Bell’s theorem determines the number of representations of a positive integer in terms of the ternary quadratic forms x2+by2+cz2 with b,c {1,2,4,8}. This number depends only on the number of representations of an int...Bell’s theorem determines the number of representations of a positive integer in terms of the ternary quadratic forms x2+by2+cz2 with b,c {1,2,4,8}. This number depends only on the number of representations of an integer as a sum of three squares. We present a modern elementary proof of Bell’s theorem that is based on three standard Ramanujan theta function identities and a set of five so-called three-square identities by Hurwitz. We use Bell’s theorem and a slight extension of it to find explicit and finite computable expressions for Tunnel’s congruent number criterion. It is known that this criterion settles the congruent number problem under the weak Birch-Swinnerton-Dyer conjecture. Moreover, we present for the first time an unconditional proof that a square-free number n 3(mod 8) is not congruent.展开更多
The field K ---- Q（x/-AT） is the only imaginary quadratic field with class number 1, in which the prime 2 splits, and we fix one of the primes p of K lying above 2. The modular elliptic curve X0 （49） has complex m...The field K ---- Q（x/-AT） is the only imaginary quadratic field with class number 1, in which the prime 2 splits, and we fix one of the primes p of K lying above 2. The modular elliptic curve X0 （49） has complex multiplication by the maximal order O of K, and we let E be the twist of Xo （49） by the quadratic extension K（v/M）/K, where M is any square free element of O with M -- i mod 4 and （M, 7） = 1. In the present note, we use surprisingly simple algebraic arguments to prove a sharp estimate for the rank of the MordeII-Weil group modulo torsion of E over the field F∞=k（Ep∞ ）, where Epic denotes the group of p∞-division points on E. Moreover, writing B for the twist of X0（49） by K（C/ET）/K, our Iwasawa-theoretic arguments also show that the weak form of the conjecture of Birch and Swinnerton-Dyer implies the non-vanishing at s =1 of the complex L-series of B over every finite layer of the unique Z2-extension of K unramified outside p. We hope to give a proof of this last non-vanishing assertion in a subsequent paper.展开更多
文摘Bell’s theorem determines the number of representations of a positive integer in terms of the ternary quadratic forms x2+by2+cz2 with b,c {1,2,4,8}. This number depends only on the number of representations of an integer as a sum of three squares. We present a modern elementary proof of Bell’s theorem that is based on three standard Ramanujan theta function identities and a set of five so-called three-square identities by Hurwitz. We use Bell’s theorem and a slight extension of it to find explicit and finite computable expressions for Tunnel’s congruent number criterion. It is known that this criterion settles the congruent number problem under the weak Birch-Swinnerton-Dyer conjecture. Moreover, we present for the first time an unconditional proof that a square-free number n 3(mod 8) is not congruent.
文摘The field K ---- Q（x/-AT） is the only imaginary quadratic field with class number 1, in which the prime 2 splits, and we fix one of the primes p of K lying above 2. The modular elliptic curve X0 （49） has complex multiplication by the maximal order O of K, and we let E be the twist of Xo （49） by the quadratic extension K（v/M）/K, where M is any square free element of O with M -- i mod 4 and （M, 7） = 1. In the present note, we use surprisingly simple algebraic arguments to prove a sharp estimate for the rank of the MordeII-Weil group modulo torsion of E over the field F∞=k（Ep∞ ）, where Epic denotes the group of p∞-division points on E. Moreover, writing B for the twist of X0（49） by K（C/ET）/K, our Iwasawa-theoretic arguments also show that the weak form of the conjecture of Birch and Swinnerton-Dyer implies the non-vanishing at s =1 of the complex L-series of B over every finite layer of the unique Z2-extension of K unramified outside p. We hope to give a proof of this last non-vanishing assertion in a subsequent paper.
