Non-Archimedean Measures Connected With Dirichlet Series

Abstract

With the aim of studying the characteristics of the electronic states at the Fermi energy. we habe inbestigated the electronic structure of NiH by means of an ab initio augmented plane wake calculation. The d band holes of the pure Ni metal are filled in the hydride. in agreement with magnetisation measurements and as suggested by a comparative study of the K absorption edges oi Ni and NiH. This feature is qualitatively similar to that found for PdH: however. the Fermi level in NiH is much closer to the top of the d bands than in PdH. This has se\eral important consequences. (i) NiH has a higher density of states with 10.78 electrons per Ryd per unit cell at the Fermi energy compared to 6.81 for PdH. This trend is i n agreement with the observed increase in the electronic specific heat from PdH, to NiH,. (ii) A larger metal d character and a smaller hydrogen s character is found in NiH compared to PdH, at the Fermi energy. The bandstructure results are used in conjunction with recent experimental estimates of the Debye and Einstein temperatures for the acoustic and optic modes respectively, to calculate the electron-phonon coupling parameter i using the model of Gaspari and Gyorffy. A large decrease of the electronic contribution to i is found from PdH to NiH. This contributes, together with the hardening of the optic modes evidenced by electrical resistivity measurements, to a drastic lowering of the superconducting transition temperature T, and should lead to the nonoccurrence of superconductivity in NiH above 1.5 K. A metal-hydrogen bonding band is found below the metal d states and lies much lower in energy than in the case of PdH; the bandstructure results are in general agreement with the calculation of Switendick. A site and angular momentum density-of-states analysis of the occupied energy bands is also presented which hopefully will stimulate photoemission and x-ray emission experiments.

The Iwasawa Main Conjectures for GL2

“…Dirichlet character ψ of conductor C prime to p, Amice-Vélu [AV75] and Vishik [Vi76] (but see also [MTT86]…”

“…Let f ∈ S ord κ (M p r , χ; A) be an ordinary p-stabilized newform, A being the ring of integers of some finite extension of Q p . Given a primitive A-valued Dirichlet character ψ of conductor C prime to p and any finite set Σ of primes, let L Σ f,ψ ∈ Λ Q,A be the p-adic L-function constructed by Amice-Vélu [AV75] and Vishik [Vi76] (see also [MTT86]) and recalled in 3.4.4.…”

“…Let L be any finite extension of Q p containing Q(f ) and the roots of x 2 − a(p, f )x + χ(p)p k−1 . In this setting, Amice-Vélu [AV75] and Vishik [Vi76] (see also [MTT86]) have constructed a p-adic L-function for f . This is a power series , where χ φ is the primitive Dirichlet character modulo p t φ of p-power order such that χ φ (1 + p) = ζ −1 ; ω is the cyclotomic character modulo p (normalized as in §2); p t φ is the conductor of ω m χ φ ; G(τ ) denotes the usual Gauss sum for a Dirichlet character τ ; Ω ± f are the canonical periods of f ; and e(φ) is an interpolation factor that involves ω m χ φ (p), χ(p), k, m, and the roots of the aforementioned polynomial.…”

A Local-Global Compatibility Conjecture in the p-adic Langlands Programme for GL_2/Q

“…2.5] (which is a variation of the classical argument of Amice-Vélu [1] and Vishik [62], and relies again on the fact that…”

Critical slope p-adic L-functions of CM modular forms

“…Let f be a cuspidal new modular eigenform of weight ≥ 2, and p a prime not dividing the level of f . It has long been known that if α is any root of the Hecke polynomial of f at p such that v p (α) < k − 1, then there is a p-adic L-function L p,α (f ) interpolating the critical L-values of f and its twists by Dirichlet characters of p-power conductor; see [MSD74,AV75,Viš76].…”