Derived Hecke Algebra and Cohomology of Arithmetic Groups

Venkatesh, Akshay

doi:10.1017/fmp.2019.6

Cited by 19 publications

(54 citation statements)

References 34 publications

(126 reference statements)

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“…This section is not used in the remainder of this paper. It serves to connect the previous construction with the discussion in [17]:…”

Section: Taylor Wiles Primesmentioning

confidence: 98%

“…The general conjectures of [17,15], transposed to the current (coherent) situation, predict that the dual of H 1 mot (Q, M g (1)) int should act on H * (X E , ω)[g]. There is a natural map…”

Section: Taylor Wiles Primesmentioning

confidence: 99%

“…For example, in the examples of modular forms attached to cubic fields, this will amount to the reduction of a unit in the cubic field at a degree one prime above q. Although explicit, the general definition is unfortunately opaque (the motivation comes from computations in [17]).…”

Section: Taylor Wiles Primesmentioning

confidence: 99%

“…In this section we define derived Hecke operators and formulate the main conjecture concerning their relationship to U g . This discussion is obtained by transcribing the theory of [17] to the present context; in this section, we just describe the conclusions of this process.…”

Section: Derived Hecke Operators and The Main Conjecturementioning

confidence: 99%

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Derived Hecke Algebra for Weight One Forms

Harris

Venkatesh

2018

Experimental Mathematics

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ABSTRACT. We study the action of the derived Hecke algebra on the space of weight one forms. By analogy with the topological case, we formulate a conjecture relating this to a certain Stark unit.We verify the truth of the conjecture numerically, for the weight one forms of level 23 and 31, and many derived Hecke operators at primes less than 200. Our computation depends in an essential way on Merel's evaluation of the pairing between the Shimura and cuspidal subgroups of J0(q).

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“…This section is not used in the remainder of this paper. It serves to connect the previous construction with the discussion in [17]:…”

Section: Taylor Wiles Primesmentioning

confidence: 98%

Section: Taylor Wiles Primesmentioning

confidence: 99%

Section: Taylor Wiles Primesmentioning

confidence: 99%

Section: Derived Hecke Operators and The Main Conjecturementioning

confidence: 99%

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Derived Hecke Algebra for Weight One Forms

Harris

Venkatesh

2018

Experimental Mathematics

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“…Altogether the mod p "derived Hecke algebra" (cf. [Ven17]) We should point out that Theorems 1.1 and 1.2 were known to some experts in the field, although they never appeared explicitly in print. Also, the A ∞ -structures we work with are only uniquely defined up a non-canonical A ∞ -isomorphism -they depend on several choices; see the end of Section 9 for instance.…”

Section: Introductionmentioning

confidence: 99%

Koszul duality for Iwasawa algebras modulo 𝑝

Sorensen

2020

Represent. Theory

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In this article we establish a version of Koszul duality for filtered rings arising from p-adic Lie groups. Our precise setup is the following. We let G be a uniform pro-p group and consider its completed group algebra Ω = k[[G]] with coefficients in a finite field k of characteristic p. It is known that Ω carries a natural filtration and grΩ = S(g) where g is the (abelian) Lie algebra of G over k. One of our main results in this paper is that the Koszul dual grΩ ! = g ∨ can be promoted to an A∞-algebra in such a way that the derived category of pseudocompact Ω-modules D(Ω) becomes equivalent to the derived category of strictly unital A∞-modules D∞( g ∨ ). In the case where G is an abelian group we prove that the A∞-structure is trivial and deduce an equivalence between D(Ω) and the derived category of differential graded modules over g ∨ which generalizes a result of Schneider for Zp.but hereD is not the derived category, it is rather squeezed in between the homotopy category and the derived category. We should emphasize that the filtered rings U considered in [Pos93] are of a completely different nature than the Iwasawa algebras we will consider in this paper. Positselski requires U to be a non-homogeneous quadratic algebra U = T (V )/(P ) for some vector space V and ideal of relations (P ) in the tensor algebra generated by a subspace P ⊂ k ⊕ V ⊕ V ⊗2 with P ∩ (k ⊕ V ) = 0. He associates the quadratic algebra A = T (V )/(Q) where Q ⊂ V ⊗2 is the image of P under projection and decrees that the natural map A → grU is an isomorphism.Analogously we will consider completed group rings Ω = Ω(G) modulo p of certain compact p-adic Lie groups G for which grΩ ≃ S(g) where g = Lie(G) is the Lie algebra, and promote the Koszul dual algebra grΩ ! ≃ g ∨ to a so-called A ∞ -algebra in such a way that D(Ω) ∼ −→ D ∞ ( g ∨ ). The notation and terminology will be explained in more detail below.

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Hecke operators on topological modular forms

Davies

2024

Advances in Mathematics

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Derived Hecke Algebra and Cohomology of Arithmetic Groups

Cited by 19 publications

References 34 publications

Derived Hecke Algebra for Weight One Forms

Derived Hecke Algebra for Weight One Forms

Koszul duality for Iwasawa algebras modulo 𝑝

Hecke operators on topological modular forms

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