Note on a theorem of Professor X

  • Between his arrival in Frankfurt in 1922 and and his proof of his famous finiteness theorem for integral points in 1929, Siegel had no publications. He did, however, write a letter to Mordell in 1926 in which he explained a proof of the finiteness of integral points on hyperelliptic curves. Recognizing the importance of this argument (and Siegel's views on publication), Mordell sent the relevant extract to be published under the pseudonym "X". The purpose of this note is to explain how to optimize Siegel's 1926 technique to obtain the following bound. Let K be a number field, S a finite set of places of K, and f∈oK,S[t] monic of degree d≥5 with discriminant Δf∈o×K,S. Then: #|{(x,y):x,y∈oK,S,y2=f(x)}|≤2rankJac(Cf)(K)⋅O(1)d3⋅([K:Q]+#|S|). This improves bounds of Evertse-Silverman and Bombieri-Gubler from 1986 and 2006, respectively. The main point underlying our improvement is that, informally speaking, we insist on "executing the descents in the presence of only one root (and not three) until the last possible moment".

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Metadaten
Author:Levent Alpöge
URN:urn:nbn:de:hebis:30:3-647354
URL:https://arxiv.org/abs/2109.14328v1
ArXiv Id:http://arxiv.org/abs/2109.14328v1
Parent Title (English):arXiv
Publisher:arXiv
Contributor(s):Carl Ludwig Siegel
Document Type:Preprint
Language:English
Date of Publication (online):2021/09/29
Date of first Publication:2021/09/29
Publishing Institution:Universitätsbibliothek Johann Christian Senckenberg
Release Date:2024/02/26
Issue:2109.14328 Version 1
Edition:Version 1
Page Number:6
HeBIS-PPN:516381016
Institutes:Informatik und Mathematik / Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Sammlungen:Universitätspublikationen
Licence (German):License LogoCreative Commons - Namensnennung 4.0