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Let p ∈ Z p\in {\mathbb {Z}} be an odd prime. We show that the fiber sequence for the cyclotomic trace of the sphere spectrum S {\mathbb {S}} admits an “eigensplitting” that generalizes known splittings on K K -theory and T C TC . We identify the summands in the fiber as the covers of Z p {\mathbb {Z}}_{p} -Anderson duals of summands in the K ( 1 ) K(1) -localized algebraic K K -theory of Z {\mathbb {Z}} . Analogous results hold for the ring Z {\mathbb {Z}} where we prove that the K ( 1 ) K(1) -localized fiber sequence is self-dual for Z p {\mathbb {Z}}_{p} -Anderson duality, with the duality permuting the summands by i ↦ p − i i\mapsto p-i (indexed mod p − 1 p-1 ). We explain an intrinsic characterization of the summand we call Z Z in the splitting T C ( Z ) p ∧ ≃ j ∨ Σ j ′ ∨ Z TC({\mathbb {Z}})^{\wedge }_{p}\simeq j \vee \Sigma j’\vee Z in terms of units in the p p -cyclotomic tower of Q p {\mathbb {Q}}_{p} .
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This content will become publicly available on April 1, 2026
Base Change and Iwasawa Main Conjectures for GL2
Abstract Let $$E$$ be an elliptic curve defined over $${\mathbb{Q}}$$ of conductor $$N$$, $$p$$ an odd prime of good ordinary reduction such that $E[p]$ is an irreducible Galois module, and $$K$$ an imaginary quadratic field with all primes dividing $Np$ split. We prove Iwasawa main conjectures for the $${\mathbb{Z}}_{p}$$-cyclotomic and $${\mathbb{Z}}_{p}$$-anticyclotomic deformations of $$E$$ over $${\mathbb{Q}}$$ and $K,$ respectively, dispensing with any of the ramification hypotheses on $E[p]$ in previous works. The strategy employs base change and the two-variable zeta element associated to $$E$$ over $$K$$, via which the sought after main conjectures are deduced from Wan’s divisibility towards a three-variable main conjecture for $$E$$ over a quartic CM field containing $$K$$ and certain Euler system divisibilities. As an application, we prove cases of the two-variable main conjecture for $$E$$ over $$K$$. The aforementioned one-variable main conjectures imply the $$p$$-part of the conjectural Birch and Swinnerton-Dyer formula for $$E$$ if $$\operatorname{ord}_{s=1}L(E,s)\leq 1$$. They are also an ingredient in the proof of Kolyvagin’s conjecture and its cyclotomic variant in our joint work with Grossi [1].
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- PAR ID:
- 10608317
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2025
- Issue:
- 8
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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