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Towards Hilbert’s tenth problem for rings of integers through Iwasawa theory and Heegner points

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Abstract

For a positive proportion of primes p and q, we prove that \({\mathbb {Z}}\) is Diophantine in the ring of integers of \({\mathbb {Q}}(\root 3 \of {p},\sqrt{-q})\). This provides a new and explicit infinite family of number fields K such that Hilbert’s tenth problem for \(O_K\) is unsolvable. Our methods use Iwasawa theory and congruences of Heegner points in order to obtain suitable rank stability properties for elliptic curves.

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  1. A pseudo isomorphism is a morphism with finite kernel and cokernel.

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Acknowledgements

Natalia Garcia-Fritz was supported by the FONDECYT Iniciación en Investigación Grant 11170192, and the CONICYT PAI Grant 79170039. Hector Pasten was supported by FONDECYT Regular Grant 1190442. We gratefully acknowledge the hospitality of the IMJ-PRG in Paris during our visit in February 2019, where the final ideas of the project were developed. We heartily thank Loïc Merel for this invitation and for several enlightening discussions on elliptic curves. The final technical details of this work were worked out and presented in May 2019 at the AIM meeting “Definability and Decidability problems in Number Theory”. We thank the AIM for their hospitality and the participants for their valuable feedback. Specially, we thank Karl Rubin and Alexandra Shlapentokh for their interest and technical remarks. Comments by Matias Alvarado, Chao Li, and Barry Mazur on an earlier version of this manuscript are gratefully acknowledged. We also thank the anonymous referee for several valuable suggestions.

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  1. Departamento de Matemáticas, Facultad de Matemáticas, Pontificia Universidad Católica de Chile, 4860 Av. Vicuña Mackenna, Macul, RM, Chile

    Natalia Garcia-Fritz & Hector Pasten

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  1. Natalia Garcia-Fritz
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  2. Hector Pasten
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Correspondence to Natalia Garcia-Fritz.

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Communicated by Wei Zhang.

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N. G.-F. was supported by the FONDECYT Iniciación en Investigación Grant 11170192, and the CONICYT PAI Grant 79170039. H.P. was supported by FONDECYT Regular Grant 1190442.

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Garcia-Fritz, N., Pasten, H. Towards Hilbert’s tenth problem for rings of integers through Iwasawa theory and Heegner points. Math. Ann. 377, 989–1013 (2020). https://doi.org/10.1007/s00208-020-01991-w

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