Hecke theory over arbitrary number fields (original) (raw)
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Hecke characters and the KKK-theory of totally real and CM number fields
2014
Let F/KF/KF/K be an abelian extension of number fields with FFF either CM or totally real and KKK totally real. If FFF is CM and the Brumer-Stark conjecture holds for F/KF/KF/K, we construct a family of G(F/K)G(F/K)G(F/K)--equivariant Hecke characters for FFF with infinite type equal to a special value of certain G(F/K)G(F/K)G(F/K)--equivariant LLL-functions. Using results of Greither-Popescu on the Brumer-Stark conjecture we construct lll-adic imprimitive versions of these characters, for primes l>2l> 2l>2. Further, the special values of these lll-adic Hecke characters are used to construct G(F/K)G(F/K)G(F/K)-equivariant Stickelberger-splitting maps in the lll-primary Quillen localization sequence for FFF, extending the results obtained in 1990 by Banaszak for K=BbbQK = \Bbb QK=BbbQ. We also apply the Stickelberger-splitting maps to construct special elements in the lll-primary piece K2n(F)lK_{2n}(F)_lK2n(F)l of K2n(F)K_{2n}(F)K2n(F) and analyze the Galois module structure of the group D(n)lD(n)_lD(n)l of divisible elements in K2n(F)lK_{2n}(F)_lK2n(F)l, for all n>0n>0n>0. If nnn is odd and coprime to lll and F=KF = KF=K is a fairly general totally real number field, we study the cyclicity of D(n)lD(n)_lD(n)l in relation to the classical conjecture of Iwasawa on class groups of cyclotomic fields and its potential generalization to a wider class of number fields. Finally, if FFF is CM, special values of our lll-adic Hecke characters are used to construct Euler systems in the odd KKK-groups with coefficients K2n+1(F,BbbZ/lk)K_{2n+1}(F, \Bbb Z/l^k)K2n+1(F,BbbZ/lk), for all n>0n>0n>0. These are vast generalizations of Kolyvagin's Euler system of Gauss sums and of the KKK-theoretic Euler systems constructed in Banaszak-Gajda when K=BbbQK = \Bbb QK=BbbQ.
Overconvergent generalised eigenforms of weight one and class fields of real quadratic fields
Advances in Mathematics, 2015
This article examines the Fourier expansions of certain non-classical p-adic modular forms of weight one: the eponymous generalised eigenforms of the title, so called because they lie in a generalised eigenspace for the Hecke operators. When this generalised eigenspace contains the theta series attached to a character of a real quadratic field K in which the prime p splits, the coe cients of the attendant generalised eigenform are expressed as p-adic logarithms of algebraic numbers belonging to an idoneous ring class field of K. This suggests an approach to "explicit class field theory" for real quadratic fields which is simpler than the one based on Stark's conjecture or its p-adic variants, and is perhaps closer in spirit to the classical theory of singular moduli.
2016
Let F be a totally real field and χ an abelian totally odd character of F. In 1988, Gross stated a p-adic analogue of Stark's conjecture that relates the value of the derivative of the p-adic L-function associated to χ and the p-adic logarithm of a p-unit in the extension of F cut out by χ. In this paper we prove Gross's conjecture when F is a real quadratic field and χ is a narrow ring class character. The main result also applies to general totally real fields for which Leopoldt's conjecture holds, assuming that either there are at least two primes above p in F , or that a certain condition relating the L-invariants of χ and χ −1 holds. This condition on L-invariants is always satisfied when χ is quadratic. Contents S. DASGUPTA, H. DARMON, and R. POLLACK 4. Galois representations 477 4.1. Representations attached to ordinary eigenforms 477 4.2. Construction of a cocycle 480 References 482
Hecke characters and the KKK-theory of totally real and CM fields
Acta Arithmetica, 2019
Using results of Greither-Popescu [19] on the Brumer-Stark conjecture we construct l-adic imprimitive versions of these characters, for primes l > 2. Further, the special values of these l-adic Hecke characters are used to construct G(F/K)-equivariant Stickelberger-splitting maps in the Quillen localization sequence for F , extending the results obtained in [1] for K = Q. We also apply the Stickelberger-splitting maps to construct special elements in K 2n (F) l and analyze the Galois module structure of the group D(n) l of divisible elements in K 2n (F) l. If n is odd, l n, and F = K is a fairly general totally real number field, we study the cyclicity of D(n) l in relation to the classical conjecture of Iwasawa on class groups of cyclotomic fields and its potential generalization to a wider class of number fields. Finally, if F is CM, special values of our l-adic Hecke characters are used to construct Euler systems in odd K-groups K 2n+1 (F, Z/l k). These are vast generalizations of Kolyvagin's Euler system of Gauss sums [33] and of the K-theoretic Euler systems constructed in [4] when K = Q.
Hecke operators with respect to the modular group of quaternions
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 1990
The classical theory of Hecke operators for (entire) elliptic modular forms is a powerful instrument in obtaining number theoretical applications and especially to reveal multiplicative properties of the Fourier coefficients. M. SUGAWA~ [20], [21] developed an approach of Hecke operators for Siegel modular forms, which was extended by H. MA~SS [15]. A comprehensive representation of this theory containing recent results can for instance be found in the books of E. FREITAG [6], chapter IV, or A.N. ANDRIANOV [2], where the more complicated theory for congruence subgroups is included. H.-W. Lu [14] started the investigation of modular forms over the Hurwitz order (cf.
2-ADIC Properties of Certain Modular Forms and Their Applications to Arithmetic Functions
It is a classical observation of Serre that the Hecke algebra acts locally nilpo- tently on the graded ring of modular forms modulo 2 for the full modular group. Here we consider the problem of classifying spaces of modular forms for which this phenomenon continues to hold. We give a number of consequences of this investigation as they relate to quadratic forms, partition functions, and central values of twisted modular L-functions.
Rationality theorems for Hecke operators on GLn
Journal of Number Theory, 2003
We define n families of Hecke operators T n k (p ) for GL n whose generating series T n k (p )u are rational functions of the form q k (u) −1 where q k is a polynomial of degree n k , and whose form is that of the kth exterior product. This work can be viewed as a refinement of work of Andrianov [1], in which he defined Hecke operators the sum of whose generating series was a rational function with nontrivial numerator and whose denominator was essentially k q k .