On the convergence of arithmetic orbifolds

Raimbault, Jean

doi:10.5802/aif.3143

Cited by 13 publications

(9 citation statements)

References 35 publications

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“…The nontrivial estimate F ( ) (Ψ) ≪ /log( ) likely follows, in a more general setting, from the methods of [19] (generalized to non-compact quotients, and using [33] to verify the hypothesis of Benjamini-Schramm convergence). We establish the following further strengthening: Theorem 1.…”

Section: Resultsmentioning

confidence: 99%

Bounds for twisted symmetric square L-functions via half-integral weight periods

Nelson

2020

Forum of Mathematics, Sigma

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We establish the first moment bound $$\begin{align*}\sum_{\varphi} L(\varphi \otimes \varphi \otimes \Psi, \tfrac{1}{2}) \ll_\varepsilon p^{5/4+\varepsilon} \end{align*}$$ for triple product L-functions, where $\Psi $ is a fixed Hecke–Maass form on $\operatorname {\mathrm {SL}}_2(\mathbb {Z})$ and $\varphi $ runs over the Hecke–Maass newforms on $\Gamma _0(p)$ of bounded eigenvalue. The proof is via the theta correspondence and analysis of periods of half-integral weight modular forms. This estimate is not expected to be optimal, but the exponent $5/4$ is the strongest obtained to date for a moment problem of this shape. We show that the expected upper bound follows if one assumes the Ramanujan conjecture in both the integral and half-integral weight cases. Under the triple product formula, our result may be understood as a strong level aspect form of quantum ergodicity: for a large prime p, all but very few Hecke–Maass newforms on $\Gamma _0(p) \backslash \mathbb {H}$ of bounded eigenvalue have very uniformly distributed mass after pushforward to $\operatorname {\mathrm {SL}}_2(\mathbb {Z}) \backslash \mathbb {H}$ . Our main result turns out to be closely related to estimates such as $$\begin{align*}\sum_{|n| < p} L(\Psi \otimes \chi_{n p},\tfrac{1}{2}) \ll p, \end{align*}$$ where the sum is over those n for which $n p$ is a fundamental discriminant and $\chi _{n p}$ denotes the corresponding quadratic character. Such estimates improve upon bounds of Duke–Iwaniec.

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Section: Resultsmentioning

confidence: 99%

Bounds for twisted symmetric square L-functions via half-integral weight periods

Nelson

2020

Forum of Mathematics, Sigma

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“…This was proven in [24] for non-uniform lattices, and in [10] in the case of all torsion-free lattices. Our proof is very similar to the proof for hyperbolic 3-manifolds appearing in [1].…”

Section: Introductionmentioning

confidence: 81%

Betti numbers of Shimura curves and arithmetic three-orbifolds

Frączyk

Raimbault

2019

Alg. Number Th.

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We show that asymptotically the first Betti number b1 of a Shimura curve satisfies the Gauss-Bonnet equality 2π(b1 − 2) = vol where vol is hyperbolic volume; equivalently 2g−2 = (1+o(1)) vol where g is the arithmetic genus. We also show that the first Betti number of a congruence hyperbolic 3-orbifolds asymptotically vanishes relatively to hyperbolic volume, that is b1/ vol → 0. This generalises results from [1] and [10] and we rely on results and techniques from these works, most importantly the notion of Benjamini-Schramm convergence of locally symmetric spaces.

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“…For arithmetic congruence groups, the adelic trace formula can be used to show that certain sequences of arithmetic groups are BS, see [18,20]. An open problem raised in these papers is the question if any sequence ( n ) of congruence subgroups in a given linear algebraic group G is already BS if the covolumes tend to infinity.…”

Section: Non-cocompact Latticesmentioning

confidence: 99%