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ACD Fotocanvas Print Editorial Board Supported by NSFC Honorary Editor General ZHOU GuangZhao (Zhou Guang Zhao) Editor General ZHU ZuoYan Institute of Hydrobiology, CAS, China Editor-in-Chief YUAN YaXiang Academy of Mathematics and Systems Science, CAS, China Associate Editors-in-Chief CHEN YongChuan Tianjin University, China GE LiMing Academy of Mathematics and Systems Science, CAS, China SHAO QiMan The Chinese University of Hong Kong, China XI NanHua Academy of Mathematics and Systems Science, CAS, China ZHANG WeiPing Nankai University, China Members BAI ZhaoJun LI JiaYu WANG YueFei University of California, Davis, USA University of Science and Technology Academy of Mathematics and Systems of China, China Science, CAS, China CAO DaoMin Academy of Mathematics and Systems LIN FangHua WU SiJue Science, CAS, China New York University, USA University of Michigan, USA CHEN XiaoJun LIU JianYa WU SiYe The Hong Kong Polytechnic University, Shandong University, China The University of Hong Kong, China China LIU KeFeng XIAO Jie University of California, Los Angeles, USA Tsinghua University, China CHEN ZhenQing Zhejiang University, China University of Washington, USA XIN ZhouPing LIU XiaoBo The Chinese University of Hong Kong, CHEN ZhiMing Peking University, China China Academy of Mathematics and Systems University of Notre Dame, USA Science, CAS, China XU Fei MA XiaoNan Capital Normal University, China CHENG ChongQing University of Denis Diderot-Paris 7, Nanjing University, China France XU Feng University of California, Riverside, USA DAI YuHong MA ZhiMing Academy of Mathematics and Systems XU JinChao Academy of Mathematics and Systems Pennsylvania State University, USA Science, CAS, China Science, CAS, China MOK NgaiMing XU XiaoPing DONG ChongYing The University of Hong Kong, China Academy of Mathematics and Systems University of California, Santa Cruz, USA PUIG Lluis Science, CAS, China CNRS, Institute of Mathematics of Jussieu, YAN Catherine H. F. DUAN HaiBao France Academy of Mathematics and Systems Texas A&M University, USA Science, CAS, China QIN HouRong YANG DaChun Nanjing University, China E WeiNan Beijing Normal University, China Princeton University, USA RINGEL Claus M. University of Bielefeld, Germany YE XiangDong Peking University, China University of Science and Technology FAN JianQing SHANG ZaiJiu of China, China Princeton University, USA Academy of Mathematics and Systems Science, CAS, China YU XingXing Georgia Institute of Technology, USA FENG Qi SHEN ZhongMin Academy of Mathematics and Systems Indiana University-Purdue University ZHANG James J. Science, CAS, China Indianapolis, USA University of Washington, USA FU JiXiang SHU Chi-Wang ZHANG JiPing Fudan University, China Brown University, USA Peking University, China GAO XiaoShan SIU Yum-Tong ZHANG Ping Academy of Mathematics and Systems Harvard University, USA Academy of Mathematics and Systems Science, CAS, China SUN LiuQuan Science, CAS, China GE GenNian Academy of Mathematics and Systems ZHANG PingWen Capital Normal University, China Science, CAS, China Peking University, China GUO XianPing SUN XiaoTao ZHANG ShouWu Sun Yat-sen University, China Academy of Mathematics and Systems Columbia University, USA Science, CAS, China HE XuMing ZHANG Xu University of Michigan, USA TAN Lei Sichuan University, China University of Angers, France HONG JiaXing ZHANG YiTang Fudan University, China TANG ZiZhou University of New Hampshire, USA Beijing Normal University, China HSU Elton P. ZHOU XiangYu Northwestern University, USA TEBOULLE Marc Academy of Mathematics and Systems Tel Aviv University, Israel JI LiZhen Science, CAS, China University of Michigan, USA WANG FengYu ZHU XiPing Beijing Normal University, China Sun Yat-sen University, China JING Bing-Yi The Hong Kong University of Science WANG HanSheng ZONG ChuanMing and Technology, China Peking University, China Peking University, China ZHANG RuiYan E ditorial Staff CHAI Zhao YANG ZhiHua [email protected] [email protected] [email protected] Cover Designer HU Yu [email protected] SCIENCE CHINA Mathematics . ARTICLES . March 2014 Vol. 57 No. 3: 649–658 doi: 10.1007/s11425-013-4705-y On non-congruent numbers with 1 modulo 4 prime factors OUYANG Yi & ZHANG ShenXing∗ Wu Wen-Tsun Key Laboratory of Mathematics, School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China Email: [email protected], [email protected] Received September 11, 2012; accepted December 28 2012; published online September 4, 2013 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-descent MSC(2010) 11G05, 11D25 Citation: Ouyang Y, Zhang S X. On non-congruent numbers with 1 modulo 4 prime factors. Sci China Math, 2014, 57: 649–658, doi: 10.1007/s11425-013-4705-y 1 Introduction The congruent number problem is about when a positive integer can be the area of a rational right triangle. A positive integer n is a non-congruent number if and only if the congruent elliptic curve E := E(n) : y2 = x3 n2x (1.1) − has Mordell-Weil rank zero. In [3,4], Feng obtained several series of non-congruent numbers for E(n) with the lowest Selmer groups. In [5], Li and Tian obtained a series of non-congruent numbers whose prime factors are 1 (mod 8) such that E(n) has second lowest Selmer groups. The essential tool of the above ≡ results is the 2-descent method of elliptic curves. In this paper, we will use this method to get a series of odd non-congruent numbers whose prime factors are 1 (mod 4) such that E(n) has second lowest ≡ Selmer groups, which include Li and Tian’s result as special cases. Suppose n is a square-free integer such that n = p p 1 (mod 8) and primes p 1 (mod 4), 1 ··· k ≡ i ≡ then by quadratic reciprocity law ( pi ) = ( pj ). pj pi Definition 1.1. Suppose n = p p 1 (mod 8) and p 1 (mod 4). The graph G(n) := (V, A) 1 ··· k ≡ i ≡ associated with n is a simple undirected graph with vertex set V := prime p n and edge set A := { | } pq : ( p )= 1 . { q − } Recall for a simple undirected graph G = (V, A), a partition V = V V is called even if for any v V 0 ∪ 1 ∈ i (i =0, 1), # v V1 i is even. G is called an odd graph if the only even partition is the trivial partition { → − } V = V . Then our main result is: ∅ ∪ ∗Corresponding author c Science China Press and Springer-Verlag Berlin Heidelberg 2013 math.scichina.com link.springer.com 650 Ouyang Y et al. Sci China Math March 2014 Vol. 57 No. 3 Theorem 1.2. Suppose n = p p 1 (mod 8) and p 1 (mod 4). If the graph G(n) is odd and 1 ··· k ≡ i ≡ δ(n) (as will be given by (4.5)) is 1, then for the congruent elliptic curve E = E(n), X 2 rankZ(E(Q))=0 and (E/Q)[2∞] ∼= (Z/2Z) . As a consequence, n is a non-congruent number. The following corollary is Li and Tian’s result [5]: Corollary 1.3. Suppose n = p1 pk and pi 1 (mod 8). If the graph G(n) is odd and the Jacobi 1+√ 1 ··· (n) ≡ symbol ( − )= 1, then for E = E , n − X 2 rankZ(E(Q))=0 and (E/Q)[2∞] ∼= (Z/2Z) . As a consequence, n is a non-congruent number. 2 Review of 2-descent method In this section, we recall the 2-descent method of computing the Selmer groups of elliptic curves. This section follows [5, pp. 232–233], also cf. [1, Section 5] and [7, Chapter X. 4]. For an isogeny ϕ : E E′ of elliptic curves defined over a number field K, one has the following → fundamental exact sequence: (ϕ) 0 E′(K)/ϕE(K) S (E/K) X(E/K)[ϕ] 0. (2.1) → → → → Moreover, if ψ : E′ E is another isogeny, for the composition ψ ϕ : E E, then the following → ◦ → diagram of exact sequences commutes (cf. [8, p.5]): 0 0 0 ι1 ι2 / / (ϕ) / / 0 E′(K)/ϕE(K) S (E/K) X(E/K)[ϕ] 0 ψ / / / / 0 / E(K)/ψϕE(K) / S(ψϕ)(E/K) / X(E/K)[ψϕ] / 0 / / (ψ) / X / 0 E(K)/ψE′(K) S (E′/K) (E′/K)[ψ] 0. 0 Now suppose n is a fixed odd positive square-free integer, K = Q, and E/Q, E′/Q, ϕ, ψ = ϕ∨ are given by (n) 2 3 2 (n) 2 3 2 E = E : y = x n x, E′ = E : y = x +4n x, − y2 y(x2 + n2) ϕ : E E′, (x, y) , d , → 7→ x2 x2 y2 y(x2 4n2) ψ : E′ E, (x, y) , − . → 7→ 4x2 8x2 (ψ) (ψϕ) Then ϕψ = [2], ψϕ = [2]. In this case ι1 and ι2 are exact. Let S˜ (E′/Q) denote the image of S (E/Q) (ψ) in S (E′/Q). Then (ϕ) (ψ) #S (E/Q) #S (E′/Q) #X(E/Q)[ϕ]= , #X(E′/Q)[ψ]= , #E′(Q)/ϕE(Q) #E(Q)/ψE′(Q) Ouyang Y et al. Sci China Math March 2014 Vol. 57 No. 3 651 and (ϕ) (ψ) #S (E/Q) #S˜ (E′/Q) #X(E/Q)[2] = · . (2.2) #E (Q)/ϕE(Q) #E(Q)/ψE (Q) ′ · ′ Similarly, (ψ) (ϕ) #S (E′/Q) #S˜ (E/Q) #X(E′/Q)[2] = · . (2.3) #E(Q)/ψE (Q) #E (Q)/ϕE(Q) ′ · ′ (ϕ) (ψ) The 2-descent method to compute the Selmer groups S (E/Q) and S (E′/Q) is as follows (cf. [7] for general elliptic curves): Let 2 S = prime factors of 2n , Q(S, 2) = b Q×/Q× :2 ord (b), p S .
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