Sparse equidistribution problems, period bounds and subconvexity
By Akshay Venkatesh. 2005
222 citations recorded by the source corpus.
We introduce a "geometric" method to bound periods of automorphic forms. The key features of this method are the use of equidistribution results in place of mean value theorems, and the systematic use of mixing and the spectral gap. Applications are given to equidistribution of sparse subsets of horocycles and to equidistribution of CM points; to subconvexity of the triple product period in the level aspect over number fields, which implies subconvexity for certain standard and Rankin-Selberg L-functions; and to bounding Fourier coefficients of automorphic forms.
Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/053880600f154708fdd276c42cce947a1d829c6d
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In Algebra and Number Theory
- V. Blomer
- P. Michel
- R. Munshi
- Peter Sarnak
- Gergely Harcos
- Akshay Venkatesh
- M. Young
- Nicolas Templier
In Geometry and Topology
- P. Michel
- Akshay Venkatesh
- V. Blomer
- Gergely Harcos
- R. Munshi
- Peter Sarnak
- Nicolas Templier
- Alex Kontorovich
In Mathematical Physics
- P. Michel
- V. Blomer
- Akshay Venkatesh
- Peter Sarnak
- R. Munshi
- Gergely Harcos
- M. Young
- Alex Kontorovich
Citations in the corpus
- The orbit method and analysis of automorphic forms (2018, result): “We briefly discuss the relationship of the ideas of this paper to some others: (i) The general philosophy that equidistribution results lead to mean value theorems for L-functions was advocated in [94], but the source of the equidistribution in this paper is quite different from that in [94].”
- Algebraic twists of modular forms and Hecke orbits (2012, background): “These result could also be interpreted in terms of equidistribution of weighted p-adic horocycles; similar questions have been studied in different contexts for rather different weights in [42, 44, 43] (e.”
- Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces (2007, result): “ may be found in [23, 60]; quantitative mixing rates are discussed in [26], and analysis of equidistribution of closed horospheres and its relevance to the theory of automorphic forms may be found in [61]. A typical instance of the second type of result is the equidistribution of Hecke points, [22] ”
- Twisted L-Functions Over Number Fields and Hilbert’s Eleventh Problem (2009, methodology): “As an application of an ingenious and flexible geometric method, Venkatesh [ V2 , Th. 6.1] proved a subconvex bound over all number fields and for all irreducible cuspidal representations”
- Central L-Values and Toric Periods for GL(2) (2009, methodology): “[4] Einseidler, M., E. Lindenstrauss, P. Michel, and A. Venkatesh.”
- The subconvexity problem for GL2 (2009, methodology): “This section exists to give a suitable language for talking about adelic equidistribution, and the norms are a somewhat cleaner version of those appearing in [67].”
- The circle method and bounds for L-functions---II: Subconvexity for twists of GL(3) L-functions (2012, methodology): “Over number fields this was established by Venkatesh [21] and Blomer and Harcos [2].”