On non-congruent numbers with 1 modulo 4 prime factors
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Abstract
In this paper, we use the 2-descent method to find a series of odd non-congruent numbers ≡ 1 (mod 8) whose prime factors are ≡ 1 (mod 4) such that the congruent elliptic curves have second lowest Selmer groups, which include Li and Tian’s result as special cases.
Keywords
non-congruent number 2-descent second 2-descentMSC(2010)
11G05 11D25Preview
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References
- 1.Birch B, Swinnerton-Dyer H P F. Notes on ellptic curves (II). J Reine Angrew Math, 1965, 218: 79–108MATHMathSciNetGoogle Scholar
- 2.Cassels J W S. Arithmetic on curves of genus 1, (IV) proof of the Hauptvermutung. J Reine Angrew Math, 1962, 211: 95–112MATHMathSciNetGoogle Scholar
- 3.Feng K. Non-congruent Numbers and Elliptic Curves with Rank Zero. Hefei: Press of University of Science and Technology of China, 2008, 25–29Google Scholar
- 4.Feng K. Non-congruent number, odd graphs and the BSD conjecture. Acta Arith, 1996, 80Google Scholar
- 5.Li D, Tian Y. On the Birch-Swinnerton-Dyer conjecture of elliptic curves ED: y 2 = x 3 − D 2 x. Acta Math Sinica, 2000, 16: 229–236CrossRefMATHMathSciNetGoogle Scholar
- 6.Serre J P. A Course in Arithmetic. Berlin: Springer-Verlag, 1973CrossRefMATHGoogle Scholar
- 7.Silverman J H. The Arithmetic of Elliptic Curves. GTM 106. New York: Springer-Verlag, 1986CrossRefGoogle Scholar
- 8.Xiong M, Zaharescu A. Selmer groups and Tate-Shararevich groups for the congruent number problem. Comment Math Helv, 2009, 84: 21–56CrossRefMATHMathSciNetGoogle Scholar
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