Primes represented by quadratic forms and the Weil abscissa of abelian profinite groups
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866911437847265280 |
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| author | Jann, Martin Kionke, Steffen |
| author_facet | Jann, Martin Kionke, Steffen |
| contents | Here we show that the Weil abscissa of the procyclic groups $\prod_{p \in S} \mathbb{Z}_p$ equals $2$ for three sets $S$: (i) the set of primes $p \equiv 1 \bmod 3$, (ii) the set of primes $p \equiv 1 \bmod 4$ and (iii) the set of primes $p \equiv 1,3 \bmod 8$. Our argument is based on the observation that integers all of whose prime factors lie in $S$ can be represented by a suitable binary quadratic form, which allows us to use a theorem of Iwaniec to exhibit a minorant for the Weil representation zeta function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_09797 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Primes represented by quadratic forms and the Weil abscissa of abelian profinite groups Jann, Martin Kionke, Steffen Number Theory Group Theory Here we show that the Weil abscissa of the procyclic groups $\prod_{p \in S} \mathbb{Z}_p$ equals $2$ for three sets $S$: (i) the set of primes $p \equiv 1 \bmod 3$, (ii) the set of primes $p \equiv 1 \bmod 4$ and (iii) the set of primes $p \equiv 1,3 \bmod 8$. Our argument is based on the observation that integers all of whose prime factors lie in $S$ can be represented by a suitable binary quadratic form, which allows us to use a theorem of Iwaniec to exhibit a minorant for the Weil representation zeta function. |
| title | Primes represented by quadratic forms and the Weil abscissa of abelian profinite groups |
| topic | Number Theory Group Theory |
| url | https://arxiv.org/abs/2602.09797 |