Triple Product Formula and Mass Equidistribution on Modular Curves of Level N

Hu, Yueke

doi:10.1093/imrn/rnw322

Cited by 13 publications

(21 citation statements)

References 22 publications

(55 reference statements)

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“…Then 1 xB(C,ε) = |ψ * ν| 2 . By (14), we have µ T ϕ (|ψ * ν| 2 ) = T ϕ(ψ * ν) 2 L 2 (G) , and so by Lemma 41,…”

Section: Positive Entropymentioning

confidence: 92%

“…Remark 18. Unlike earlier works such as [23,26,14], we have allowed arbitrary central characters in Theorem 17. We note that the case of the argument in which the conductor of the central character is as large as possible relative to that of the representation is a bit more technically challenging than the others; see (26) and following.…”

Section: 2mentioning

confidence: 99%

“…Remark 19. Cases of Theorem 17 in which m..m is highly unbalanced, such as the most traditional case m..m = 0..n analogous to Theorem 6, are easier: they follow, sometimes with a power savings, from the triple product formula, the convexity bound for triple product L-functions, and nontrivial local estimates as in [26,14]. Cases in which m..m is balanced, such as the case m..m = −N..N illustrated in §1.1, do not follow from such local arguments and require the new ideas introduced here.…”

Section: 2mentioning

confidence: 99%

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Microlocal lifts and quantum unique ergodicity on GL2(ℚp)

Nelson

2018

Alg. Number Th.

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“…Then 1 xB(C,ε) = |ψ * ν| 2 . By (14), we have µ T ϕ (|ψ * ν| 2 ) = T ϕ(ψ * ν) 2 L 2 (G) , and so by Lemma 41,…”

Section: Positive Entropymentioning

confidence: 92%

Section: 2mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

See 1 more Smart Citation

Microlocal lifts and quantum unique ergodicity on GL2(ℚp)

Nelson

2018

Alg. Number Th.

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“…In fact, the main result of [31] gave an unconditional power savings bound D f (g) ≪ g C −δ1 0 log(Ck) −δ2 for some positive constants δ 1 , δ 2 , where as before, C 0 denotes the largest integer such that C 2 0 |C. Further extensions of this result to the case when g is not of full level were obtained in [16].…”

mentioning

confidence: 82%

Some analytic aspects of automorphic forms on GL(2) of minimal type

Nelson

Saha

2019

Comment. Math. Helv.

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Let π be a cuspidal automorphic representation of PGL 2 (A Q ) of arithmetic conductor C and archimedean parameter T , and let φ be an L 2 -normalized automorphic form in the space of π. The sup-norm problem asks for bounds on φ ∞ in terms of C and T . The quantum unique ergodicity (QUE) problem concerns the limiting behavior of the L 2 -mass |φ| 2 (g) dg of φ. All previous work on these problems in the conductor-aspect has focused on the case that φ is a newform.In this work, we study these problems for a class of automorphic forms that are not newforms. Precisely, we assume that for each prime divisor p of C, the local component πp is supercuspidal (and satisfies some additional technical hypotheses), and consider automorphic forms φ for which the local components φp ∈ πp are "minimal" vectors. Such vectors may be understood as non-archimedean analogues of lowest weight vectors in holomorphic discrete series representations of PGL 2 (R).For automorphic forms as above, we prove a sup-norm bound that is sharper than what is known in the newform case. In particular, if π∞ is a holomorphic discrete series of lowest weight k, we obtain the optimal bound C 1/8−ǫ k 1/4−ǫ ≪ǫ |φ|∞ ≪ǫ C 1/8+ǫ k 1/4+ǫ . We prove also that these forms give analytic test vectors for the QUE period, thereby demonstrating the equivalence between the strong QUE and the subconvexity problems for this class of vectors. This finding contrasts the known failure of this equivalence [31] for newforms of powerful level.

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“…Remark. The general case of the conjecture is likely accessible by brute-force analysis of matrix coefficients as in [34,19]. It also seems likely to follow by effectizing the proof of (2.25).…”

Section: Lemma the Conclusion Of The Conjecture Holds Under The Addimentioning

confidence: 96%

Subconvex equidistribution of cusp forms: Reduction to Eisenstein observables

Nelson¹

2019

Duke Math. J.

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Let π traverse a sequence of cuspidal automorphic representations of GL 2 with large prime level, unramified central character and bounded infinity type. For G ∈ {GL 1 , PGL 2 }, let H(G) denote the assertion that subconvexity holds for G-twists of the adjoint L-function of π, with polynomial dependence upon the conductor of the twist. We show that H(GL 1 ) implies H(PGL 2 ).In geometric terms, H(PGL 2 ) corresponds roughly to an instance of arithmetic quantum unique ergodicity with a power savings in the error term, H(GL 1 ) to the special case in which the relevant sequence of measures is tested against an Eisenstein series.

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Triple Product Formula and Mass Equidistribution on Modular Curves of Level N

Abstract: It was shown in previous works that the measure associated to holomorphic newforms of weight k and level q will tend weakly to the Haar measure on modular curve of level 1, as qk → ∞. In this paper we proved that this phenomenon is also true on modular curves of general level N.3

Cited by 13 publications

References 22 publications

Microlocal lifts and quantum unique ergodicity on GL2(ℚp)

Microlocal lifts and quantum unique ergodicity on GL2(ℚp)

Some analytic aspects of automorphic forms on GL(2) of minimal type

Subconvex equidistribution of cusp forms: Reduction to Eisenstein observables

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