Rankin Triple Products and Quantum Chaos

By Thomas C. Watson. 2008

162 citations recorded by the source corpus.

We prove explicit Harris-Kudla type formulas for triples of Maass forms, holomorphic forms, and combinations thereof, on the hyperbolic plane modulo congruence groups and co-compact lattices arising from Eichler orders of quaternion algebras. These formulas relate the central value of the corresponding Rankin triple product L-function to a squared trilinear period integral. Assuming subconvexity estimates for these L-values, we prove Quantum Unique Ergodicity on such quotients; the relevant Lindelof hypotheses imply a quantitative form of QUE, with an optimal rate. In connection with the Berry/Hejhal Random Wave conjecture, we prove decay of third moments in the high energy limit, making use of a subconvexity result of Iwaniec/Ivic/Jutila and Kim-Shahidi's result on cuspidality of the symmetric cube.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/01e8c5865d3ec02adeae7be46ffff44ca0b60650

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Citations in the corpus

Equidistribution Of Cusp Forms In The Level Aspect (2010, methodology): “(4) The identity relating μf (φ) to L(φ× f × f, 12 ) that Soundararajan’s method takes as input is given by Watson [41] when f and φ are newforms of the same (squarefree) level.”
On the Fourier coefficients of modular forms of half-integral weight (2009, unknown): “For q | N χ and q | N + , ˜ ω χ,q (u) = χ q (d) 2 for u = a b c d ∈ U 0,q (χ).”
Quantum unique ergodicity for SL_2(Z)\H (2009, background): “For more complete accounts of the quantum unique ergodicity problem the reader may consult [7], [9], [11], [13] and references therein; a comprehensive introduction to the theory of Maass forms is provided in [4].”
The quantum unique ergodicity conjecture for thin sets (2013, methodology): “ U= ET, but it would be interesting to study the holomorphic case too. It is well-known that the QUE conjecture can be approached either via bounds for triple product L-functions via Watson’s formula [W], or alternatively by shifted convolution sums via Poincare series. These two methods are apparen”
Distribution of mass of holomorphic cusp forms (2012, background): “estricted to ﬁxed compact regions within Γ\Hthis has been shown by Spinu [Sp]. In the level aspect, a best-possible result on average has been proved in [Bl1]. All these results have Watson’s formula [Wa2] as a starting point that translates the L4-norm into a mean value of certain triple product L-”
Automorphic period and the central value of Rankin--Selberg L-function (2012, methodology): “At a split place, in practical computation it seems easier to use the normalization via Whittaker model instead of the matrix coefficient integration, for example, the archimedean computation by Watson for the triple product L-function ([42]).”
Weak subconvexity for central values of $L$-functions (2008, unknown): “For the case of a Maass cusp form, a beautiful formula of Watson (see Theorem 3 of [33]) shows that (here φ has been normalized so that 〈φ, φ〉 = 1) |〈φFk, Fk〉|(2) = 1 8 L∞( 1 2 , f × f × φ)L( 12 , f × f × φ) Λ(1, sym2f)2Λ(1, sym2φ) where L(s, f × f × φ) is the triple product L-function of Example 2,”
Mass equidistribution for Hecke eigenforms (2008, background): “N T ] 9 S ep 2 00 8 MASS EQUIDISTRIBUTION FOR HECKE EIGENFORMS”
The orbit method and analysis of automorphic forms (2018, methodology): “sed in terms of matrix coefﬁcient integrals as follows (see x18): jhpr ˙ (v);uij2 = Z s2H hˇ(s)v;vihu;˙(s)ui: (1.22) Such integrals had previously been estimated in some low-rank examples (see, e.g., [101, 104, 11, 67, 58, 59]) via explicit calculation. To analyze (1.22), ﬁrst decompose as in x1.7 t”