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<p>Let <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <mi>N</mi> </math></EquationSource> <EquationSource Format="TEX">$N$</EquationSource> </InlineEquation> be a prime and <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <mi>ϕ</mi> </math></EquationSource> <EquationSource Format="TEX">$\phi $</EquationSource> </InlineEquation> be a Hecke-Maaß cuspidal newform for the Hecke congruence subgroup <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Γ</mi> <mn>0</mn> </msub> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$\Gamma _{0}(N)$</EquationSource> </InlineEquation> in <InlineEquation ID="IEq5"> <EquationSource Format="MATHML"><math> <msub> <mo>SL</mo> <mi>n</mi> </msub> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$\operatorname{SL}_{n}(\mathbb{R})$</EquationSource> </InlineEquation>. We define a notion of the bulk <InlineEquation ID="IEq6"> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Ω</mi> <mi>N</mi> </msub> </math></EquationSource> <EquationSource Format="TEX">$\Omega _{N}$</EquationSource> </InlineEquation> of the space <InlineEquation ID="IEq7"> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Γ</mi> <mn>0</mn> </msub> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">∖</mi> <msub> <mo>SL</mo> <mi>n</mi> </msub> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mo>SO</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$\Gamma _{0}(N) \backslash \operatorname{SL}_{n}(\mathbb{R})/ \operatorname{SO}(n)$</EquationSource> </InlineEquation> with respect to a compact set <InlineEquation ID="IEq8"> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msub> <mo>SL</mo> <mi>n</mi> </msub> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$\Omega \subset \operatorname{SL}_{n}(\mathbb{R})$</EquationSource> </InlineEquation>. For any prime <InlineEquation ID="IEq9"> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> <EquationSource Format="TEX">$n$</EquationSource> </InlineEquation>, we prove sub-baseline bounds for the sup-norm of <InlineEquation ID="IEq10"> <EquationSource Format="MATHML"><math> <mi>ϕ</mi> </math></EquationSource> <EquationSource Format="TEX">$\phi $</EquationSource> </InlineEquation> restricted to <InlineEquation ID="IEq11"> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Ω</mi> <mi>N</mi> </msub> </math></EquationSource> <EquationSource Format="TEX">$\Omega _{N}$</EquationSource> </InlineEquation>. Conditionally on GRH, we generalise this result to all <InlineEquation ID="IEq12"> <EquationSource Format="MATHML"><math> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </math></EquationSource> <EquationSource Format="TEX">$n \geq 2$</EquationSource> </InlineEquation>. The methods involve a new reduction theory with level structure, based on generalisations of Atkin-Lehner operators.</p>

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The sup-norm problem for newforms of large level on \(\operatorname{PGL}(n)\)

  • 期刊论文

  • 发表日期：

    2026年3月3日

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  • Radu Toma

摘要

Let N $N$ be a prime and ϕ $\phi $ be a Hecke-Maaß cuspidal newform for the Hecke congruence subgroup Γ 0 ( N ) $\Gamma _{0}(N)$ in SL n ( R ) $\operatorname{SL}_{n}(\mathbb{R})$ . We define a notion of the bulk Ω N $\Omega _{N}$ of the space Γ 0 ( N ) ∖ SL n ( R ) / SO ( n ) $\Gamma _{0}(N) \backslash \operatorname{SL}_{n}(\mathbb{R})/ \operatorname{SO}(n)$ with respect to a compact set Ω ⊂ SL n ( R ) $\Omega \subset \operatorname{SL}_{n}(\mathbb{R})$ . For any prime n $n$ , we prove sub-baseline bounds for the sup-norm of ϕ $\phi $ restricted to Ω N $\Omega _{N}$ . Conditionally on GRH, we generalise this result to all n ≥ 2 $n \geq 2$ . The methods involve a new reduction theory with level structure, based on generalisations of Atkin-Lehner operators.