## The archimedean theory of the exterior square $L$-functions over $\mathbb {Q}$

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- by Stephen D. Miller and Wilfried Schmid
- J. Amer. Math. Soc.
**25**(2012), 465-506 - DOI: https://doi.org/10.1090/S0894-0347-2011-00719-4
- Published electronically: December 6, 2011

## Abstract:

The analytic properties of automorphic $L$-functions have historically been obtained either through integral representations (the “Rankin-Selberg method”) or properties of the Fourier expansions of Eisenstein series (the “Langlands-Shahidi method”). We introduce a method based on pairings of automorphic distributions that appears to be applicable to a wide variety of $L$-functions, including all which have integral representations. In some sense our method could be considered a completion of the Rankin-Selberg method because of its common features. We consider a particular but representative example, the exterior square $L$-functions on $GL(n)$, by constructing a pairing which we compute as a product of this $L$-function times an explicit ratio of Gamma functions. We use this to deduce that exterior square $L$-functions, when multiplied by the Gamma factors predicted by Langlands, are holomorphic on $\mathbb {C}-\{0,1\}$ with at most simple poles at 0 and 1, proving a conjecture of Langlands which has not been obtained by the existing two methods.## References

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## Bibliographic Information

**Stephen D. Miller**- Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854-8019
- Email: miller@math.rutgers.edu
**Wilfried Schmid**- Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
- Email: schmid@math.harvard.edu
- Received by editor(s): November 24, 2010
- Received by editor(s) in revised form: September 12, 2011
- Published electronically: December 6, 2011
- Additional Notes: The first author was partially supported by NSF grant DMS-0901594 and an Alfred P. Sloan Foundation Fellowship

The second author was partially supported by DARPA grant HR0011-04-1-0031 and NSF grant DMS-0500922 - © Copyright 2011 by Stephen D. Miller and Wilfried Schmid
- Journal: J. Amer. Math. Soc.
**25**(2012), 465-506 - MSC (2010): Primary 11F55, 11F66
- DOI: https://doi.org/10.1090/S0894-0347-2011-00719-4
- MathSciNet review: 2869024

Dedicated: In memory of Joseph Shalika