Characteristic functions of p-adic integral operators

Pavel Etingof; David Kazhdan

doi:10.1090/conm/780/15685

Characteristic functions of p-adic integral operators

Pavel Etingof
, David Kazhdan

Einstein Institute of Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

Let P ∈ Q_p [x, y], s ∈ C with sufficiently large real part, and consider the integral operator ∫ 1 (A_P,sf)(y):= 1 − p⁻¹ |P (x, y)|^sf(x)|dx| Z_p on L²(Z_p). We show that if P is homogeneous of degree d then for each character χ of Z^×p the characteristic function det(1 − uA_P,s,χ) of the restriction A_P,s,χ of A_P,s to the eigenspace L²(Z_p)_χ is the q-Wronskian of a set of solutions of a (possibly confluent) q-hypergeometric equation, where q = p^−1−ds . In particular, the nonzero eigenvalues of A_P,s,χ are the reciprocals of the zeros of such q-Wronskian.

Original language	English
Title of host publication	Hypergeometry, Integrability and Lie Theory - Virtual Conference Hypergeometry, Integrability and Lie Theory, 2020
Editors	Erik Koelink, Stefan Kolb, Nicolai Reshetikhin, Bart Vlaar
Publisher	American Mathematical Society
Pages	1-27
Number of pages	27
ISBN (Print)	9781470465209
DOIs	https://doi.org/10.1090/conm/780/15685
State	Published - 2022
Event	Virtual conference on Hypergeometry, Integrability and Lie Theory, 2020 - Virtual, Online Duration: 7 Dec 2020 → 11 Dec 2020

Publication series

Name	Contemporary Mathematics
Volume	780
ISSN (Print)	0271-4132
ISSN (Electronic)	1098-3627

Conference

Conference	Virtual conference on Hypergeometry, Integrability and Lie Theory, 2020
City	Virtual, Online
Period	7/12/20 → 11/12/20

Bibliographical note

Publisher Copyright:
© 2022 American Mathematical Society.

Access to Document

10.1090/conm/780/15685

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Etingof, P., & Kazhdan, D. (2022). Characteristic functions of p-adic integral operators. In E. Koelink, S. Kolb, N. Reshetikhin, & B. Vlaar (Eds.), Hypergeometry, Integrability and Lie Theory - Virtual Conference Hypergeometry, Integrability and Lie Theory, 2020 (pp. 1-27). (Contemporary Mathematics; Vol. 780). American Mathematical Society. https://doi.org/10.1090/conm/780/15685

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abstract = "Let P ∈ Qp [x, y], s ∈ C with sufficiently large real part, and consider the integral operator ∫ 1 (AP,sf)(y):= 1 − p−1 |P (x, y)|sf(x)|dx| Zp on L2(Zp). We show that if P is homogeneous of degree d then for each character χ of Z×p the characteristic function det(1 − uAP,s,χ) of the restriction AP,s,χ of AP,s to the eigenspace L2(Zp)χ is the q-Wronskian of a set of solutions of a (possibly confluent) q-hypergeometric equation, where q = p−1−ds . In particular, the nonzero eigenvalues of AP,s,χ are the reciprocals of the zeros of such q-Wronskian.",

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Etingof, P & Kazhdan, D 2022, Characteristic functions of p-adic integral operators. in E Koelink, S Kolb, N Reshetikhin & B Vlaar (eds), Hypergeometry, Integrability and Lie Theory - Virtual Conference Hypergeometry, Integrability and Lie Theory, 2020. Contemporary Mathematics, vol. 780, American Mathematical Society, pp. 1-27, Virtual conference on Hypergeometry, Integrability and Lie Theory, 2020, Virtual, Online, 7/12/20. https://doi.org/10.1090/conm/780/15685

Characteristic functions of p-adic integral operators. / Etingof, Pavel; Kazhdan, David.
Hypergeometry, Integrability and Lie Theory - Virtual Conference Hypergeometry, Integrability and Lie Theory, 2020. ed. / Erik Koelink; Stefan Kolb; Nicolai Reshetikhin; Bart Vlaar. American Mathematical Society, 2022. p. 1-27 (Contemporary Mathematics; Vol. 780).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AB - Let P ∈ Qp [x, y], s ∈ C with sufficiently large real part, and consider the integral operator ∫ 1 (AP,sf)(y):= 1 − p−1 |P (x, y)|sf(x)|dx| Zp on L2(Zp). We show that if P is homogeneous of degree d then for each character χ of Z×p the characteristic function det(1 − uAP,s,χ) of the restriction AP,s,χ of AP,s to the eigenspace L2(Zp)χ is the q-Wronskian of a set of solutions of a (possibly confluent) q-hypergeometric equation, where q = p−1−ds . In particular, the nonzero eigenvalues of AP,s,χ are the reciprocals of the zeros of such q-Wronskian.

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Characteristic functions of p-adic integral operators

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