## Eigenfunctions of the Weil representation of unitary groups of one variable

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- by Tonghai Yang PDF
- Trans. Amer. Math. Soc.
**350**(1998), 2393-2407

## Abstract:

In this paper, we construct explicit eigenfunctions of the local Weil representation on unitary groups of one variable in the $p$-adic case when $p$ is odd. The idea is to use the lattice model, and the results will be used to compute special values of certain Hecke $L$-functions in separate papers. We also recover Moen’s results on when a local theta lifting from $U(1)$ to itself does not vanish.## References

- Michael Harris, Stephen S. Kudla, and William J. Sweet,
*Theta dichotomy for unitary groups*, J. Amer. Math. Soc.**9**(1996), no. 4, 941–1004. MR**1327161**, DOI 10.1090/S0894-0347-96-00198-1 - Stephen S. Kudla,
*Splitting metaplectic covers of dual reductive pairs*, Israel J. Math.**87**(1994), no. 1-3, 361–401. MR**1286835**, DOI 10.1007/BF02773003 - Jian-shu Li,
*Theta series*, Lecture notes at University of Maryland, College Park, 1992. - S. Minakshi Sundaram,
*On non-linear partial differential equations of the parabolic type*, Proc. Indian Acad. Sci., Sect. A.**9**(1939), 479–494. MR**0000088** - Courtney Moen,
*The dual pair $(\textrm {U}(1),\textrm {U}(1))$ over a $p$-adic field*, Pacific J. Math.**158**(1993), no. 2, 365–386. MR**1206444** - Colette Mœglin, Marie-France Vignéras, and Jean-Loup Waldspurger,
*Correspondances de Howe sur un corps $p$-adique*, Lecture Notes in Mathematics, vol. 1291, Springer-Verlag, Berlin, 1987 (French). MR**1041060**, DOI 10.1007/BFb0082712 - R. Ranga Rao,
*On some explicit formulas in the theory of Weil representation*, Pacific J. Math.**157**(1993), no. 2, 335–371. MR**1197062** - Jonathan D. Rogawski,
*The multiplicity formula for $A$-packets*, The zeta functions of Picard modular surfaces, Univ. Montréal, Montreal, QC, 1992, pp. 395–419. MR**1155235** - F. Rodriguez Villegas and T.H. Yang,
*Special values of L-functions of powers of canonical Hecke characters at the central point*, to appear in Duke Math. J. - Lawrence C. Washington,
*Introduction to cyclotomic fields*, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. MR**718674**, DOI 10.1007/978-1-4684-0133-2 - André Weil,
*Sur la formule de Siegel dans la théorie des groupes classiques*, Acta Math.**113**(1965), 1–87 (French). MR**223373**, DOI 10.1007/BF02391774 - Tonghai Yang,
*Theta liftings and Hecke L-functions*, J. Reine Angew. Math.**485**(1997), 25-53. - —,
*Nonvanishing of central Hecke L-value and rank of associated elliptic curves*, to appear in Compositio Math.

## Additional Information

**Tonghai Yang**- Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
- MR Author ID: 606823
- Email: thyang@math.lsa.umich.edu
- Received by editor(s): November 7, 1995
- Received by editor(s) in revised form: July 3, 1996
- Additional Notes: Partially supported by NSF grant DMS-9304580
- © Copyright 1998 Americal Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**350**(1998), 2393-2407 - MSC (1991): Primary 11F27, 11E45
- DOI: https://doi.org/10.1090/S0002-9947-98-01915-1
- MathSciNet review: 1407714