Sum formula for 𝑆𝐿₂ over a totally real number field

Bruggeman, Roelof W.; Miatello, Roberto J.

doi:10.1090/memo/0919

Cited by 7 publications

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“…With this choice the norm f of f ∈ L 2 ξ (Γ\G, χ) does not change if we consider f as an element of L 2 ξ (Γ 1 \G, χ) for a subgroup Γ 1 of finite index in Γ. As discussed in §2.3.4 in [6], the Fourier expansion of one automorphic form in V ̟ determines the Fourier expansion of any automorphic form in V ̟ . This is valid at all cusps, not only at the cusp ∞ considered in [6] and [7].…”

Section: Fourier Coefficientsmentioning

confidence: 99%

Eigenvalues of Hecke operators on Hilbert modular groups

Bruggeman

Miatello

2013

Asian Journal of Mathematics

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Abstract. Let F be a totally real field, let I be a nonzero ideal of the ring of integers O F of F , let Γ 0 (I) be the congruence subgroup of Hecke type of G = d j=1 SL 2 (R) embedded diagonally in G, and let χ be a character of Γ 0 (I) of the form χFor a finite subset P of prime ideals p not dividing I, we consider the ring H I , generated by the Hecke operators T (p 2 ), p ∈ P (see §3.2) acting on (Γ, χ)-automorphic forms on G.Given the cuspidal space L 2,cusp ξ Γ 0 (I)\G, χ , we let V̟ run through an orthogonal system of irreducible G-invariant subspaces so that each V̟ is invariant under H I . For each 1 ≤ j ≤ d, let λ̟ = (λ ̟,j ) be the vector formed by the eigenvalues of the Casimir operators of the d factors of G on V̟, and for each p ∈ P , we take λ̟,p ∈ Jp :is the eigenvalue on V̟ of the Hecke operator T (p 2 ). If for some prime p the Hecke operator T (p) can be defined then its eigenvalue on V̟ is real and equal to λ̟,p or −λ̟,p.For each family of expanding boxes t → Ωt, as in (3) in R d , and fixed interval Jp in Jp, for each p ∈ P , we consider the counting functionHere c r (̟) denotes the normalized Fourier coefficient of order r at ∞ for the elements of V̟, with r ∈ O ′ F p O ′ F for every p ∈ P . In the main result in this paper, Theorem 1.1, we give, under some mild conditions on the Ωt, the asymptotic distribution of the function N (Ωt; (Jp) p∈P ), as t → ∞. We show that at the finite places outside I the eigenvalues of the Hecke operator T (p 2 ) are equidistributed compatibly with the Sato-Tate measure, whereas at the archimedean places the eigenvalues λ̟ are equidistributed with respect to the Plancherel measure.As a consequence, if we pick an infinite place l and we prescribe λ ̟,j ∈ Ω j for all infinite places j = l and λ̟,p ∈ Jp for all finite places p in P for fixed sets Ω j and fixed intervals Jp ⊂ Jp with positive measure and then allow λ ̟,l to run over larger and larger regions, then there are infinitely many representations ̟ in such a set, and their positive density is as described in Theorem 1.1.

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Section: Fourier Coefficientsmentioning

confidence: 99%

Eigenvalues of Hecke operators on Hilbert modular groups

Bruggeman

Miatello

2013

Asian Journal of Mathematics

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“…It is worth mentioning that the results in this paper ought to generalise naturally to cusp forms on GL 2 over arbitrary number fields F . In [BrMia09], Bruggeman and Miatello prove a form of the Kuznetsov formula for GL 2 over a totally real field and use this to prove weighted Weyl law for cusp forms. Similarly, in [Mag13], Maga proves a semi-adèlic version of the Kuznetsov formula for GL 2 over an arbitrary number field.…”

Section: Introductionmentioning

confidence: 99%

Density theorems for exceptional eigenvalues for congruence subgroups

Humphries

2018

Alg. Number Th.

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“…The second form plays a primary role in the investigation of Kuznetsov on sums of Kloosterman sums in the direction of the Linnik-Selberg conjecture. I Along the classical lines, the Kuznetsov trace formula has been studied and generalized by many authors (see, for example, [Bru1,Bru2,Pro,DI,BM2]). Their ideas of generalizing the formula to the non-spherical case are essentially the same as Kuznetsov. It should however be noted that the pair of Poincaré series is chosen and spectrally decomposed in the space of a given K-type.…”

Section: Introductionmentioning

confidence: 99%

On the Kuznetsov trace formula for PGL2(C)

2017

Journal of Functional Analysis

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In this note, using a representation theoretic method of Cogdell and Piatetski-Shapiro, we prove the Kuznetsov trace formula for an arbitrary discrete group Γ in PGL 2 pCq that is cofinite but not cocompact. An essential ingredient is a kernel formula, recently proved by the author, on Bessel functions for PGL 2 pCq. This approach avoids the difficult 2010 Mathematics Subject Classification. 11F72, 11M36.

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Sum formula for 𝑆𝐿₂ over a totally real number field

Cited by 7 publications

References 0 publications

Eigenvalues of Hecke operators on Hilbert modular groups

Eigenvalues of Hecke operators on Hilbert modular groups

Density theorems for exceptional eigenvalues for congruence subgroups

On the Kuznetsov trace formula for PGL2(C)

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