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 Hecke’s Integral Formula for Relative Quadratic Extensions of Algebraic Number Fields

2008 ◽  
Vol 189 ◽  
pp. 139-154 ◽  
Author(s):  
Shuji Yamamoto
Keyword(s):  
Zeta Function ◽  
Eisenstein Series ◽  
Algebraic Number ◽  
Integral Formula ◽  
Zeta Functions ◽  
Quadratic Extension ◽  
Number Fields ◽  
Algebraic Number Fields ◽  
Limit Formula ◽  
Quadratic Extensions

AbstractLet K/F be a quadratic extension of number fields. After developing a theory of the Eisenstein series over F, we prove a formula which expresses a partial zeta function of K as a certain integral of the Eisenstein series. As an application, we obtain a limit formula of Kronecker’s type which relates the 0-th Laurent coefficients at s = 1 of zeta functions of K and F.

scholarly journals On the capitulation of the $2$-ideal classes of the field Q(\sqrt{pq_1q_2}, i) of type (2, 2, 2)

2019 ◽  
Vol 38 (4) ◽  
pp. 127-135 ◽  
Author(s):  
Abdelmalek Azizi ◽  
Abdelkader Zekhnini ◽  
Mohammed Taous
Keyword(s):  
Class Group ◽  
Quadratic Extensions ◽  
Genus Field

We study the capitulation of the 2-ideal classes of the field k =Q(\sqrt{p_1p_2q}, \sqrt{-1}), where p_1\equiv p_2\equiv-q\equiv1 \pmod 4  are different primes, in its three quadratic extensions contained in its absolute genus field k^{*} whenever the 2-class group of $\kk$ is of type $(2, 2, 2)$.

scholarly journals EPSTEIN ZETA-FUNCTIONS, SUBCONVEXITY, AND THE PURITY CONJECTURE

2018 ◽  
Vol 19 (2) ◽  
pp. 581-596 ◽  
Author(s):  
Valentin Blomer
Keyword(s):  
Eisenstein Series ◽  
Quadratic Forms ◽  
Critical Line ◽  
Zeta Functions ◽  
Subconvexity Bounds

Subconvexity bounds on the critical line are proved for general Epstein zeta-functions of $k$-ary quadratic forms. This is related to sup-norm bounds for unitary Eisenstein series on $\text{GL}(k)$ associated with the maximal parabolic of type $(k-1,1)$, and the exact sup-norm exponent is determined to be $(k-2)/8$ for $k\geqslant 4$. In particular, if $k$ is odd, this exponent is not in $\frac{1}{4}\mathbb{Z}$, which is relevant in the context of Sarnak’s purity conjecture and shows that it can in general not directly be generalized to Eisenstein series.

scholarly journals Heegner points and Eisenstein series

2011 ◽  
Vol 23 (6) ◽  
Author(s):  
Nicolas Templier
Keyword(s):  
Eisenstein Series ◽  
Heegner Points

Generating the Mapping Class Group

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Exact Sequence ◽  
Simplicial Complex ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Detailed Structure ◽  
Inductive Step ◽  
Closed Curves ◽  
Combinatorial Object ◽  
Generating Sets

This chapter considers the Dehn–Lickorish theorem, which states that when g is greater than or equal to 0, the mapping class group Mod(Sɡ) is generated by finitely many Dehn twists about nonseparating simple closed curves. The theorem is proved by induction on genus, and the Birman exact sequence is introduced as the key step for the induction. The key to the inductive step is to prove that the complex of curves C(Sɡ) is connected when g is greater than or equal to 2. The simplicial complex C(Sɡ) is a useful combinatorial object that encodes intersection patterns of simple closed curves in Sɡ. More detailed structure of C(Sɡ) is then used to find various explicit generating sets for Mod(Sɡ), including those due to Lickorish and to Humphries.

A Primer on Mapping Class Groups (PMS-49)

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Graduate Students ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Class Group ◽  
Mapping Class ◽  
Torelli Group ◽  
Surface Bundles ◽  
Surface Homeomorphisms ◽  
Low Dimensional

The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students. It begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn–Nielsen–Baer–theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.

Electoral Integrity in America

2018 ◽  
Keyword(s):  
News Media ◽  
Class Group ◽  
Public Trust ◽  
Red Flags ◽  
Fake News ◽  
Electoral Fraud ◽  
State Regulations ◽  
Electoral Laws ◽  
American Elections

The contemporary era raises a series of red flags about electoral integrity in America. Problems include plummeting public trust, exacerbated by President Trump’s claims of massive electoral fraud. Confidence in the impartiality and reliability of information from the news media has eroded. And Russian meddling has astutely exploited both these vulnerabilities, heightening fears that the 2016 contest was unfair. This book brings together a first-class group of expert academics and practitioners to analyze challenges facing contemporary elections in America. Contributors analyze evidence for a series of contemporary challenges facing American elections, including the weaknesses of electoral laws, overly restrictive electoral registers, gerrymandering district boundaries, fake news, the lack of transparency, and the hodgepodge of inconsistent state regulations. The conclusion sets these issues in comparative context and draws out the broader policy lessons for improving electoral integrity and strengthening democracy.

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