Functiones et Approximatio Commentarii Mathematici

Hypothesis H and the prime number theorem for automorphic representations

Jie Wu and Yangbo Ye

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Abstract

For any unitary cuspidal representations $\pi_n$ of $GL_n(\mathbb{Q}_\mathbb{A})$, $n=2,3,4$, respectively, consider two automorphic representations $\Pi$ and $\Pi'$ of $GL_6(\mathbb{Q}_\mathbb{A})$, where $\Pi_p\cong\wedge^2\pi_{4,p}$ for $p\neq 2,3$ and $\pi_{4,p}$ not supercuspidal ($\pi_{4, p}$ denotes the local component of $\pi_4$), and $\Pi'=\pi_2\boxtimes\pi_3$. First, Hypothesis H for $\Pi$ and $\Pi'$ is proved. Then contributions from prime powers are removed from the prime number theorem for cuspidal representations $\pi$ and $\pi'$ of $GL_m(\mathbb{Q}_\mathbb{A})$ and $GL_{m'}(\mathbb{Q}_\mathbb{A})$, respectively. The resulting prime number theorem is unconditional when $m,m'\leq 4$ and is under Hypothesis H otherwise.

Article information

Source
Funct. Approx. Comment. Math., Volume 37, Number 2 (2007), 461-471.

Dates
First available in Project Euclid: 18 December 2008

Permanent link to this document
https://projecteuclid.org/euclid.facm/1229619665

Digital Object Identifier
doi:10.7169/facm/1229619665

Mathematical Reviews number (MathSciNet)
MR2364718

Zentralblatt MATH identifier
1230.11065

Subjects
Primary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields

Keywords
Hypothesis H functoriality prime number theorem

Citation

Wu, Jie; Ye, Yangbo. Hypothesis H and the prime number theorem for automorphic representations. Funct. Approx. Comment. Math. 37 (2007), no. 2, 461--471. doi:10.7169/facm/1229619665. https://projecteuclid.org/euclid.facm/1229619665


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References

  • Henry H. Kim, Functoriality for the exterior square of $GL_4$ and symmetric fourth of $GL_2$, With appendix 1 by D. Ramakrishnan and appendix 2 by H. Kim & P. Sarnak, J. Amer. Math. Soc. 16 (2003), 139--183.
  • Henry H. Kim, A note on Fourier coefficients of cusp forms on $GL_n$, Forum Math. 18 (2006), 115--119.
  • Henry H. Kim & Freydoon Shahidi, Functorial products for $GL_2\times GL_3$ and symmetric cube for $GL_2$, Ann. of Math. 155 (2002), 837--893.
  • Jianya Liu, Yonghui Wang, & Yangbo Ye, A proof of Selberg's orthogonality for automorphic $L$-functions, Manu. Math. 118 (2005), 135-149.
  • Jianya Liu & Yangbo Ye, Superposition of zeros of distinct $L$-functions, Forum Math. 14 (2002), 419-455.
  • Jianya Liu & Yangbo Ye, Selberg's orthogonality conjecture for automorphic $L$-functions, Amer. J. Math. 127 (2005), 837-849.
  • Jianya Liu & Yangbo Ye, Zeros of automorphic $L$-functions and noncyclic base change, in: Number Theory: Tradition and Modernization, edited by W. Zhang and Y. Tanigawa, Springer, Berlin, 2006, 119-152.
  • Jianya Liu & Yangbo Ye, Perron's formula and the prime number theorem for automorphic $L$-functions, Pure and Applied Mathematics Quarterly, 3 (2007), 481--497.
  • Wenzhi Luo, Zeév Rudnick & Peter Sarnak, On the generalized Ramanujan conjecture for $GL(n)$, Proceedings of Symposia In Pure Mathematics, vol. 66, part 2, 1999, 301--310.
  • Zeév Rudnick & Peter Sarnak, Zeros of principal $L$-functions and random matrix theory, Duke Math. J. 81 (1996), 269--322.