Geodesic
Local $\varepsilon$-conjecture and $p$-adic differential equations
Documenta mathematica, Tome 29 (2024) no. 5, pp. 1125-1156

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Laurent Berger attached a p-adic differential equation Nrig​(M) with a Frobenius structure to an arbitrary de Rham (φ,Γ)-module M over a Robba ring. In this article, we compare the local epsilon conjecture for the cyclotomic deformation of M with that of Nrig​(M). We first define an isomorphism between the fundamental lines of their cyclotomic deformations using the second author’s results on the big exponential map. As a main result of the article, we show that this isomorphism enables us to reduce the local epsilon conjecture for the cyclotomic deformation of M to that of Nrig​(M). The result can be regarded as a refined interpolation formula of the big exponential map.
DOI : 10.4171/dm/973
Classification : 11F80, 11F85, 11R23
Mots-clés : p-adic Hodge theory, (φ,Γ)-module, Bloch–Kato’s exponential, Perrin-Riou’s exponential 
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     author = {Tetsuya Ishida and Kentaro Nakamura},
     title = {Local $\varepsilon$-conjecture and $p$-adic differential equations},
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     doi = {10.4171/dm/973},
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Tetsuya Ishida; Kentaro Nakamura. Local $\varepsilon$-conjecture and $p$-adic differential equations. Documenta mathematica, Tome 29 (2024) no. 5, pp. 1125-1156. doi: 10.4171/dm/973

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