scholarly journals Congruences for critical values of higher derivatives of twisted Hasse–Weil L-functions, III

2021 ◽  
pp. 1-26
Author(s):  
WERNER BLEY ◽  
DANIEL MACIAS CASTILLO
Keyword(s):  
Prime Number ◽  
Numerical Computation ◽  
Galois Module ◽  
Height Pairing ◽  
Relevant Case ◽  
Congruence Relations ◽  
Weil Group ◽  
Higher Derivatives ◽  
Tamagawa Number Conjecture ◽  
Derivatives Of

Abstract Let A be an abelian variety defined over a number field k, let p be an odd prime number and let $F/k$ be a cyclic extension of p-power degree. Under not-too-stringent hypotheses we give an interpretation of the p-component of the relevant case of the equivariant Tamagawa number conjecture in terms of integral congruence relations involving the evaluation on appropriate points of A of the ${\rm Gal}(F/k)$ -valued height pairing of Mazur and Tate. We then discuss the numerical computation of this pairing, and in particular obtain the first numerical verifications of this conjecture in situations in which the p-completion of the Mordell–Weil group of A over F is not a projective Galois module.

scholarly journals On Mordell–Weil groups and congruences between derivatives of twisted Hasse–Weil L-functions

2018 ◽  
Vol 2018 (734) ◽  
pp. 187-228
Author(s):  
David Burns ◽  
Daniel Macias Castillo ◽  
Christian Wuthrich
Keyword(s):  
Prime Number ◽  
Abelian Variety ◽  
Galois Extension ◽  
Field Extension ◽  
Galois Structure ◽  
Weil Group ◽  
Relevant Field ◽  
Tamagawa Number Conjecture ◽  
Derivatives Of ◽  
Positive Rank

AbstractLetAbe an abelian variety defined over a number fieldkand letFbe a finite Galois extension ofk. Letpbe a prime number. Then under certain not-too-stringent conditions onAandFwe compute explicitly the algebraic part of thep-component of the equivariant Tamagawa number of the pair(h^{1}(A_{/F})(1),\mathbb{Z}[{\rm Gal}(F/k)]). By comparing the result of this computation with the theorem of Gross and Zagier we are able to give the first verification of thep-component of the equivariant Tamagawa number conjecture for an abelian variety in the technically most demanding case in which the relevant Mordell–Weil group has strictly positive rank and the relevant field extension is both non-abelian and of degree divisible byp. More generally, our approach leads us to the formulation of certain precise families of conjecturalp-adic congruences between the values ats=1of derivatives of the Hasse–WeilL-functions associated to twists ofA, normalised by a product of explicit equivariant regulators and periods, and to explicit predictions concerning the Galois structure of Tate–Shafarevich groups. In several interesting cases we provide theoretical and numerical evidence in support of these more general predictions.

The Coarse Scale Velocity

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Velocity Field ◽  
Building Blocks ◽  
Coarse Scale ◽  
Divergence Equation ◽  
Higher Derivatives ◽  
Order Of Magnitude ◽  
Derivatives Of

This chapter deals with the coarse scale velocity. It begins the proof of Lemma (10.1) by choosing a double mollification for the velocity field. Here ∈ᵥ is taken to be as large as possible so that higher derivatives of velement are less costly, and each vsubscript Element has frequency smaller than λ‎ so elementv⁻¹ must be smaller than λ‎ in order of magnitude. Each derivative of vsubscript Element up to order L costs a factor of Ξ‎. The chapter proceeds by describing the basic building blocks of the construction, the choice of elementv and the parametrix expansion for the divergence equation.

scholarly journals On the Zeros of Higher Derivatives of Hardy'sZ-Function

1999 ◽  
Vol 75 (2) ◽  
pp. 262-278 ◽  
Author(s):  
Kohji Matsumoto ◽  
Yoshio Tanigawa
Keyword(s):  
Higher Derivatives ◽  
Derivatives Of

An Introduction to Finite Differencing

1998 ◽  
Author(s):  
T. N. Krishnamurti ◽  
H. S. Bedi ◽  
V. M. Hardiker
Keyword(s):  
Finite Difference ◽  
Finite Differences ◽  
Single Point ◽  
Finite Differencing ◽  
Finite Difference Approximations ◽  
Difference Approximations ◽  
Student Reading ◽  
Higher Derivatives ◽  
Derivatives Of

This chapter on finite differencing appears oddly placed in the early part of a text on spectral modeling. Finite differences are still traditionally used for vertical differencing and for time differencing. Therefore, we feel that an introduction to finite-differencing methods is quite useful. Furthermore, the student reading this chapter has the opportunity to compare these methods with the spectral method which will be developed in later chapters. One may use Taylor’s expansion of a given function about a single point to approximate the derivative(s) at that point. Derivatives in the equation involving a function are replaced by finite difference approximations. The values of the function are known at discrete points in both space and time. The resulting equation is then solved algebraically with appropriate restrictions. Suppose u is a function of x possessing derivatives of all orders in the interval (x — n∆x, x + n∆x). Then we can obtain the values of u at points x ± n∆ x, where n is any integer, in terms of the value of the function and its derivatives at point x, that is, u(x) and its higher derivatives.

Symbolic Powers of Regular Primes

1981 ◽  
Vol 33 (6) ◽  
pp. 1331-1337 ◽  
Author(s):  
Yasunori Ishibashi
Keyword(s):  
Prime Ideal ◽  
Finite Type ◽  
Polynomial Ring ◽  
Type A ◽  
Symbolic Power ◽  
Perfect Field ◽  
Symbolic Powers ◽  
Higher Derivatives ◽  
Differential Filtration ◽  
Derivatives Of

In a recent paper [6], P. Seibt has obtained the following result: Let k be a field of characteristic 0, k[T1, … , Tr] the polynomial ring in r indeterminates over k, and let P be a prime ideal of k[T1, … , Tr]. Then a polynomial F belongs to the n-th symbolic power P(n) of P if and only if all higher derivatives of F from the 0-th up to the (n – l)-st order belong to P.In this work we shall naturally generalize this result so as to be valid for primes of the polynomial ring over a perfect field k. Actually, we shall get a generalization as a corollary to a theorem which asserts: For regular primes P in a k-algebra R of finite type, a certain differential filtration of R associated with P coincides with the symbolic power filtration (P(n))n≧0.

scholarly journals A look into the cosmological consequences of a dark energy model with higher derivatives of H in the framework of Chameleon Brans–Dicke cosmology

2019 ◽  
Vol 28 (11) ◽  
pp. 1950149 ◽  
Author(s):  
Antonio Pasqua ◽  
Surajit Chattopadhyay ◽  
Aroonkumar Beesham
Keyword(s):  
Dark Energy ◽  
Dark Energy Model ◽  
Late Time ◽  
Statefinder Parameters ◽  
Eos Parameter ◽  
Higher Derivatives ◽  
De Formula ◽  
Derivatives Of ◽  
The Universe ◽  
Far Future

In this paper, we study some relevant cosmological features of a Dark Energy (DE) model with Granda–Oliveros cut-off, which is just a specific case of Nojiri–Odintsov holographic DE [S. Nojiri and S. D. Odintsov, Gen. Relativ. Gravit. 38 (2006) 1285] unifying phantom inflation with late-time acceleration, in the framework of Chameleon Brans–Dicke (BD) cosmology. Choosing a particular ansatz for some of the quantities involved, we derive the expressions of some important cosmological quantities, like the Equation of State (EoS) parameter of DE [Formula: see text], the effective EoS parameter [Formula: see text], the pressure of DE [Formula: see text] and the deceleration parameter [Formula: see text]. Moreover, we study the behavior of statefinder parameters [Formula: see text] and [Formula: see text], of the cosmographic parameters [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] and of the squared speed of the sound [Formula: see text] for both case corresponding to noninteracting and interacting Dark sectors. We also plot the quantities we have derived and we calculate their values for [Formula: see text] (i.e. for the beginning of the universe history), for [Formula: see text] (i.e. for far future) and for the present time, indicated with [Formula: see text]. The EoS parameters have been tested against various observational values available in the literature.

scholarly journals THE -MODULAR LOCAL LANGLANDS CORRESPONDENCE AND LOCAL CONSTANTS

2019 ◽  
pp. 1-51 ◽  
Author(s):  
R. Kurinczuk ◽  
N. Matringe
Keyword(s):  
Prime Number ◽  
Local Field ◽  
Equivalence Classes ◽  
Modular Representations ◽  
Langlands Correspondence ◽  
Weil Group ◽  
Local Langlands Correspondence ◽  
Residual Characteristic

Let  $F$ be a non-archimedean local field of residual characteristic  $p$ , $\ell \neq p$ be a prime number, and  $\text{W}_{F}$ the Weil group of  $F$ . We classify equivalence classes of  $\text{W}_{F}$ -semisimple Deligne  $\ell$ -modular representations of  $\text{W}_{F}$ in terms of irreducible  $\ell$ -modular representations of  $\text{W}_{F}$ , and extend constructions of Artin–Deligne local constants to this setting. Finally, we define a variant of the  $\ell$ -modular local Langlands correspondence which satisfies a preservation of local constants statement for pairs of generic representations.

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