
• A. Greenleaf, M. Pramanik and W. Tang, Oscillatory integral operators with homogeneous polynomial phases in several variables, J. Funct. Anal. 224 (2007), 444–487.
• A. Greenleaf and A. Seeger, Oscillatory integral operators with low-order degeneracies, Duke Math. J. 112 (2002), 397–420.
• J. Igusa, An introduction to the theory of local zeta functions, AMS/IP Stud. Adv. Math. 14, American Mathematical Society, Providence, RI; International Press, Cambridge, MA, 2000.
• P. Jacobs, The distribution $|f|^{\lambda}$, oscillating integrals and principal value integrals, J. Anal. Math. 81 (2000), 343–372.
• A. Kaneko, Newton diagrams, singular points and oscillating integrals, in Japanese, Lecture notes in Sophia university, No. 11, 1981.
• T. Kimura, Introduction to prehomogeneous vector spaces, Transl. Math. Monogr. 215, American Mathematical Society, Providence, RI, 2003.
• Y. Matsui and K. Takeuchi, A geometric degree formula for $A$-discriminants and Euler obstructions of toric varieties, Adv. Math. 226 (2011), 2040–2064.
• Y. Matsui and K. Takeuchi, Milnor fibers over singular toric varieties and nearby cycle sheaves, Tohoku Math. J. 63 (2011), 113–136.
• Y. Matsui and K. Takeuchi, Monodromy at infinity, Newton polyhedra and constructible sheaves, Math. Z. 268 (2011), 409–439.
• J. Nicaise, An introduction to $p$-adic and motivic zeta functions and the monodromy conjecture, Algebraic and analytic aspects of zeta functions and L-functions, 141–166, MSJ Mem. 21, Math. Soc. Japan, Tokyo, 2010.
• T. Oda, Convex bodies and algebraic geometry. An introduction to the theory of toric varieties, Ergeb. Math. Grenzgeb. (3) 15, Springer-Verlag, Berlin, 1988.
• M. Oka, Non-degenerate complete intersection singularity, Hermann, Paris, 1997.
• M. Sato, T. Kawai and M. Kashiwara, Microfunctions and pseudo-differential equations, Hyperfunctions and pseudo-differential equations (Proc. Conf., Katata, 1971; dedicated to the memory of André Martineau), pp. 265–529, Lecture Notes in Math. 287, Springer, Berlin, 1973.
• H. Soga, Conditions against rapid decrease of oscillatory integrals and their applications to inverse scattering problems, Osaka J. Math. 23 (1986), 441–456.
• K. Takeuchi, Monodromy at infinity of $A$-hypergeometric functions and toric compactifications, Math. Ann. 348 (2010), 815–831.
• A. N. Varchenko, Newton polyhedra and estimation of oscillating integrals, Funct. Anal. Appl. 10 (1977), 175–196.