Hybrid subconvexity bounds for $$L(1/2,\hbox {sym}^{2} f \otimes \chi )$$

2021 ◽  
Author(s):  
Fei Hou
Keyword(s):  
Subconvexity Bounds

scholarly journals Subconvexity bounds for Rankin–Selberg L-functions for congruence subgroups

2006 ◽  
Vol 121 (2) ◽  
pp. 204-223 ◽  
Author(s):  
Yuk-Kam Lau ◽  
Jianya Liu ◽  
Yangbo Ye
Keyword(s):  
Congruence Subgroups ◽  
Subconvexity Bounds

Hybrid subconvexity for class group 𝐿-functions and uniform sup norm bounds of Eisenstein series

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Asbjørn Christian Nordentoft
Keyword(s):  
Eisenstein Series ◽  
Class Group ◽  
Heegner Points ◽  
Quadratic Extensions ◽  
Norm Bound ◽  
Subconvexity Bounds

AbstractIn this paper, we study hybrid subconvexity bounds for class group 𝐿-functions associated to quadratic extensions K/\mathbb{Q} (real or imaginary). Our proof relies on relating the class group 𝐿-functions to Eisenstein series evaluated at Heegner points using formulas due to Hecke. The main technical contribution is the uniform sup norm bound for Eisenstein series E(z,1/2+it)\ll_{\varepsilon}y^{1/2}(\lvert t\rvert+1)^{1/3+\varepsilon}, y\gg 1, extending work of Blomer and Titchmarsh. Finally, we propose a uniform version of the sup norm conjecture for Eisenstein series.

scholarly journals Bad reduction of genus curves with CM jacobian varieties

2017 ◽  
Vol 153 (12) ◽  
pp. 2534-2576
Author(s):  
Philipp Habegger ◽  
Fabien Pazuki
Keyword(s):  
Number Field ◽  
Complex Multiplication ◽  
Abelian Extension ◽  
Reduction Theory ◽  
Jacobian Varieties ◽  
Subconvexity Bounds ◽  
Derivatives Of ◽  
Logarithmic Derivatives ◽  
Faltings Height

We show that a genus $2$ curve over a number field whose jacobian has complex multiplication will usually have stable bad reduction at some prime. We prove this by computing the Faltings height of the jacobian in two different ways. First, we use a known case of the Colmez conjecture, due to Colmez and Obus, that is valid when the CM field is an abelian extension of the rationals. It links the height and the logarithmic derivatives of an $L$-function. The second formula involves a decomposition of the height into local terms based on a hyperelliptic model. We use the reduction theory of genus $2$ curves as developed by Igusa, Liu, Saito, and Ueno to relate the contribution at the finite places with the stable bad reduction of the curve. The subconvexity bounds by Michel and Venkatesh together with an equidistribution result of Zhang are used to bound the infinite places.

Subconvexity bounds for automorphicL-functions

2009 ◽  
Vol 9 (1) ◽  
pp. 95-124 ◽  
Author(s):  
A. Diaconu ◽  
P. Garrett
Keyword(s):  
Arbitrary Number ◽  
Error Term ◽  
Power Saving ◽  
Number Fields ◽  
Cusp Forms ◽  
Subconvexity Bounds

AbstractWe break the convexity bound in thet-aspect forL-functions attached to cusp formsffor GL2(k) over arbitrary number fieldsk. The argument uses asymptotics with error term with a power saving, for second integral moments over spectral families of twistsL(s,f⊗χ) by Grossencharacters χ, from our previous paper on integral moments.

scholarly journals Motohashi’s fourth moment identity for non-archimedean test functions and applications

2020 ◽  
Vol 156 (5) ◽  
pp. 1004-1038 ◽  
Author(s):  
Valentin Blomer ◽  
Peter Humphries ◽  
Rizwanur Khan ◽  
Micah B. Milinovich
Keyword(s):  
Upper Bounds ◽  
Fourth Moment ◽  
Auxiliary Result ◽  
Dirichlet Polynomial ◽  
Test Functions ◽  
Test Function ◽  
The Riemann Zeta Function ◽  
Sixth Moment ◽  
Riemann Zeta ◽  
Subconvexity Bounds

Motohashi established an explicit identity between the fourth moment of the Riemann zeta function weighted by some test function and a spectral cubic moment of automorphic $L$-functions. By an entirely different method, we prove a generalization of this formula to a fourth moment of Dirichlet $L$-functions modulo $q$ weighted by a non-archimedean test function. This establishes a new reciprocity formula. As an application, we obtain sharp upper bounds for the fourth moment twisted by the square of a Dirichlet polynomial of length $q^{1/4}$. An auxiliary result of independent interest is a sharp upper bound for a certain sixth moment for automorphic $L$-functions, which we also use to improve the best known subconvexity bounds for automorphic $L$-functions in the level aspect.

scholarly journals Subconvexity bounds for tripleL-functions and representation theory

2010 ◽  
Vol 172 (3) ◽  
pp. 1679-1718 ◽  
Author(s):  
Joseph Bernstein ◽  
Andre Reznikov
Keyword(s):  
Representation Theory ◽  
Subconvexity Bounds
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