Analytic isolation of newforms of given level

Nelson, Paul David

doi:10.1007/s00013-017-1039-y

Cited by 7 publications

(3 citation statements)

References 17 publications

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“…Our argument is inspired by those of Eichler [15] and Shimizu [64] as well as Gross's formulation [23, (12.13)] of Eichler's trace formula. It was shown in [48] that for N 2, ρ reg (f ) defines the orthogonal projection onto the span of the orthonormal set F N defined in §2.5.…”

Section: Proof Of the Algebraic Inputmentioning

confidence: 99%

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Quantum variance on quaternion algebras, I

Nelson

2016

Preprint

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We determine the quantum variance of a sequence of families of automorphic forms on a compact quotient arising from a non-split quaternion algebra. Our results compare to those obtained on SL 2 (Z)\ SL 2 (R) in work of Luo, Sarnak and Zhao, whose method required a cusp. Our method uses the theta correspondence to reduce the problem to the estimation of metaplectic Rankin-Selberg convolutions. We apply it here to the first non-split case.8 This is the special case |λ(n)| 2 = d|n λ(n 2 /d 2 ) of the Hecke multiplicatively exploited by Luo-Sarnak-Zhao. 9 The analysis of triple product averages mentioned above exploits the related identity L(ϕ × ϕ × Ψ, 1/2) = L(sym 2 ϕ × Ψ, 1/2)L(Ψ, 1/2).

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Section: Proof Of the Algebraic Inputmentioning

confidence: 99%

“…Completion of the calculation. Assume now that B is as in §2, that (48) is satisfied for all p / ∈ S := {2}, and that φ ′ 2 , φ ′′ 2 are as in Lemma 12. These assumptions fully determine φ ′ , φ ′′ .…”

Section: Rallis Inner Product Formulamentioning

confidence: 99%

Quantum variance on quaternion algebras, I

Nelson

2016

Preprint

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“…We overcome (1) in Theorem 3.1 by developing a Petersson formula for newforms for S κ pΓ 0 pN q, ǫq. There has been much recent interest in such formulas, see for example [BBD `17], [Nel17], [PY18], and [You17]. Theorem 3.1 is a generalization of [BBD `17, Prop.…”

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Bounds for traces of Hecke operators and applications to modular and elliptic curves over a finite field

Petrow

2018

Alg. Number Th.

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We give an upper bound for the trace of a Hecke operator acting on the space of holomorphic cusp forms with respect to certain congruence subgroups. Such an estimate has applications to the analytic theory of elliptic curves over a finite field, going beyond the Riemann hypothesis over finite fields. As the main tool to prove our bound on traces of Hecke operators, we develop a Petersson formula for newforms for general nebentype characters.

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A generalized cubic moment and the Petersson formula for newforms

Petrow

Young

2018

Math. Ann.

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Using a cubic moment, we prove a Weyl-type subconvexity bound for the quadratic twists of a holomorphic newform of square-free level, trivial nebentypus, and arbitrary even weight. This generalizes work of Conrey and Iwaniec in that the newform that is being twisted may have arbitrary square-free level, and also that the quadratic character may have even conductor. One of the new tools developed in this paper is a more general Petersson formula for newforms of square-free level.

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Analytic isolation of newforms of given level

Abstract: Abstract. We describe a method for understanding averages over newforms on Γ 0 (q) in terms of averages over all forms of some level. The method is simplest when q is divisible by the cubes of its prime divisors.

Cited by 7 publications

References 17 publications

Quantum variance on quaternion algebras, I

Quantum variance on quaternion algebras, I

Bounds for traces of Hecke operators and applications to modular and elliptic curves over a finite field

A generalized cubic moment and the Petersson formula for newforms

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