An Equivariant Main Conjecture for Branched ℤ_p-Towers of Finite Graphs and Applications

Abstract

Let p be a prime and let Y/X be a finite abelian branched cover of finite connected graphs. We consider a compatible branched ℤ_p-tower above Y whose total Galois group over X is abelian. We construct an equivariant Picard Iwasawa module over the completed group ring of the total Galois group and give an explicit presentation of this module. On the analytic side, we construct an equivariant Artin--Ihara L-element at the top of the tower and isolate a modified factor whose value at u=1 is well defined. Our main result computes the Fitting ideal of the equivariant Picard Iwasawa module in terms of these special values and augmentation ideals determined by the inertia groups. As applications, we recover the main conjecture for branched ℤ_p-towers from our earlier work and obtain finite-level Fitting-ideal and annihilation formulas which can be viewed as graph-theoretic analogues of the conjectures of Kurihara and Brumer--Stark.

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BibTeX RIS

BibTeX

@misc{gambheera2026equivariant,
  title = {{An Equivariant Main Conjecture for Branched $\mathbb Z_p$-Towers of Finite Graphs and Applications}},
  author = {Rusiru Gambheera},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.27551},
  eprint = {2610.27551},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.27551},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - An Equivariant Main Conjecture for Branched ℤ_p-Towers of Finite Graphs and Applications
AU  - Rusiru Gambheera
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.27551
UR  - https://arxiv.org/abs/2610.27551
AB  - Let p be a prime and let Y/X be a finite abelian branched cover of finite connected graphs. We consider a compatible branched ℤ_p-tower above Y whose total Galois group over X is abelian. We construct an equivariant Picard Iwasawa module over the completed group ring of the total Galois group and give an explicit presentation of this module. On the analytic side, we construct an equivariant Artin--Ihara L-element at the top of the tower and isolate a modified factor whose value at u=1 is well defined. Our main result computes the Fitting ideal of the equivariant Picard Iwasawa module in terms of these special values and augmentation ideals determined by the inertia groups. As applications, we recover the main conjecture for branched ℤ_p-towers from our earlier work and obtain finite-level Fitting-ideal and annihilation formulas which can be viewed as graph-theoretic analogues of the conjectures of Kurihara and Brumer--Stark.
ER  -

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