论文标题
椭圆形曲线的等级增长$ s_4 $和$ a_4 $ $ $ QUARTIC的理性扩展
Rank growth of elliptic curves in $S_4$ and $A_4$ quartic extensions of the rationals
论文作者
论文摘要
我们研究了椭圆曲线从$ \ mathbb {q} $到$ s_4 $和$ a_4 $ QUARTIC EXTENTIONS $ K/\ MATHBB {Q} $的等级增长。特别是，我们对固定的$ e $和变化$ k $的数量$ \ mathrm {rk}（e/k） - \ mathrm {rk}（e/\ mathbb {q}）$感兴趣。当$ \ mathrm {rk}（e/\ mathbb {q}）\ leq 1 $，$ e $均处于其他条件下，我们证明，有很多$ s_4 $ s_4 $ QUARTENIONS $ K/\ MATHBB {Q} $在$上不提高等级，即$ \ Math.k） \ mathrm {rk}（e/\ mathbb {q}）= 0 $。为此，我们展示了如何在某些二次扩展中控制$ e $的2-核等级，这反过来又有助于控制$ s_4 $和$ a_4 $ a_4 $四分位扩展名的$ \ mathbb {q} $的等级。
We investigate the rank growth of elliptic curves from $\mathbb{Q}$ to $S_4$ and $A_4$ quartic extensions $K/\mathbb{Q}$. In particular, we are interested in the quantity $\mathrm{rk}(E/K) - \mathrm{rk}(E/\mathbb{Q})$ for fixed $E$ and varying $K$. When $\mathrm{rk}(E/\mathbb{Q}) \leq 1$, with $E$ subject to some other conditions, we prove there are infinitely many $S_4$ quartic extensions $K/\mathbb{Q}$ over which $E$ does not gain rank, i.e. such that $\mathrm{rk}(E/K) - \mathrm{rk}(E/\mathbb{Q}) = 0$. To do so, we show how to control the 2-Selmer rank of $E$ in certain quadratic extensions, which in turn contributes to controlling the rank in families of $S_4$ and $A_4$ quartic extensions of $\mathbb{Q}$.
