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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2512.20483 |
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| _version_ | 1866917166657306624 |
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| author | Creech, Steven Twiss, Henry Wei, Zhining Zenz, Peter |
| author_facet | Creech, Steven Twiss, Henry Wei, Zhining Zenz, Peter |
| contents | We let $f$ be a half-integral weight modular form of weight $κ>4$ on $Γ_0(4)$ that is an eigenfunction of all Hecke operators $T_n$, so that $T_nf = Λ_f(n)n^{\frac{κ-1}{2}}f$. Let $\|f\|$ denote the Petersson norm of $f$. We study a weighted second moment of the central value of the $L$-function associated to $f$ over an orthogonal basis $H_κ(4)$ of $S_κ(Γ_0(4))$. This corresponds to studying the following sum:
$$\sum_{f\in H_κ(4)}\frac{Λ_f(n)\vert L(1/2,f)\vert^2}{\|f\|^2}.$$
Using the relative trace formula, we obtain an asymptotic formula for the second moment. We then use the method of amplification to get the subconvexity bound
$$L(1/2,f)\ll_{\varepsilon} (κ^2)^{\frac{1}{4}-\frac{1}{40}+\varepsilon}.$$
This is the first subconvexity result for half-integral weight modular forms in the weight aspect. We also apply our second moment result to get a quantitative simultaneous non-vanishing result for central values of $L$-functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_20483 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Second Moment of Central Values of Half-Integral Weight Modular Forms and Subconvexity Creech, Steven Twiss, Henry Wei, Zhining Zenz, Peter Number Theory We let $f$ be a half-integral weight modular form of weight $κ>4$ on $Γ_0(4)$ that is an eigenfunction of all Hecke operators $T_n$, so that $T_nf = Λ_f(n)n^{\frac{κ-1}{2}}f$. Let $\|f\|$ denote the Petersson norm of $f$. We study a weighted second moment of the central value of the $L$-function associated to $f$ over an orthogonal basis $H_κ(4)$ of $S_κ(Γ_0(4))$. This corresponds to studying the following sum: $$\sum_{f\in H_κ(4)}\frac{Λ_f(n)\vert L(1/2,f)\vert^2}{\|f\|^2}.$$ Using the relative trace formula, we obtain an asymptotic formula for the second moment. We then use the method of amplification to get the subconvexity bound $$L(1/2,f)\ll_{\varepsilon} (κ^2)^{\frac{1}{4}-\frac{1}{40}+\varepsilon}.$$ This is the first subconvexity result for half-integral weight modular forms in the weight aspect. We also apply our second moment result to get a quantitative simultaneous non-vanishing result for central values of $L$-functions. |
| title | Second Moment of Central Values of Half-Integral Weight Modular Forms and Subconvexity |
| topic | Number Theory |
| url | https://arxiv.org/abs/2512.20483 |