Isogenies of CM Elliptic Curves
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909533525245952 |
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| author | Assing, Edgar Li, Yingkun Wang, Tian Xia, Jiacheng |
| author_facet | Assing, Edgar Li, Yingkun Wang, Tian Xia, Jiacheng |
| contents | Given two CM elliptic curves over a number field and a natural number $m$, we establish a polynomial lower bound (in terms of $m$) for the number of rational primes $p$ such that the reductions of these elliptic curves modulo a prime above $p$ are $m$-isogenous. The proof relies on higher Green functions and theorems of Gross-Zagier and Gross-Kohnen-Zagier. A crucial observation is that the Fourier coefficients of incoherent Eisenstein series can be approximated by those of coherent Eisenstein series of increasing level. Another key ingredient is an explicit upper bound for the Petersson norm of an arbitrary elliptic modular form in terms of finitely many of its Fourier coefficients at the cusp infinity, which is a result of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_05685 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Isogenies of CM Elliptic Curves Assing, Edgar Li, Yingkun Wang, Tian Xia, Jiacheng Number Theory 11F30, 11G15, 14K02 Given two CM elliptic curves over a number field and a natural number $m$, we establish a polynomial lower bound (in terms of $m$) for the number of rational primes $p$ such that the reductions of these elliptic curves modulo a prime above $p$ are $m$-isogenous. The proof relies on higher Green functions and theorems of Gross-Zagier and Gross-Kohnen-Zagier. A crucial observation is that the Fourier coefficients of incoherent Eisenstein series can be approximated by those of coherent Eisenstein series of increasing level. Another key ingredient is an explicit upper bound for the Petersson norm of an arbitrary elliptic modular form in terms of finitely many of its Fourier coefficients at the cusp infinity, which is a result of independent interest. |
| title | Isogenies of CM Elliptic Curves |
| topic | Number Theory 11F30, 11G15, 14K02 |
| url | https://arxiv.org/abs/2503.05685 |