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We prove the Rankin-Selberg convolution of two cuspidal automorphic representations are automorphic whenever one of them arises from an irreducible representation of an abelian-by-nilpotent Galois extension, which extends the previous result of Arthur-Clozel. Moreover, if one of such representations is of dimension at most 2 and another representation arises from a nearly nilpotent extension or a Galois extension of degree at most 59, the automorphy of the Rankin-Selberg convolution has been derived. As an application, we show that certain quotients of L-functions associated to non-CM elliptic curves are automorphic, which generalises a result of M. R. Murty and V. K. Murty.

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The Chebotarev Density Theorem and the Pair Correlation Conjecture

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Authors

M. Ram Murty ¹, V. Kumar Murty ², Peng-Jie Wong ¹

Affiliations
1 Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario K7L 3N6, CA
2 Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, CA

Source

Journal of the Ramanujan Mathematical Society, Vol 33, No 4 (2018), Pagination: 399-426

Abstract

In this note, we formulate pair correlation conjectures and refine the effective version of the Chebotarev density theorem established by the first two authors. Also, we apply our result to study Artin’s primitive ischolar_main conjecture and the Lang-Trotter conjectures and obtain shaper error terms.

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Zeros of Dedekind Zeta Functions and Holomorphy of Artin L-Functions

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Authors

Peng-Jie Wong ¹

Affiliations
1 Department of Mathematics, Queen’s University, Kingston, Ontario K7L 3N6, CA

Source

Journal of the Ramanujan Mathematical Society, Vol 34, No 2 (2019), Pagination: 253-261

Abstract

For any Galois extension K/k of number fields, we show that every Artin L-function for Gal(K/k) is holomorphic at s = s0 ≠= 1 whenever the quotient ζK (s)/ζk (s) of Dedekind zeta functions has a zero of order at most max{2, p₂ − 2} at s = s0 (here p₂ stands for the second smallest prime divisor of [K : k]). This result gives a refinement of the previous work of Foote and V. K. Murty.

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