scholarly journals SOME ARITHMETIC PROPERTIES OF AN ELLIPTIC DEDEKIND SUM

2020 ◽  
Vol 46 (1) ◽  
pp. 35-54
Author(s):  
Genki Shibukawa
Keyword(s):  
Dedekind Sum ◽  
Arithmetic Properties

The (α, β)-ramification invariants of a number field

2021 ◽  
Vol 71 (1) ◽  
pp. 251-263
Author(s):  
Guillermo Mantilla-Soler
Keyword(s):  
Zeta Function ◽  
Number Field ◽  
Residue Class ◽  
Interesting Application ◽  
Dedekind Zeta Function ◽  
Arithmetic Properties

Abstract Let L be a number field. For a given prime p, we define integers α p L $ \alpha_{p}^{L} $ and β p L $ \beta_{p}^{L} $ with some interesting arithmetic properties. For instance, β p L $ \beta_{p}^{L} $ is equal to 1 whenever p does not ramify in L and α p L $ \alpha_{p}^{L} $ is divisible by p whenever p is wildly ramified in L. The aforementioned properties, although interesting, follow easily from definitions; however a more interesting application of these invariants is the fact that they completely characterize the Dedekind zeta function of L. Moreover, if the residue class mod p of α p L $ \alpha_{p}^{L} $ is not zero for all p then such residues determine the genus of the integral trace.

scholarly journals Spectral curves are transcendental

2021 ◽  
Vol 111 (1) ◽  
Author(s):  
H. W. Braden
Keyword(s):  
Spectral Curve ◽  
Spectral Curves ◽  
Arithmetic Properties ◽  
A Charge

AbstractSome arithmetic properties of spectral curves are discussed: the spectral curve, for example, of a charge $$n\ge 2$$ n ≥ 2 Euclidean BPS monopole is not defined over $$\overline{\mathbb {Q}}$$ Q ¯ if smooth.

scholarly journals On a sum analogous to Dedekind sum and its mean square value formula

2002 ◽  
Vol 32 (1) ◽  
pp. 47-55 ◽  
Author(s):  
Zhang Wenpeng
Keyword(s):  
Asymptotic Property ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Mean Square ◽  
Dedekind Sum ◽  
The Mean

The main purpose of this paper is using the mean value theorem of DirichletL-functions to study the asymptotic property of a sum analogous to Dedekind sum, and give an interesting mean square value formula.

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.

scholarly journals Arithmetic properties of partition triples with odd parts distinct

2015 ◽  
Vol 11 (06) ◽  
pp. 1791-1805 ◽  
Author(s):  
Liuquan Wang
Keyword(s):  
Infinite Family ◽  
Arithmetic Properties ◽  
Partition Triples

Let pod -3(n) denote the number of partition triples of n where the odd parts in each partition are distinct. We find many arithmetic properties of pod -3(n) involving the following infinite family of congruences: for any integers α ≥ 1 and n ≥ 0, [Formula: see text] We also establish some arithmetic relations between pod (n) and pod -3(n), as well as some congruences for pod -3(n) modulo 7 and 11.

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