On the conjecture of Birch and Swinnerton-Dyer for an elliptic curve of rank 3
On the conjecture of Birch and Swinnerton-Dyer for an elliptic curve of rank 3
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关于 Birch 和 Swinnerton-Dyer 对 3 阶椭圆曲线的猜想
DOI：
10.1090/s0025-5718-1985-0777279-x
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发表时间：
1985
影响因子：
2
通讯作者：
D. Zagier
中科院分区：
文献类型：
--
作者：
J. Buhler;B. Gross;D. Zagier
椭圆曲线y2 = 4x 328 x + 25在Q上秩为3。假设这条曲线的WeilTaniyama猜想，我们表明，它的L-系列L（s）有一个三重零在s = 1和计算lim，_I L（s）/（s 1）3至28个小数位;其值与产品的调节器和真实的周期，根据Birch-Swinnerton-Dyer猜想，如果III是平凡的。本文的目的是对椭圆曲线（1）E：y2 = 4x 328 x + 25用数值方法（高精度）验证Birch和Swinnerton-Dyer的猜想. E的导体是5077，这显然是Q上秩为3的曲线的最小导体。由于以前的精确数值验证是针对秩为0或1的模曲线进行的，现在可以从理论上证实[2]，[4]，因此似乎需要测试更大秩曲线的猜想。我们假设大家对椭圆曲线理论有一定的了解，好的参考文献有[3]和[5]。1.正则高度函数。BirchSwinnerton-Dyer配方中的主要成分之一是调节剂，即，E（Q）上表示正则高度配对的矩阵的行列式？R关于E（Q）/E（Q）tors的Z-基在这一节中，我们描述如何计算点Pe E（Q）的正则高度。我们首先回顾一下定义。E的全局最小模型具有形式（2）y2 +y=x3- 7 x +6，通过将（1）中的y替换为2 y + 1并除以4获得;该方程具有判别式Ai = 5077。如果P E E（Q），则P的朴素高度被定义为（3）h（P）= logmax（lal，B），x（P）= a/B，B > 0，（a，B）= 1（这里我们对E使用模型（1）还是（2）并不重要，因为x坐标是相同的）;规范高度是E（Q）上的唯一二次型h？R使得h（P）h（P）是有界的，并且正则高度对是相应的双线性形式（P，P '）=（（h（P + P'）h（P）h（P '））. h的定义直接意味着公式h（P）= limn n 2 h（P），但这不便于计算。可用的公式是（4）h（P）= logb + F（x（P）），接收于1984年3月20日;修订于1984年6月11日。1980年数学学科分类初级14 K 07，14 G10。? 01985 American Mathematical Society 0025-5718/85 $1.00 + $.25每页473此内容于2016年8月30日星期二06：32：02 UTC从 157.55.39.215下载所有使用http://about.jstor.org/terms约束474 JOE.贝内迪特·布勒格罗斯和唐B。其中B表示如（3）中的x（P）的分母，并且F（x）是由00 F（x）= log定义的实值函数|X| + E4-n-1 logz，（5）n~~l=0（5）~+145049x4 + 14 X250 x + 49 x，1Xn Xn 4x 328 xn + 25在x = 0附近，（5）中的前两项变为无穷大，但我们可以将它们联合收割机得到1 ~~注意，将xn+1与xn关联的公式是针对P E E将x（2 P）与x（P）关联的公式，使得xn = x（2 P）。特别地，x，l > e3 = 1.946.对于n > 1，其中e1 < e2 < e3表示多项式4x 3 28 x + 25的根，因此zn介于1和1.328.之间。并且log Zn在0和0.284之间...因此，（5）或（6）中的级数收敛得非常快，我们可以计算h（P）到任何所需的精度。公式（4）是Tate [6]计算高度的通用公式的特殊化;事实上，F（x（P））是Tate关于（P，P）的无限分量的公式，而ord，（B）log p（p撇）给出了典范高度的p分量（即使对于不好的归约的素数p = 5077，因为Neron模型在p处的纤维是不可约的）。然而，Tate的结果，虽然在文献中引用，尚未发表，所以我们给出了一个直接的证明（4）在我们的情况下。根据这个定义，它足以表明（4）右边的表达式与h（P）相差一个有界的量，并且如果P被2 P代替，则乘以4。通过已经引用的公式，用2 P代替P，用x（2 P）= a*/B * 代替x（P）= a/B*，其中a* = a4 + 14 a2 b 250 ab 3 + 49 b4，B* = 4a 3 b28 ab 3 + 25 b4。我们认为B* 是x（2 P）的正分母。实际上，用g.c.d. 's表明，对于任何整数a，B，（a，B）= 1，除非a 92 B（mod 5077），在这种情况下为50771（a*，B*），（a*，B*）= 1。但这在这里不可能发生，因为4x 3 28 x + 25 = 4（x 92）2（x + 184）+ 5077（20 x 1227）将被5077整除，但不能被50772整除，如果x是92（mod 5077），因此，不可能是一个平方。(This是对5077处的Neron模型只有一个组件这一事实的基本重述。）另一方面，在（5）中，用2 P代替P，用x，n+1，Zn+代替xn，zn，所以
The elliptic curve y2 = 4x3 28x + 25 has rank 3 over Q. Assuming the WeilTaniyama conjecture for this curve, we show that its L-series L(s) has a triple zero at s = 1 and compute lim, _I L(s)/(s 1)3 to 28 decimal places; its value agrees with the product of the regulator and real period, in accordance with the Birch-Swinnerton-Dyer conjecture if III is trivial. The object of this note is to verify the conjecture of Birch and Swinnerton-Dyer numerically (to high accuracy) for the elliptic curve (1) E :y2 = 4x3 28x + 25. The conductor of E is 5077, which is apparently the smallest conductor for a curve of rank 3 over Q. Since previous accurate numerical verifications were done for modular curves of rank 0 or 1, and these can now be confirmed theoretically [2], [4], it seemed desirable to test the conjecture for a curve of larger rank. We assume some familiarity with the theory of elliptic curves; good references are [3] and [5]. 1. The Canonical Height Function. One of the main ingredients in the BirchSwinnerton-Dyer formula is the regulator, i.e., the determinant of the matrix expressing the canonical height pairing on E(Q) ? R with respect to a Z-basis of E(Q)/E(Q) torsIn this section we describe how to calculate the canonical height of a point P e E(Q). We first recall the definition. The global minimal model for E has the form (2) y2 +y=x3-7x+6, obtained by replacing y by 2y + 1 in (1) and dividing by 4; this equation has discriminant A\ = 5077. If P E E(Q), then the naive height of P is defined as (3) h(P) = logmax(lal, b), x(P) = a/b, b > 0, (a, b) = 1 (here it does not matter whether we use model (1) or (2) for E, as the x-coordinates are the same); the canonical height is the unique quadratic form h on E(Q) ? R such that h (P) h (P) is bounded, and the canonical height pairing is the associated bilinear form (P, P') = ((h(P + P') h(P) h(P')). The definition of h immediately implies the formula h (P) = limn n 2h (nP), but this is not convenient for calculations. A formula which is usable is (4) h (P) = logb + F(x(P)), Received March 20, 1984; revised June 11, 1984. 1980 Mathematics Subject Classification. Primary 14K07, 14G10. ?01985 American Mathematical Society 0025-5718/85 $1.00 + $.25 per page 473 This content downloaded from 157.55.39.215 on Tue, 30 Aug 2016 06:32:02 UTC All use subject to http://about.jstor.org/terms 474 JOE. P. BUHLER, BENEDICT H. GROSS AND DON B. ZAGIER where b denotes the denominator of x(P) as in (3) and F(x) is the real-valued function defined by 00 F(x) = log|x| + E 4-n-1logz, (5) n~~~~~~~~~~~~~~l=0 (5) ~+14 50 49 x4 + 14X2 50x + 49 x,1 Xn Xn 4x3 28xn+ 25 Near x = 0 the first two terms in (5) become infinite, but we can combine them to obtain 1 ~~~~~~~~~~~~00 (6) F(x)= log(x4 + 14X2-50x + 49) + E 4-n-1logz tt= 1 a formula which now makes sense for all x. Note that the formula relating xn+l to xn is the formula relating x(2P) to x(P) for P E E, so that xn = x(2 P). In particular, x,l > e3 = 1.946... for n > 1, where e1 < e2 < e3 denote the roots of the polynomial 4x3 28x + 25, so zn lies between 1 and 1.328 ... and log zn between 0 and 0.284.... Therefore the series in (5) or (6) converges very rapidly and we can calculate h ( P) to any desired degree of accuracy. Formula (4) is the specialization to our case of a general recipe of Tate [6] for computing heights; indeed, F(x(P)) is Tate's formula for the infinite component of ( P, P) while ord,(b)log p (p prime) gives the p-component of the canonical height (even for the prime p = 5077 of bad reduction, since the fiber of the Neron model at p is irreducible). However, Tate's result, although quoted in the literature, has not yet been published, so we give a direct proof of (4) in our case. By virtue of the definition, it will suffice to show that the expression on the right-hand side of (4) differs by a bounded amount from h (P) and is multiplied by 4 if P is replaced by 2P. By the formula already cited, replacing P by 2P replaces x(P) = a/b by x(2P) = a*/b*, where a* = a4 + 14a2b250ab3 + 49b4, b* = 4a3b28ab3 + 25b4. We claim that b* is the exact denominator of x(2P). Indeed, an elementary calculation with g.c.d.'s shows that (a*, b*) = 1 for any integers a, b with (a, b) = 1 unless a 92b (mod5077), in which case 50771(a*, b*). But this cannot happen here, since 4x3 28x + 25 = 4(x 92)2(x + 184) + 5077(20x 1227) would be divisible by 5077 but not by 50772 if x were 92 (mod 5077) and hence, could not be a square. (This is an elementary restatement of the fact that the Neron model at 5077 has only one component.) On the other hand, replacing P by 2P replaces xn, zn by x,n+1, Zn+ in (5), so