Non-vanishing for quartic Hecke $L$-functions and ranks of elliptic curves
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866911560776024064 |
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| author | Castillo, Cruz de Faveri, Alexandre Dunn, Alexander |
| author_facet | Castillo, Cruz de Faveri, Alexandre Dunn, Alexander |
| contents | We show that a positive proportion of Hecke $L$-functions attached to the quartic residue symbols $\big( \frac{\cdot}{q} \big)_4$ for squarefree $q \in \mathbb{Z}[i]$ do not vanish at the central point. Our method also extends to the Hecke characters associated to quartic twists of the congruent number curve $E : y^2 = x^3 - x$. In particular, we prove that the elliptic curve $E^{(q)} : y^2 = x^3 - qx$ has Mordell-Weil rank $0$ over $\mathbb{Q}(i)$ for a positive proportion of squarefree $q \in \mathbb{Z}[i]$ ordered by norm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_01316 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Non-vanishing for quartic Hecke $L$-functions and ranks of elliptic curves Castillo, Cruz de Faveri, Alexandre Dunn, Alexander Number Theory Algebraic Geometry 11R42 (Primary) 11L05, 11G40, 11F30, 11R11, 11G05 (Secondary) We show that a positive proportion of Hecke $L$-functions attached to the quartic residue symbols $\big( \frac{\cdot}{q} \big)_4$ for squarefree $q \in \mathbb{Z}[i]$ do not vanish at the central point. Our method also extends to the Hecke characters associated to quartic twists of the congruent number curve $E : y^2 = x^3 - x$. In particular, we prove that the elliptic curve $E^{(q)} : y^2 = x^3 - qx$ has Mordell-Weil rank $0$ over $\mathbb{Q}(i)$ for a positive proportion of squarefree $q \in \mathbb{Z}[i]$ ordered by norm. |
| title | Non-vanishing for quartic Hecke $L$-functions and ranks of elliptic curves |
| topic | Number Theory Algebraic Geometry 11R42 (Primary) 11L05, 11G40, 11F30, 11R11, 11G05 (Secondary) |
| url | https://arxiv.org/abs/2604.01316 |