Dold-Gauss Congruences, Norm Descent, and Rational Rigidity
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| Format: | Preprint |
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2025
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| _version_ | 1866915522366406656 |
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| author | Bal, Hartosh Singh |
| author_facet | Bal, Hartosh Singh |
| contents | We develop a Witt--Hadamard calculus for Euler products that unifies the classical Gauss congruences with their modern refinement, the Dold congruences. Within this framework we prove \emph{norm descent}: Dold congruences are functorial under finite extensions and preserved by prime--ideal norms $N_{K/\mathbb{Q}}$, yielding integer ghosts from algebraic ones. We extend the theory from $\mathbb{Z}$ to Dedekind domains, and show that integrality is stable under both Hadamard and Witt products. Two rigidity theorems lie at the core: a \emph{cyclotomic residues theorem}, asserting that if the logarithmic derivative has only cyclotomic poles then integrality forces rationality; and a stronger \emph{Dold$^{+}$ rigidity theorem}, showing that any algebraic series satisfying refined Dold congruences is necessarily rational. These results sharpen the Gauss--Dold picture: ordinary congruences enforce integrality, while the strengthened form collapses algebraic cases to rational ones. Applications include prime--ideal ladders in number fields and exact product laws for dynamical zeta functions, illustrated for subshifts of finite type and circle doubling. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_25038 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dold-Gauss Congruences, Norm Descent, and Rational Rigidity Bal, Hartosh Singh Number Theory Combinatorics Primary 11B37, 11A07, 37C25, Secondary 05A17, 13F35, 11R18 We develop a Witt--Hadamard calculus for Euler products that unifies the classical Gauss congruences with their modern refinement, the Dold congruences. Within this framework we prove \emph{norm descent}: Dold congruences are functorial under finite extensions and preserved by prime--ideal norms $N_{K/\mathbb{Q}}$, yielding integer ghosts from algebraic ones. We extend the theory from $\mathbb{Z}$ to Dedekind domains, and show that integrality is stable under both Hadamard and Witt products. Two rigidity theorems lie at the core: a \emph{cyclotomic residues theorem}, asserting that if the logarithmic derivative has only cyclotomic poles then integrality forces rationality; and a stronger \emph{Dold$^{+}$ rigidity theorem}, showing that any algebraic series satisfying refined Dold congruences is necessarily rational. These results sharpen the Gauss--Dold picture: ordinary congruences enforce integrality, while the strengthened form collapses algebraic cases to rational ones. Applications include prime--ideal ladders in number fields and exact product laws for dynamical zeta functions, illustrated for subshifts of finite type and circle doubling. |
| title | Dold-Gauss Congruences, Norm Descent, and Rational Rigidity |
| topic | Number Theory Combinatorics Primary 11B37, 11A07, 37C25, Secondary 05A17, 13F35, 11R18 |
| url | https://arxiv.org/abs/2509.25038 |