We prove non-trivial upper bounds for general bilinear forms with trace functions of bountiful sheaves,where the supports of two variables can be arbitrary subsets in F_(p) of suitable sizes.This essentially recovers ...We prove non-trivial upper bounds for general bilinear forms with trace functions of bountiful sheaves,where the supports of two variables can be arbitrary subsets in F_(p) of suitable sizes.This essentially recovers the Polya-Vinogradov range,and also applies to symmetric powers of Kloosterman sums and Frobenius traces of elliptic curves.In the case of hyper-Kloosterman sums,we can beat the Pólya-Vinogradov barrier by combining additive combinatorics with a deep result of Kowalski,Michel and Sawin(2017) on sum-products of Kloosterman sheaves.Two Sato-Tate distributions of Kloosterman sums and Frobenius traces of elliptic curves in sparse families are also concluded.展开更多
Let f1,...,fkand g1,...,qkbe non-CM newforms of square-free levels.Denote byλ_(sym)jf_(i)(n)the coefficients of the Dirichlet expansion of L(sym^(j)f_(i),s)andν1,...,νkthe distinct positive integers such thatλ_(sy...Let f1,...,fkand g1,...,qkbe non-CM newforms of square-free levels.Denote byλ_(sym)jf_(i)(n)the coefficients of the Dirichlet expansion of L(sym^(j)f_(i),s)andν1,...,νkthe distinct positive integers such thatλ_(sym)jf_(i)(νi)≠0.In this paper,we obtain that there exist infinitely many positive integers m such that 0<λ_(sym)jf_(1)(m+ν1)|<|λ_(sym)jf_(2)(m+ν2)|<…<|λ_(sym)jf_(k)(m+νk)|.For coefficients of the Dirichlet expansion of L(sym^(j1)f×sym^(j2)g,s),we have a similar result.展开更多
基金supported by National Natural Science Foundation of China (Grant Nos. 12025106 and 11971370)。
文摘We prove non-trivial upper bounds for general bilinear forms with trace functions of bountiful sheaves,where the supports of two variables can be arbitrary subsets in F_(p) of suitable sizes.This essentially recovers the Polya-Vinogradov range,and also applies to symmetric powers of Kloosterman sums and Frobenius traces of elliptic curves.In the case of hyper-Kloosterman sums,we can beat the Pólya-Vinogradov barrier by combining additive combinatorics with a deep result of Kowalski,Michel and Sawin(2017) on sum-products of Kloosterman sheaves.Two Sato-Tate distributions of Kloosterman sums and Frobenius traces of elliptic curves in sparse families are also concluded.
基金Supported by the National Key Research and Development Program of China(Grant No.2021YFA1000700)NSFC(Grant No.12031008)。
文摘Let f1,...,fkand g1,...,qkbe non-CM newforms of square-free levels.Denote byλ_(sym)jf_(i)(n)the coefficients of the Dirichlet expansion of L(sym^(j)f_(i),s)andν1,...,νkthe distinct positive integers such thatλ_(sym)jf_(i)(νi)≠0.In this paper,we obtain that there exist infinitely many positive integers m such that 0<λ_(sym)jf_(1)(m+ν1)|<|λ_(sym)jf_(2)(m+ν2)|<…<|λ_(sym)jf_(k)(m+νk)|.For coefficients of the Dirichlet expansion of L(sym^(j1)f×sym^(j2)g,s),we have a similar result.
