$\mathcal L$-invariants and deep congruences between newforms
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arXiv
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| Format: | Preprint |
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2026
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| author | Conti, Andrea Gräf, Peter Mathias |
| author_facet | Conti, Andrea Gräf, Peter Mathias |
| contents | We study congruences modulo powers of a prime $p$ between pairs of $p$-new modular Hecke eigenforms of level $Γ_0(p)$ and same weight $k$. Based on explicit computations, we conjecture that every such eigenform $f$ admits a twin to which it is congruent modulo a surprisingly high power of $p$, whose exponent is close to the opposite of the valuation of the $\mathcal L$-invariant of $f$, and whose Atkin--Lehner sign is opposite to that of $f$. This is a new phenomenon that is not explained by the known results on the $p$-adic variation of eigenforms. Inspired by the global picture, we formulate a local conjecture describing congruences between semistable representations of fixed weight, varying $\mathcal L$-invariant, and opposite Atkin--Lehner signs. We give some theoretical evidence towards our conjectures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_15211 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $\mathcal L$-invariants and deep congruences between newforms Conti, Andrea Gräf, Peter Mathias Number Theory 11F33, 11F80, 11F85 We study congruences modulo powers of a prime $p$ between pairs of $p$-new modular Hecke eigenforms of level $Γ_0(p)$ and same weight $k$. Based on explicit computations, we conjecture that every such eigenform $f$ admits a twin to which it is congruent modulo a surprisingly high power of $p$, whose exponent is close to the opposite of the valuation of the $\mathcal L$-invariant of $f$, and whose Atkin--Lehner sign is opposite to that of $f$. This is a new phenomenon that is not explained by the known results on the $p$-adic variation of eigenforms. Inspired by the global picture, we formulate a local conjecture describing congruences between semistable representations of fixed weight, varying $\mathcal L$-invariant, and opposite Atkin--Lehner signs. We give some theoretical evidence towards our conjectures. |
| title | $\mathcal L$-invariants and deep congruences between newforms |
| topic | Number Theory 11F33, 11F80, 11F85 |
| url | https://arxiv.org/abs/2602.15211 |