The subconvexity problem for GL2

By P. Michel, Akshay Venkatesh. 2009

257 citations recorded by the source corpus.

Generalizing and unifying prior results, we solve the subconvexity problem for the L-functions of GL 1 and GL 2 automorphic representations over a fixed number field, uniformly in all aspects. A novel feature of the present method is the softness of our arguments; this is largely due to a consistent use of canonically normalized period relations, such as those supplied by the work of Waldspurger and Ichino–Ikeda.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/73f5ec1aa98216927cb7d070ca3390324150aad8

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Citations in the corpus

The circle method and bounds for L-functions---II: Subconvexity for twists of GL(3) L-functions (2012, background): “For number fields, Michel and Venkatesh [16] have established such a hybrid subconvex bound with an unspecified exponent.”
The quantum unique ergodicity conjecture for thin sets (2013, methodology): “ries, Zagier computes (see p.430 of [Za]) that (5.42) hE(z,1 2 +s1)E(z,1 2 +s2),E(z,1 2 +s)ireg = c θ(1 2 +s)θ( 1 2 +s1)θ( 2 +s2) Y δ1,δ2∈{±1} ζ(1 2 +s+δ1s1 +δ2s2)Γ 1 2 +s+δ1s1 +δ2s2 2 . Lemma 5.3([MV]). Suppose that F and Gare smooth functions on Γ\H, Gwith compact support, Fof moderate growth wit”
The fourth moment of Dirichlet L-functions along a coset and the Weyl bound (2019, methodology): “Using canonical inner products on Whittaker models as in [MV], we obtain particularly pleasant normalizations of Fourier expansions and Bruggeman-Kuznetsov formulas.”
Algebraic twists of modular forms and Hecke orbits (2012, background): “Kuznetzov formula (see [30] for instance). This will make it possible to cover the generalization mentioned above in a uniform way, to extend the results to automorphic forms over number ﬁelds (as in [29]) and to generalize the equidistribution results described in section 1.5 to adelic quotients as”
Distribution of periodic torus orbits and Duke's theorem for cubic fields (2007, background): “For n = 3 the bound (7) follows from the work of Burgess [6] if K is abelian and (essentially) from the deep work of Duke, Friedlander and Iwaniec if K is cubic not abelian ([3, 14,37]).”
Sub-Weyl subconvexity for Dirichlet $L$-functions to prime power moduli (2014, methodology): “From a generalist point of view (such as Selberg class), it is generally believed [17] that the analytic behavior of L-functions is controlled in a universal fashion by the conductors C(π).”
On signs of Fourier coefficients of cusp forms (2011, background): “Using a subconvexity bound would slightly push the limit: Michel and Venkatesh [17] have recently proved that one can replace 1/4+ in (2·1) by η for some η which is slightly smaller than 1/4.”
On modular signs (2009, methodology): “ of any information on y. We note that, more generally, if we have L(f,1/2+it) ≪ (k2N(1+|t|)2)η, for t ∈ R, where η > 0, then we get (2.1) for x > Q2η+ε (the recent work of Michel and Venkatesh [15] provides such a uniform result for some – very small – η < 1/4, and the convexity bound stat”
The orbit method and analysis of automorphic forms (2018, unknown): “, [84, 11, 94, 56]).”
Families of L -Functions and Their Symmetry (2014, background): “Other examples are the GL2 subconvexity results which are proved by deforming the given form in a family (see [57] and [78] for accounts).”
Twisted L-Functions Over Number Fields and Hilbert’s Eleventh Problem (2009, methodology): “Remark 10. This should be compared with Lemma 2. The rst bound with unspecied exponents is contained in [ MV , (9.3)].”