论文标题
平均而言，Selmer组的尺寸和椭圆曲线家族的排名
On average sizes of Selmer groups and ranks in families of elliptic curves having marked points
论文作者
论文摘要
我们确定$ 2 $ - 和$ 3 $ -SELMER组的平均尺寸/界限，各个椭圆曲线的家族具有明显的分数，从而证实了几例POONEN启发式方法。结果，我们推断出所有这些家庭的椭圆曲线的平均等级都是有限的。我们的证明是统一的，并利用涉及各种形式的$ 2 \ times 2 \ times 2 \ times 2 $和$ 3 \ times 3 \ times 3 $矩阵的参数化，我们在上一篇论文中研究了。 我们还推断出，$ y^2 = ax^4 + bx^2 z^2 + cz^4 $的$ 100 \％$属的属属，$ a，b，c \ in \ in \ mathbb {z} $，当订购为$ \ max \ max \ max \ max \ {| b | b |^2，| ac | | | | | | \} $，失败。其他即将到来的应用程序包括证明，整数的积极比例分别为（不是）两个理性立方的总和，以及$ \ Mathbb {p}^1 \ times \ times \ mathbb {p}^1 $ by $ \ mathbb {q} $ + \ timebb {q} $超过{q} $ fift hasse prince principle princip prince principle principle of $ \ mathbb {p}^1 \ times \ mathbb {p}^1 $。
We determine average sizes/bounds for the $2$- and $3$-Selmer groups in various families of elliptic curves with marked points, thus confirming several cases of the Poonen--Rains heuristics. As a consequence, we deduce that the average ranks of the elliptic curves in all of these families are bounded. Our proofs are uniform and make use of parametrizations involving various forms of $2 \times 2 \times 2 \times 2$ and $3 \times 3 \times 3$ matrices that we studied in a previous paper. We also deduce that $100\%$ of genus one curves of the form $y^2 = Ax^4 + Bx^2 z^2 + Cz^4$ with $A, B, C \in \mathbb{Z}$, when ordered by $\max\{|B|^2,|AC|\}$, fail the Hasse principle. Other forthcoming applications include proofs that a positive proportion of integers are (respectively, are not) the sum of two rational cubes, and a positive proportion of genus one curves in $\mathbb{P}^1 \times \mathbb{P}^1$ over $\mathbb{Q}$ fail the Hasse principle.
