a p-adic zeta function, or more generally a p-adic L-function, is a function analogous to the Riemann zeta function, or more general L-functions, but...

p, the p-adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; p-adic numbers...

In each case, there is a main conjecture linking the tower to a p-adic L-function. In 2002, Christopher Skinner and Eric Urban claimed a proof of a...

main conjecture of Iwasawa theory is a deep relationship between p-adic L-functions and ideal class groups of cyclotomic fields, proved by Kenkichi Iwasawa...

generalisation of that phenomenon. In that case results have been obtained for p-adic L-functions, which describe certain Galois modules. The statistics of the zero...

(2004), Fontaine's rings and p-adic L-functions (PDF) Koblitz, Neal (1984), p-adic Numbers, p-adic Analysis, and Zeta-Functions, Graduate Texts in Mathematics...

χℓ form a strictly compatible system of ℓ-adic representations. The p-adic cyclotomic character is the p-adic Tate module of the multiplicative group...

Motivic zeta function of a motive Multiple zeta function, or Mordell–Tornheim zeta function of several variables p-adic zeta function of a p-adic number Prime...

Euler system p-adic L-function Arithmetic geometry Complex multiplication Abelian variety of CM-type Chowla–Selberg formula Hasse–Weil zeta function...

Fellowship. Hida received in 1992 for his research on p-adic L-functions of algebraic groups and p-adic Hecke rings the Spring Prize of the Mathematical Society...

1960s meant that Hasse–Weil L-functions could be regarded as Artin L-functions for the Galois representations on l-adic cohomology groups. Bad reduction...

the p-adic valuation or p-adic order of an integer n is the exponent of the highest power of the prime number p that divides n. It is denoted ν p ( n...

with a thesis on p-adic L-functions attached to elliptic curves, a Ph.D. in Mathematics in 1990 with a thesis on p-Adic L-functions and Galois groups...

a p-adic L-function with the eigenvalues of an operator, so can be thought of as an analogue of the Hilbert–Pólya conjecture for p-adic L-functions. Several...

Ramanujan", Séminaire Delange-Pisot-Poitou, 14 Swinnerton-Dyer, H. P. F. (1973), "On l-adic representations and congruences for coefficients of modular forms"...

A p-adic language is defined as the set of strings L η ( p ) = { n 1 n 2 n 3 … | 0 ≤ n k < p  and  0. n 1 n 2 n 3 … > η } {\displaystyle L_{\eta }(p...

(PDF) on 2008-05-10. Neukirch et al. (2008) p. 626–627 Iwasawa, Kenkichi (1972). Lectures on p-adic L-functions. Annals of Mathematics Studies. Vol. 74....

system of polynomial equations over number fields, finite fields, p-adic fields, or function fields, i.e. fields that are not algebraically closed excluding...

the p-adic completions of Q . {\displaystyle \mathbb {Q} .} Let v be a valuation of K and let L be a field extension of K. An extension of v (to L) is...

1960 using p-adic methods), and the remaining conjecture, the analogue of the Riemann hypothesis was proved by Pierre Deligne (1974) using ℓ-adic cohomology...

much of the work on p-adic L-functions. Crystalline cohomology, from the point of view of number theory, fills a gap in the l-adic cohomology information...

both analytic functions. The work of Lubotzky and Mann, combined with Michel Lazard's solution to Hilbert's fifth problem over the p-adic numbers, shows...

to define the p-adic zeta function. The simplest form of Kummer's congruence states that B h h ≡ B k k ( mod p )  whenever  h ≡ k ( mod p − 1 ) {\displaystyle...

The Ihara zeta function was first defined by Yasutaka Ihara in the 1960s in the context of discrete subgroups of the two-by-two p-adic special linear...

construction of p-adic L-functions for modular forms on GL(2) over any number field. He gave a formula for the explicit sums of arithmetic functions expressing...

important results on the l-adic representations attached to modular forms, and the conjectural functional equations of L-functions. Deligne also focused...

group is zero). If X is a smooth proper scheme over a field K then the ℓ-adic cohomology groups of its geometric fibre are Galois modules for the absolute...

q} elements, and Frobq is the geometric Frobenius acting on ℓ {\displaystyle \ell } -adic étale cohomology with compact supports of X ¯ {\displaystyle...

general L-functions are also defined by Euler products, involving, at each finite place, the determinant of the Frobenius endomorphism acting on l-adic cohomology...

contributions include works on p-adic L functions and real-analytic automorphic forms. His work on p-adic L-functions, later recognised as an aspect of...