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

Garcia-Fritz, Natalia; Pasten, Hector

doi:10.1007/s00208-020-01991-w

Cited by 13 publications

(10 citation statements)

References 66 publications

(91 reference statements)

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“…considering polynomials of arbitrary degree in the input). The latter is widely believed to be unsolvable (for unconditional results, see [Den75, DL78, Ph88, Sh89, V89, SS89] and more recently [GFP20]; conditional results have been obtained for every K -[Po02, CPhZ05, Sh08, MR10, MR18]).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Existential decidability for addition and divisibility in holomorphy subrings of global fields

Martinez-Ranero¹,

Utreras²,

Vidaux³

2020

Preprint

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Existential decidability for addition and divisibility in holomorphy subrings of global fields

Martinez-Ranero¹,

Utreras²,

Vidaux³

2020

Preprint

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“…The general case remains open, although it is known that this would follow from standard conjectures on elliptic curves [29, 33]. The available unconditional results are proved in [18, 19, 37, 45, 46, 49] and most recently in [23, 30]. All recent progress on this problem was possible thanks to the elliptic curve criteria from [9, 39, 48].…”

Section: The Dprm Property For Listable Structuresmentioning

confidence: 99%

Notes on the DPRM Property for Listable Structures

Pasten¹

2021

J. symb. log.

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A celebrated result by Davis, Putnam, Robinson, and Matiyasevich shows that a set of integers is listable if and only if it is positive existentially definable in the language of arithmetic. We investigate analogues of this result over structures endowed with a listable presentation. When such an analogue holds, the structure is said to have the DPRM property. We prove several results addressing foundational aspects around this problem, such as uniqueness of the listable presentation, transference of the DPRM property under interpretation, and its relation with positive existential bi-interpretability. A first application of our results is the rigorous proof of (strong versions of) several folklore facts regarding transference of the DPRM property. Another application of the theory we develop is that it will allow us to link various Diophantine conjectures to the question of whether the DPRM property holds for global fields. This last topic includes a study of the number of existential quantifiers needed to define a Diophantine set.

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“…The general case remains open, although it is known that this would follow from standard conjectures on elliptic curves [28,32]. The available unconditional results are proved in [18,17,36,44,47,43] and most recently in [29] and [22]. All recent progress on this problem was possible thanks to the elliptic curve criteria from [38,8,46].…”

Section: 5mentioning

confidence: 99%

Notes on the DPRM property for listable structures

Pasten¹

2020

Preprint

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The celebrated DPRM theorem by M. Davis, H. Putnam, J. Robinson, and Y. Matiyasevich shows that a set of integers is listable (also called recursively enumerable, or computably enumerable) if and only if it is Diophantine, i.e., positive existentially definable over the language of arithmetic. We present some foundational material to investigate analogues of the DPRM theorem over other structures. This allows us to give rigorous proofs of various folklore results in the literature. In addition, we investigate different conjectures that point in the direction that the analogue of the DPRM theorem for global fields does not hold.

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

Cited by 13 publications

References 66 publications

Existential decidability for addition and divisibility in holomorphy subrings of global fields

Existential decidability for addition and divisibility in holomorphy subrings of global fields

Notes on the DPRM Property for Listable Structures

Notes on the DPRM property for listable structures

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