Zeta elements for p-ordinary modular forms over imaginary quadratic fields and applications
Abstract
Let p≥ 5 be a prime, and let f be a p-ordinary cuspidal newform of even weight k = 2r ≥ 2 and level N, with p nmid N and trivial nebentype. In this article, we prove Kato's Iwasawa main conjecture for f, formulated in terms of the p-adic L-function of f, under the sole assumptions that the associated Galois representation of f satisfies the big-image hypothesis and that its residual representation is absolutely irreducible. This generalises the main result of Burungale--Castella--Skinner, which treats elliptic curves over ℚ with good ordinary reduction at p. Our proof follows their general strategy. A key new ingredient is the construction, over an imaginary quadratic field in which p splits, of a zeta element attached to f, together with a proof of its explicit reciprocity laws, generalising the corresponding work of Burungale--Skinner--Tian--Wan. These results provide the higher-weight analogue of a crucial input in the argument of Burungale--Castella--Skinner and allow their method to be carried out for higher-weight modular forms.
Related
- Imaginary elements and crystals for quantum groups
- Maximal unramified 3-extensions of imaginary quadratic fields and \textrm SL\sb 2(\mathbb Z\sb 3)
- On the Iwasawa invariants of admissible quadratic twists of elliptic curves at ordinary Eisenstein primes
- Ternary Perfect Forms Over Imaginary Quadratic Field
- p-class groups in the cyclotomic ℤp-extension of imaginary quadratic fields
BibTeX
@misc{xu2026zeta,
title = {{Zeta elements for $p$-ordinary modular forms over imaginary quadratic fields and applications}},
author = {Ruichen Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38558},
eprint = {2609.38558},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38558},
}
RIS
TY - UNPB
TI - Zeta elements for p-ordinary modular forms over imaginary quadratic fields and applications
AU - Ruichen Xu
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.38558
UR - https://arxiv.org/abs/2609.38558
AB - Let p≥ 5 be a prime, and let f be a p-ordinary cuspidal newform of even weight k = 2r ≥ 2 and level N, with p nmid N and trivial nebentype. In this article, we prove Kato's Iwasawa main conjecture for f, formulated in terms of the p-adic L-function of f, under the sole assumptions that the associated Galois representation of f satisfies the big-image hypothesis and that its residual representation is absolutely irreducible. This generalises the main result of Burungale--Castella--Skinner, which treats elliptic curves over ℚ with good ordinary reduction at p. Our proof follows their general strategy. A key new ingredient is the construction, over an imaginary quadratic field in which p splits, of a zeta element attached to f, together with a proof of its explicit reciprocity laws, generalising the corresponding work of Burungale--Skinner--Tian--Wan. These results provide the higher-weight analogue of a crucial input in the argument of Burungale--Castella--Skinner and allow their method to be carried out for higher-weight modular forms.
ER -