scholarly journals The Chowla–Selberg formula for abelian CM fields and Faltings heights

2015 ◽  
Vol 152 (3) ◽  
pp. 445-476 ◽  
Author(s):  
Adrian Barquero-Sanchez ◽  
Riad Masri
Keyword(s):  
Gamma Function ◽  
Abelian Varieties ◽  
Modular Function ◽  
Rational Numbers ◽  
Explicit Formulas ◽  
Arithmetic Properties ◽  
Hilbert Modular ◽  
Cm Points ◽  
Analogous Function

In this paper we establish a Chowla–Selberg formula for abelian CM fields. This is an identity which relates values of a Hilbert modular function at CM points to values of Euler’s gamma function ${\rm\Gamma}$ and an analogous function ${\rm\Gamma}_{2}$ at rational numbers. We combine this identity with work of Colmez to relate the CM values of the Hilbert modular function to Faltings heights of CM abelian varieties. We also give explicit formulas for products of exponentials of Faltings heights, allowing us to study some of their arithmetic properties using the Lang–Rohrlich conjecture.

Special Values of Class Group L-Functions for CM Fields

2010 ◽  
Vol 62 (1) ◽  
pp. 157-181
Author(s):  
Riad Masri
Keyword(s):  
Modular Function ◽  
Special Values ◽  
Relative Class ◽  
Hilbert Modular ◽  
Cm Points ◽  
Relative Class Number ◽  
Rational Multiple ◽  
Totally Real ◽  
Elliptic Units ◽  
Hilbert Modular Variety

Abstract. Let H be the Hilbert class field of a CM number field K with maximal totally real subfield F of degree n over ℚ. We evaluate the second term in the Taylor expansion at s = 0 of the Galoisequivariant L-function ΘS∞(s) associated to the unramified abelian characters of Gal(H/K). This is an identity in the group ring C[Gal(H/K)] expressing Θ (n)S∞ (0) as essentially a linear combination of logarithms of special values ﹛Ψ (zσ)﹜, where Ψ: ℍn →ℝ is a Hilbert modular function for a congruence subgroup of SL2 (OF) and ﹛zσ : σ ∈ Gal(H/K)﹜ are CM points on a universal Hilbert modular variety. We apply this result to express the relative class number hH/hK as a rational multiple of the determinant of an (hK − 1) × (hK − 1) matrix of logarithms of ratios of special values Ψ (zσ), thus giving rise to candidates for higher analogs of elliptic units. Finally, we obtain a product formula for Ψ (zσ) in terms of exponentials of special values of L-functions.

Local Bounds for Torsion Points on Abelian Varieties

2008 ◽  
Vol 60 (3) ◽  
pp. 532-555 ◽  
Author(s):  
Pete L. Clark ◽  
Xavier Xarles
Keyword(s):  
Abelian Varieties ◽  
Residue Field ◽  
Rational Numbers ◽  
Abelian Surface ◽  
Torsion Subgroup ◽  
Good Reduction ◽  
Torsion Points ◽  
Ramification Index ◽  
Numerical Invariants ◽  
Split Torus

AbstractWe say that an abelian variety over a p-adic field K has anisotropic reduction (AR) if the special fiber of its Néronminimal model does not contain a nontrivial split torus. This includes all abelian varieties with potentially good reduction and, in particular, those with complex or quaternionic multiplication. We give a bound for the size of the K-rational torsion subgroup of a g-dimensional AR variety depending only on g and the numerical invariants of K (the absolute ramification index and the cardinality of the residue field). Applying these bounds to abelian varieties over a number field with everywhere locally anisotropic reduction, we get bounds which, as a function of g, are close to optimal. In particular, we determine the possible cardinalities of the torsion subgroup of an AR abelian surface over the rational numbers, up to a set of 11 values which are not known to occur. The largest such value is 72.

scholarly journals Further Results on a Curious Arithmetic Function

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Long Chen ◽  
Kaimin Cheng ◽  
Tingting Wang
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Arithmetic Function ◽  
Rational Numbers ◽  
Arithmetic Properties ◽  
Adic Valuation ◽  
Positive Integers

Let p be an odd prime number and n be a positive integer. Let vpn, N∗, and Q+ denote the p-adic valuation of the integer n, the set of positive integers, and the set of positive rational numbers, respectively. In this paper, we introduce an arithmetic function fp:N∗⟶Q+ defined by fpn≔n/pvpn1−vpn for any positive integer n. We show several interesting arithmetic properties about that function and then use them to establish some curious results involving the p-adic valuation. Some of these results extend Farhi’s results from the case of even prime to that of odd prime.

scholarly journals Modular embeddings of Teichmüller curves

2016 ◽  
Vol 152 (11) ◽  
pp. 2269-2349 ◽  
Author(s):  
Martin Möller ◽  
Don Zagier
Keyword(s):  
Modular Forms ◽  
Fuchsian Groups ◽  
Triangle Groups ◽  
Flat Surfaces ◽  
Arithmetic Properties ◽  
Hilbert Modular Surfaces ◽  
Hilbert Modular ◽  
Genus Two ◽  
Teichmuller Curves ◽  
Teichmüller Curves

Fuchsian groups with a modular embedding have the richest arithmetic properties among non-arithmetic Fuchsian groups. But they are very rare, all known examples being related either to triangle groups or to Teichmüller curves. In Part I of this paper we study the arithmetic properties of the modular embedding and develop from scratch a theory of twisted modular forms for Fuchsian groups with a modular embedding, proving dimension formulas, coefficient growth estimates and differential equations. In Part II we provide a modular proof for an Apéry-like integrality statement for solutions of Picard–Fuchs equations. We illustrate the theory on a worked example, giving explicit Fourier expansions of twisted modular forms and the equation of a Teichmüller curve in a Hilbert modular surface. In Part III we show that genus two Teichmüller curves are cut out in Hilbert modular surfaces by a product of theta derivatives. We rederive most of the known properties of those Teichmüller curves from this viewpoint, without using the theory of flat surfaces. As a consequence we give the modular embeddings for all genus two Teichmüller curves and prove that the Fourier developments of their twisted modular forms are algebraic up to one transcendental scaling constant. Moreover, we prove that Bainbridge’s compactification of Hilbert modular surfaces is toroidal. The strategy to compactify can be expressed using continued fractions and resembles Hirzebruch’s in form, but every detail is different.

Completely normal elements in some finite abelian extensions

2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Ja Koo ◽  
Dong Shin
Keyword(s):  
Function Fields ◽  
Modular Function ◽  
Rational Numbers ◽  
Quadratic Fields ◽  
Cyclotomic Fields ◽  
Imaginary Quadratic Fields ◽  
Abelian Extensions ◽  
Class Fields ◽  
Normal Element ◽  
Ray Class Fields

AbstractWe present some completely normal elements in the maximal real subfields of cyclotomic fields over the field of rational numbers, relying on the criterion for normal element developed in [Jung H.Y., Koo J.K., Shin D.H., Normal bases of ray class fields over imaginary quadratic fields, Math. Z., 2012, 271(1–2), 109–116]. And, we further find completely normal elements in certain abelian extensions of modular function fields in terms of Siegel functions.

scholarly journals Special values of the Gamma function at CM points

2014 ◽  
Vol 36 (3) ◽  
pp. 355-373 ◽  
Author(s):  
M. Ram Murty ◽  
Chester Weatherby
Keyword(s):  
Gamma Function ◽  
Special Values ◽  
Cm Points

Introduction and Statement of Main Results

2012 ◽  
Author(s):  
Xinyi Yuan ◽  
Shou-Wu Zhang ◽  
Wei Zhang
Keyword(s):  
Abelian Varieties ◽  
Main Idea ◽  
Shimura Curves ◽  
Modular Curves ◽  
Heegner Points ◽  
Cm Points ◽  
Totally Real ◽  
Totally Real Fields ◽  
Derivatives Of ◽  
Original Formula

This chapter states the main result of this book regarding Shimura curves and abelian varieties as well as the main idea of the proof of a complete Gross–Zagier formula on quaternionic Shimura curves over totally real fields. It begins with a discussion of the original formula proved by Benedict Gross and Don Zagier, which relates the Néeron–Tate heights of Heegner points on X⁰(N) to the central derivatives of some Rankin–Selberg L-functions under the Heegner condition. In particular, it considers the Gross–Zagier formula on modular curves and abelian varieties parametrized by Shimura curves. It then decribes CM points and the Waldspurger formula before concluding with an outline of our proof, along with the notation and terminology.

On adelic automorphic forms with respect to a quadratic extension

2000 ◽  
Vol 62 (1) ◽  
pp. 29-43 ◽  
Author(s):  
Ze-Li Dou
Keyword(s):  
Modular Forms ◽  
Automorphic Forms ◽  
Quadratic Extension ◽  
Algebraic Number Field ◽  
Hilbert Modular Forms ◽  
Arithmetic Properties ◽  
Theta Lift ◽  
Hilbert Modular ◽  
Algebraic Foundation ◽  
Totally Real

Let E/F be a totally real quadratic extension of a totally real algebraic number field. The author has in an earlier paper considered automorphic forms defined with respect to a quaternion algebra BE over E and a theta lift from such quaternionic forms to Hilbert modular forms over F. In this paper we construct adelic forms in the same setting, and derive explicit formulas concerning the action of Hecke operators. These formulas give an algebraic foundation for further investigations, in explicit form, of the arithmetic properties of the adelic forms and of the associated zeta and L-functions.

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