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Ordinary parts of admissible representations of padic reductive groups II. Derived functors
, 2010
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Nonvanishing modulo p of Hecke L–values and application
 LFUNCTIONS AND GALOIS REPRESENTATIONS EDITED BY DAVID BURNS , KEVIN BUZZARD , JAN NEKOVÁR
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On the freeness of the integral cohomology groups of HilbertBlumenthal varieties as Hecke modules
"... Let F be a totally real field of degree d ≥ 1. Let f be a Hilbert modular cusp form defined over F of level N ⊂ OF and parallel weight (k, k,..., k) with k ≥ 2. Assume that f is a normalized newform and a common eigenform of all the Hecke operators. Such a Hilbert modular form will be called a primi ..."
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Let F be a totally real field of degree d ≥ 1. Let f be a Hilbert modular cusp form defined over F of level N ⊂ OF and parallel weight (k, k,..., k) with k ≥ 2. Assume that f is a normalized newform and a common eigenform of all the Hecke operators. Such a Hilbert modular form will be called a primitive
HECKE FIELDS OF HILBERT MODULAR ANALYTIC FAMILIES
"... Abstract. For a non CM slope 0 analytic family of Hilbert modular forms, we prove that the Hecke field over Q[µp∞] grows indefinitely large over any infinite set of arithmetic points with fixed weight by increasing ppower level for a prime p over a rational prime p. The odd prime condition: p> 2 ..."
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Abstract. For a non CM slope 0 analytic family of Hilbert modular forms, we prove that the Hecke field over Q[µp∞] grows indefinitely large over any infinite set of arithmetic points with fixed weight by increasing ppower level for a prime p over a rational prime p. The odd prime condition: p> 2 made in [H11] is also eliminated in this paper for the strong horizontal theorem. Fix a prime p and field embeddings C ι∞ ← ↪ Q ιp ↩ → Qp ⊂ Cp, where Q is an algebraic closure of Q. Fix a finite totally real extension F/Q inside Q with integer ring O. We define a prime ideal of O by p = {α ∈ O: ιp(α)p < 1} and fix an Oideal N prime to p. Let Sκ(N, ɛ; C) denote the space of weight κ adelic Hilbert cusp forms f: GL2(F)\GL2(FA) → C of level N with Neben character ɛ modulo N, where N is a nonzero ideal of O. Here the weight κ = (κ1, κ2) is the Hodge weight of the rank 2 pure motive M(f) with coefficient in the Hecke field Q(f) associated to any Hecke eigenform f ∈ Sκ(N, ɛ; C) (see [BR]). For each field embedding σ: F ↩ → Q, M(f) ⊗F,ι∞◦σ C has Hodge weight (κ1,σ, κ2,σ) and (κ2,σ, κ1,σ), and the motivic weight κ1,σ+κ2,σ is independent of σ. We normalize the weight imposing κ1,σ ≤ κ2,σ. This normalization is
DRAFT: PADIC DEFORMATIONS OF COHOMOLOGY CLASSES OF SUBGROUPS OF GL(N,Z): THE NONORDINARY CASE
, 2000
"... Our paper [ASBarcelona] presented a control theorem on the ordinary part of a padic deformation of the cohomology of congruence subgroups of GL(n,Z). In the current work, we present a different approach to this problem which enables us to work with a much larger part of the nontorsion part of the ..."
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Our paper [ASBarcelona] presented a control theorem on the ordinary part of a padic deformation of the cohomology of congruence subgroups of GL(n,Z). In the current work, we present a different approach to this problem which enables us to work with a much larger part of the nontorsion part of the cohomology, including
TRANSCENDENCE OF HECKE OPERATORS IN THE BIG HECKE ALGEBRA
"... Abstract. We prove a transcendence result of Hecke operator T (l) over the ring generated by diamond operators over Q in a nonCM component of the cyclotomic ordinary big padic Hecke algebra and discuss its conjectural implication of a modulo p version to the vanishing of the cyclotomic µinvariant ..."
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Abstract. We prove a transcendence result of Hecke operator T (l) over the ring generated by diamond operators over Q in a nonCM component of the cyclotomic ordinary big padic Hecke algebra and discuss its conjectural implication of a modulo p version to the vanishing of the cyclotomic µinvariant. Take the field Q of all numbers in C algebraic over Q. Fix a prime p and a field embedding Q ip ↪ → Qp ⊂ Cp. Fix a totally real number field F inside Q with integer ring O (as the base field for Hilbert modular forms). We choose and fix an Oideal N prime to p (as the level of modular form). Let Sk(Np r, ;C) denote the space of (parallel) weight k adelic Hilbert cusp forms of level Npr with Neben character modulo Npr. Thus is the central character of the automorphic representation generated by each Hecke eigenform in the space. We regard as a character of the strict ray class group ClF (Np ∞) = lim←−m ClF (Npm) module Np∞. Here (k − 1, 0) and (0, k − 1) are the padic