A note on logarithmic growth of solutions of $p$-adic differential equations without solvability

For a $p$-adic differential equation solvable in an open disc (in a $p$-adic sense), around 1970, Dwork proves that the solutions satisfy a certain growth condition on the boundary. Dwork also conjectures that a similar phenomenon should be observed without assuming the solvability. In this paper, we verify Dwork’s conjecture in the rank two case, which is the first non-trivial result on the conjecture. The proof is an application of Kedlaya’s decomposition theorem of $p$-adic differential equations defined over annulus.

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This work is supported by JSPS KAKENHI Grant-in-Aid for Young Scientists (B) JP17K14161.

Received 25 January 2018

Accepted 9 September 2018

Published 27 November 2019