Towards Hilbert’s tenth problem for rings of integers through Iwasawa theory and Heegner points
نویسندگان
چکیده
منابع مشابه
Hilbert’s Tenth Problem over Rings of Integers of Number Fields
This is a term paper for Thomas Scanlon’s Math 229, Model Theory, at UC Berkeley, Fall 2006. The goal of this survey is to understand Bjorn Poonen’s theorem about elliptic curves and Hilbert’s Tenth problem over rings of integers of number fields and to record and exposit a few ideas which may be useful in an attack with Poonen’s theorem as a starting point.
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Let E be an elliptic curve over Q and let p ≥ 5 be a prime of good supersingular reduction for E. Let K be an imaginary quadratic field satisfying a modified “Heegner hypothesis” in which p splits, write K∞ for the anticyclotomic Zp-extension of K and let Λ denote the Iwasawa algebra of K∞/K. By extending to the supersingular case the Λ-adic Kolyvagin method originally developed by Bertolini in...
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Our aim in this paper is to present a framework in which the results of Waldspurger and Gross–Zagier can be viewed simultaneously. This framework may also be useful in understanding recent work of Zhang, Xue, Cornut, Vatsal, and Darmon. It involves a blending of techniques from representation theory and automorphic forms with those from the arithmetic of modular curves. I hope readers from one ...
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Let F ⊆ K be number fields, and let OF and OK be their rings of integers. If there exists an elliptic curve E over F such that rkE(F ) = rkE(K) = 1, then there exists a diophantine definition of OF over OK .
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We prove that if p ≡ 4, 7 (mod 9) is prime and 3 is not a cube modulo p, then both of the equations x + y = p and x + y = p have a solution with x, y ∈ Q.
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ژورنال
عنوان ژورنال: Mathematische Annalen
سال: 2020
ISSN: 0025-5831,1432-1807
DOI: 10.1007/s00208-020-01991-w