Modular forms with non-vanishing central values and linear independence of Fourier coefficients
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910509128744960 |
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| author | Banerjee, Debargha Majumder, Priyanka |
| author_facet | Banerjee, Debargha Majumder, Priyanka |
| contents | In this article, we are interested in modular forms with non-vanishing central critical values and linear independence of Fourier coefficients of modular forms. The main ingredient is a generalization of a theorem due to VanderKam to modular symbols of higher weights. We prove that for sufficiently large primes $p$, Hecke operators $T_1, T_2, \ldots, T_D$ act linearly independently on the winding elements inside the space of weight $2k$ cuspidal modular symbol $\mathbb{S}_{2k}(Γ_0(p))$ with $k\geq 1$ for $D^2\ll p$. This gives a bound on the number of newforms with non-vanishing arithmetic $L$-functions at their central critical points and linear independence on the reductions of these modular forms for prime modulo $l\not=p$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_00900 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Modular forms with non-vanishing central values and linear independence of Fourier coefficients Banerjee, Debargha Majumder, Priyanka Number Theory In this article, we are interested in modular forms with non-vanishing central critical values and linear independence of Fourier coefficients of modular forms. The main ingredient is a generalization of a theorem due to VanderKam to modular symbols of higher weights. We prove that for sufficiently large primes $p$, Hecke operators $T_1, T_2, \ldots, T_D$ act linearly independently on the winding elements inside the space of weight $2k$ cuspidal modular symbol $\mathbb{S}_{2k}(Γ_0(p))$ with $k\geq 1$ for $D^2\ll p$. This gives a bound on the number of newforms with non-vanishing arithmetic $L$-functions at their central critical points and linear independence on the reductions of these modular forms for prime modulo $l\not=p$. |
| title | Modular forms with non-vanishing central values and linear independence of Fourier coefficients |
| topic | Number Theory |
| url | https://arxiv.org/abs/2307.00900 |