Arithmetic & Homotopic Galois Theory IRN | On the integral structure of Galois Lie algebras

October 5, 2026	On the integral structure of Galois Lie algebras Ishii Shun, RIMS Kyoto University, JP RIMS Kyoto (Building 15 Room 201) + Zoom · JP: 15:30 · FR: 08:30 The lower central series of the pro-\(p\) fundamental group of a once-punctured CM elliptic curve induces, via the outer Galois action, the so-called weight filtration on the absolute Galois group of the base field. We call the associated graded Lie algebra the Galois Lie algebra. In this talk, we study the integral structure of this Lie algebra. After reviewing earlier results, we present recent computations aimed at detecting elements of order \(p\) in its abelianization, focusing on \(p=73\) for elliptic curves with complex multiplication by \(\mathbb{Q}(\sqrt{-1})\) and \(p=277\) for those with complex multiplication by \(\mathbb{Q}(\sqrt{-3})\). Our approach is modelled on earlier work of McCallum and Sharifi and combines congruences among certain derivations of a free Lie algebra on two generators with calculations of cup products in Galois cohomology.
S. Ishii, On the kernels of the pro-p outer Galois representations associated to once-punctured CM elliptic curves, preprint (2023), [arXiv]. W. G. McCallum and R. T. Sharifi, A cup product in the Galois cohomology of number fields, Duke Math. J. 120(2) (2003), 269-310.
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October 5, 2026

On the integral structure of Galois Lie algebras

Ishii Shun, RIMS Kyoto University, JP RIMS Kyoto (Building 15 Room 201) + Zoom · JP: 15:30 · FR: 08:30

The lower central series of the pro-\(p\) fundamental group of a once-punctured CM elliptic curve induces, via the outer Galois action, the so-called weight filtration on the absolute Galois group of the base field. We call the associated graded Lie algebra the Galois Lie algebra.

In this talk, we study the integral structure of this Lie algebra. After reviewing earlier results, we present recent computations aimed at detecting elements of order \(p\) in its abelianization, focusing on \(p=73\) for elliptic curves with complex multiplication by \(\mathbb{Q}(\sqrt{-1})\) and \(p=277\) for those with complex multiplication by \(\mathbb{Q}(\sqrt{-3})\). Our approach is modelled on earlier work of McCallum and Sharifi and combines congruences among certain derivations of a free Lie algebra on two generators with calculations of cup products in Galois cohomology.

S. Ishii, On the kernels of the pro-p outer Galois representations associated to once-punctured CM elliptic curves, preprint (2023), [arXiv].
W. G. McCallum and R. T. Sharifi, A cup product in the Galois cohomology of number fields, Duke Math. J. 120(2) (2003), 269-310.