Expository notes on Spectral Reciprocity with Explicit Transform
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914206240997376 |
|---|---|
| author | Gu, Haonan |
| author_facet | Gu, Haonan |
| contents | We assemble three basic analytic inputs -- the Kuznetsov trace formula on $\mathrm{SL}_2(\mathbb Z)$ with explicit continuous spectrum, the $\mathrm{GL}_3$ Voronoi formula, and $t$-aspect second-moment bounds for $L(1/2+it,φ)$ -- into a single framework for a smoothed $\mathrm{GL}_3$ spectral average. For a fixed Hecke-Maass cusp form $φ$ on $\mathrm{SL}_3(\mathbb Z)$, we evaluate a weight-$0$ spectral average of $L(1/2,φ\times f_j)$ over the $\mathrm{GL}_2$ Maass spectrum. In the Kuznetsov normalization where the diagonal transform has density $t\tanh(πt)$, the diagonal contributes exactly $2\mathcal H_0[h]$; the off-diagonal and the continuous spectrum are bounded with power savings consistent with the currently best unconditional second-moment bounds in the $\mathrm{GL}_3$ $t$-aspect. The argument is organized into a sequence of steps: normalizations and approximate functional equations, insertion into Kuznetsov, Voronoi summation on $\mathrm{GL}_3$, a bilinear estimate for the off-diagonal, evaluation of the diagonal and the Eisenstein contribution, moment refinements and parameter optimization, and finally plateau-smooth spectral windows and standard generalizations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_12118 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Expository notes on Spectral Reciprocity with Explicit Transform Gu, Haonan Number Theory We assemble three basic analytic inputs -- the Kuznetsov trace formula on $\mathrm{SL}_2(\mathbb Z)$ with explicit continuous spectrum, the $\mathrm{GL}_3$ Voronoi formula, and $t$-aspect second-moment bounds for $L(1/2+it,φ)$ -- into a single framework for a smoothed $\mathrm{GL}_3$ spectral average. For a fixed Hecke-Maass cusp form $φ$ on $\mathrm{SL}_3(\mathbb Z)$, we evaluate a weight-$0$ spectral average of $L(1/2,φ\times f_j)$ over the $\mathrm{GL}_2$ Maass spectrum. In the Kuznetsov normalization where the diagonal transform has density $t\tanh(πt)$, the diagonal contributes exactly $2\mathcal H_0[h]$; the off-diagonal and the continuous spectrum are bounded with power savings consistent with the currently best unconditional second-moment bounds in the $\mathrm{GL}_3$ $t$-aspect. The argument is organized into a sequence of steps: normalizations and approximate functional equations, insertion into Kuznetsov, Voronoi summation on $\mathrm{GL}_3$, a bilinear estimate for the off-diagonal, evaluation of the diagonal and the Eisenstein contribution, moment refinements and parameter optimization, and finally plateau-smooth spectral windows and standard generalizations. |
| title | Expository notes on Spectral Reciprocity with Explicit Transform |
| topic | Number Theory |
| url | https://arxiv.org/abs/2512.12118 |