Sifting for small split primes of an imaginary quadratic field in a given ideal class
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909279158534144 |
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| author | Gaudet, Louis M. |
| author_facet | Gaudet, Louis M. |
| contents | Let $D>3$, $D\equiv3\;(4)$ be a prime, and let $\mathcal{C}$ be an ideal class in the field $\mathbb{Q}(\sqrt{-D})$. In this article, we give a new proof that $p(D,\mathcal{C})$, the smallest norm of a split prime $\mathfrak{p}\in\mathcal{C}$, satisfies $p(D,\mathcal{C})\ll D^L$ for some absolute constant $L$. Our proof is sieve theoretic. In particular, this allows us to avoid the use of log-free zero-density estimates (for class group $L$-functions) and the repulsion properties of exceptional zeros, two crucial inputs to previous proofs of this result. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_01610 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sifting for small split primes of an imaginary quadratic field in a given ideal class Gaudet, Louis M. Number Theory Let $D>3$, $D\equiv3\;(4)$ be a prime, and let $\mathcal{C}$ be an ideal class in the field $\mathbb{Q}(\sqrt{-D})$. In this article, we give a new proof that $p(D,\mathcal{C})$, the smallest norm of a split prime $\mathfrak{p}\in\mathcal{C}$, satisfies $p(D,\mathcal{C})\ll D^L$ for some absolute constant $L$. Our proof is sieve theoretic. In particular, this allows us to avoid the use of log-free zero-density estimates (for class group $L$-functions) and the repulsion properties of exceptional zeros, two crucial inputs to previous proofs of this result. |
| title | Sifting for small split primes of an imaginary quadratic field in a given ideal class |
| topic | Number Theory |
| url | https://arxiv.org/abs/2408.01610 |