Explicit doubling integrals for $\widetilde {\mathrm {Sp}_2}(\mathbb {Q}_2)$ using “good test vectors”

Author:
Christian A. Zorn

Journal:
Represent. Theory **14** (2010), 285-323

MSC (2010):
Primary 22E50; Secondary 11F70

DOI:
https://doi.org/10.1090/S1088-4165-10-00371-7

Published electronically:
March 15, 2010

MathSciNet review:
2608965

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Abstract: In a previous paper (see http:/www.math.ohio-state.edu/~czorn/works.html), we computed examples of the doubling integral for constituents of the unramified principal series of $\mathrm {Sp}_2(F)$ and $\widetilde {\textrm {Sp}_2}(F)$ where $F$ was a non-dyadic field. These computations relied on certain “good test vectors” and “good theta test sections” motivated by the non-vanishing of theta lifts. In this paper, we aim to prove a partial analog for $\widetilde {\textrm {Sp}_2}(\mathbb {Q}_2)$. However, due to several complexities, we compute the doubling integral only for certain irreducible principal series representations induced from characters with ramified quadratic twists. We develop some $2$-adic analogs for the machinery in the paper mentioned above; however, these tend to be more delicate and have more restrictive hypotheses than the non-dyadic case. Ultimately, this paper and the one mentioned above develop several computations intended to be used for future research into the non-vanishing of theta lifts.

- W. Casselman.
*Introduction to the theory of admissible representations of p-adic reductive groups*. Preprint. Accessed at http://www.math.ubc.ca/~cass/research.html. - Stephen Gelbart, Ilya Piatetski-Shapiro, and Stephen Rallis,
*Explicit constructions of automorphic $L$-functions*, Lecture Notes in Mathematics, vol. 1254, Springer-Verlag, Berlin, 1987. MR**892097** - Stephen S. Kudla,
*Seesaw dual reductive pairs*, Automorphic forms of several variables (Katata, 1983) Progr. Math., vol. 46, Birkhäuser Boston, Boston, MA, 1984, pp. 244–268. MR**763017** - S. Kudla. On the Theta Correspondence. Lectures at European School of Group Theory, Beilngries 1996. Accessed at http://www.math.toronto.edu/~skudla/ssk.research.html.
- Stephen S. Kudla, Michael Rapoport, and Tonghai Yang,
*Modular forms and special cycles on Shimura curves*, Annals of Mathematics Studies, vol. 161, Princeton University Press, Princeton, NJ, 2006. MR**2220359** - S. Rallis,
*On the Howe duality conjecture*, Compositio Math.**51**(1984), no. 3, 333–399. MR**743016** - R. Ranga Rao,
*On some explicit formulas in the theory of Weil representation*, Pacific J. Math.**157**(1993), no. 2, 335–371. MR**1197062** - J.-P. Serre,
*A course in arithmetic*, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French; Graduate Texts in Mathematics, No. 7. MR**0344216** - T. A. Springer,
*Reductive groups*, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 3–27. MR**546587** - Marko Tadić,
*Jacquet modules and induced representations*, Math. Commun.**3**(1998), no. 1, 1–17 (English, with English and Croatian summaries). MR**1648862** - Tonghai Yang,
*An explicit formula for local densities of quadratic forms*, J. Number Theory**72**(1998), no. 2, 309–356. MR**1651696**, DOI https://doi.org/10.1006/jnth.1998.2258 - Tonghai Yang,
*Local densities of 2-adic quadratic forms*, J. Number Theory**108**(2004), no. 2, 287–345. MR**2098640**, DOI https://doi.org/10.1016/j.jnt.2004.05.002 - C. Zorn. Computing local $L$-factors for the unramified principal series of $\textrm {Sp}_2(F)$ and its metaplectic cover. Univ. of Maryland Thesis, 2007.
- C. Zorn. Reducibility of the principal series for $\widetilde {\textrm {Sp}_2}(F)$ over a $p$-adic field. Canadian Journal of Mathematics, to appear. Available online at http://www.math.ohio-state.edu/~czorn/works.html.
- C. Zorn. Explicit computations of the doubling integral for $\textrm {Sp}_2(F)$ and $\widetilde {\textrm {Sp}_2}(F)$. Preprint. Available online at http://www.math.ohio-state.edu/~czorn/works.html.

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Additional Information

**Christian A. Zorn**

Affiliation:
Department of Mathematics, The Ohio State University, 231 W. 18th Ave., Columbus, Ohio 43210

Email:
czorn@math.ohio-state.edu

Received by editor(s):
January 9, 2009

Received by editor(s) in revised form:
December 7, 2009

Published electronically:
March 15, 2010

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.