Averages of Twisted -Functions

Jackson, J. E.; Knightly, Andrew

doi:10.1017/s1446788715000142

Cited by 3 publications

(5 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“….5] (which considered Maass forms instead of holomorphic forms). Duke [8] was the first to study first and second moments of this type, though he did not consider the dependence on D. The case of even weight k, k ≥ 4, could probably be derived with some more refined estimates from work of Jackson and Knightly [21] or Kohnen and Sengupta [26], but there are convergence problems in both of their approaches for k = 2.…”

Section: Corollary 17 We Havementioning

confidence: 99%

Rankin–Selberg L-functions and the reduction of CM elliptic curves

2015

View full text Add to dashboard Cite

Section: Corollary 17 We Havementioning

confidence: 99%

Rankin–Selberg L-functions and the reduction of CM elliptic curves

2015

View full text Add to dashboard Cite

“…Remarks 3.2. (1) When N > 1, it is shown in [JK,§9] that the sum of the weights is nonzero when N + k is sufficiently large. When N = 1, this can only be verified under certain extra conditions mentioned in Theorem 3.3 below.…”

Section: Weighted Equidistribution Of Hecke Eigenvalues Imentioning

confidence: 99%

“…Now suppose N = 1, χ 2 = 1, and s = 1 2 . Then there is an extra main term in [JK,Theorem 1.1], so that in place of (3.4), we have…”

Section: Weighted Equidistribution Of Hecke Eigenvalues Imentioning

confidence: 99%

“…706]: the notation χ 1,v is not defined, χ v does not define a character of the additive group X, and it is asserted that the integral g(χ v ) defined there, which clearly has absolute value ≤ 1, coincides with the classical Gauss sum which has absolute value q c/2 . A detailed treatment of the local test function with the desired spectral properties (and giving the same main term on the geometric side in [RR1]) is given in [JK,(3.11)-(3.12)]. For our purpose, it is enough to know that…”

Section: Low-lying Zeros IImentioning

confidence: 99%

See 1 more Smart Citation

Weighted Distribution of Low-lying Zeros of GL(2) -functions

Knightly¹,

Reno²

2019

Can. J. Math.-J. Can. Math.

Self Cite

View full text Add to dashboard Cite

We show that if the zeros of an automorphic L-function are weighted by the central value of the L-function or a quadratic imaginary base change, then for certain families of holomorphic GL(2) newforms, it has the effect of changing the distribution type of low-lying zeros from orthogonal to symplectic, for test functions whose Fourier transforms have sufficiently restricted support. However, if the L-value is twisted by a nontrivial quadratic character, the distribution type remains orthogonal. The proofs involve two vertical equidistribution results for Hecke eigenvalues weighted by central twisted L-values. One of these is due to Feigon and Whitehouse, and the other is new and involves an asymmetric probability measure that has not appeared in previous equidistribution results for GL(2).

show abstract

“…The first moment over the full basis of cusp forms (or over the space of primitive forms of prime level and small weight) has been studied intensively in different aspects. See, for example, [1], [3], [7], [8], [13], and [14]. In case of prime power level and weight 2 the best known error term for M 1 (l, 0, it) with respect to parameters N, T and l is obtained in [3].…”

Section: Introductionmentioning

confidence: 99%