Hybrid subconvexity bounds for twisted L-functions on GL(3)


Let q be a large prime, and χ the quadratic character modulo q. Let ϕ be a self-dual Hecke-Maass cusp form for SL(3, ℤ), and uj a Hecke-Maass cusp form for Г0(q) ⊆ SL(2, ℤ) with spectral parameter tj. We prove, for the first time, some hybrid subconvexity bounds for the twisted L-functions on GL(3), such as

$$L\left( {1/2,\phi \times {u_j} \times \chi } \right){ \ll _{\phi ,\varepsilon }}{\left( {q\left( {1 + \left| {{t_j}} \right|} \right)} \right)^{3/2 - \theta + \varepsilon }},L\left( {1/2 + it,\phi \times \chi } \right){ \ll _{\phi ,\varepsilon }}{\left( {q\left( {1 + \left| t \right|} \right)} \right)^{3/4 - \theta /2 + \varepsilon }},$$

for any ε > 0, where θ = 1/23 is admissible. The proofs depend on the first moment of a family of L-functions in short intervals. In order to bound this moment, we first use the approximate functional equations, the Kuznetsov formula, and the Voronoi formula to transform it to a complicated summation; and then we apply different methods to estimate it, which give us strong bounds in different aspects. We also use the stationary phase method and the large sieve inequalities.

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This work was supported by National Natural Science Foundation of China (Grant No. 11531008) and the Ministry of Education of China (Grant No. IRT 16R43). The author thanks Professors Dorian Goldfeld, Jianya Liu, Zeév Rudnick, and Wei Zhang for their valuable advice and constant encouragement. He also thanks Professor Matthew Young for explaining some details in his paper [29]. He also thanks the anonymous referees and editors for their kind comments and valuable suggestions. This work was finished when the author was visiting Columbia University. He is grateful to the China Scholarship Council (CSC) for supporting his studies at Columbia University, and also thanks the Department of Mathematics at Columbia University for its hospitality.

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Correspondence to Bingrong Huang.

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Huang, B. Hybrid subconvexity bounds for twisted L-functions on GL(3). Sci. China Math. 64, 443–478 (2021). https://doi.org/10.1007/s11425-017-9428-6

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  • L-functions
  • subconvexity
  • GL(3)
  • twisted
  • quadratic character


  • 11F66
  • 11F67
  • 11M41