Uniform Bound for Hecke L-Functions

By Matti Jutila, Yoichi Motohashi. 2005

74 citations recorded by the source corpus.

Our principal aim in the present article is to establish a uniform hybrid bound for individual values on the critical line of Hecke $L$-functions associated with cusp forms over the full modular group. This is rendered in the statement that for $t\ge0$ $$ \eqalignno{H_j(\txt{1\over2}+it)&\ll(\kappa_j+t)^{1/3+\epsilon},& (1.1)\cr H_{j,k}(\txt{1\over2}+it)&\ll (k+t)^{1/3+\epsilon},&(1.2)\cr} $$ with the common notation to be made precise in the course of discussion. Talks on this and relevant results were delivered by the authors at MF Oberwolfach on September 20, 2004, and at General Assembry of Math. Soc. Japan on March 30, 2005.

Subfields: Algebra and Number Theory, Artificial Intelligence, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d392e48d1ab2a8bcec86e8d169a5eaa1860aa257

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In Algebra and Number Theory

1. V. Blomer
2. P. Michel
3. M. Young
4. Gergely Harcos
5. Akshay Venkatesh
6. Peter Sarnak
7. R. Munshi
8. Y. Lau

In Artificial Intelligence

1. M. Young
2. V. Blomer
3. P. Michel
4. Gergely Harcos
5. Akshay Venkatesh
6. A. Reznikov
7. Jianya Liu
8. Peter Sarnak

In Mathematical Physics

1. V. Blomer
2. P. Michel
3. M. Young
4. Gergely Harcos
5. Akshay Venkatesh
6. Peter Sarnak
7. R. Munshi
8. Y. Lau

Citations in the corpus

• The quantum unique ergodicity conjecture for thin sets (2013, background): “ made to be very small since TRm/Y ≤ R2. Note the similarity with (8.57). Then with the truncation k≪ R(TY)ε, setting σ= 1/2, and using L(1 2 + iT,f) ≪ (T+ k)1/3+ε, another result of Jutila-Motohashi [JM], the bound on the holomorphic forms becomes equivalent to that of the Maass forms, except we ca”
• A new bound K2/3+ɛ for Rankin-Selberg ℒ-functions for Hecke congruence subgroups (2010, background): “In [18], Jutila and Motohashi pointed out that their method might be generalized to get a subconvexity bound for our Rankin-Selberg L-functions for the full modular group in the k aspect with the merit of being uniform in small |t|.”
• Equidistribution, L-functions and ergodic theory: on some problems of Yu. Linnik (2006, background): “In this lecture, we mainly discuss the recent progress made on the q-aspect, although the other aspects are very interesting, both for applications and for conceptual reasons (see [6], [42], [46], [70]).”
• The spectral decomposition of shifted convolution sums (2007, background): “from Eisenstein series rather than cusp forms and their spectral decomposition is very explicitly known by the work of Motohashi [Mo1] (cf. [ JM , Lemma 4]).”
• The subconvexity problem for GL2 (2009, methodology): “The resulting formula relates, on the left-hand side,∫ ν∈iR ξ(1/2 + t1 + ν)ξ(1/2− t1 + ν)ξ(1/2 + t2 + λ− ν)ξ(1/2− t2 + λ− ν) to, on the right hand side,∑ M Λ(1/2 + t1 + t2, ϕ)Λ(1/2 + t1 − t2, ϕ)Λ(1/2 + λ, ϕ) + Eisenstein and degenerate terms This is an example of a formula of Motohashi [52]; it is p”
• Nodal Domains of Maass Forms I (2012, background): “pothesis. For our purpose, what we need is a bound of the form L(1 2 +it;f) ˝e (1+jtj)AtB f for some finite A but with B < 1 12 . The best subconvex bounds for L(s;f) known give B = 1 3 (see [Iv01],[JM05]) which is far short of 1 12 . For this reason we simply assume the full Lindelof Hypothesis. ”