Hida族中的解析秩一阶传播
Analytic rank one propagation in Hida families
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中文总结 AI 辅助
该研究在算术代数几何相关猜想成立的假设下，证明了非 CM 新形式对应的 Hida 族中解析秩一阶的传播性质，为 Greenberg 的极小性猜想提供了证据。
中文摘要 AI 辅助
设$f$为权$k\geq4$、水平$N$且 Nebentypus 平凡的非 CM 新形式。设$p\nmid N$为$f$的一个奇素数，且为普通素数，记$\boldsymbol f^{(p)}$为经过$f$的$p$进 Hida 族。在算术代数几何的两个一般猜想的特定实例成立的假设下（$p$进 Abel-Jacobi 映射的单射性、Gillett-Soulé 意义下的阿基米德高度配对的正定性），我们证明，对于上述除有限个外的所有$p$，若$f$的解析秩为$1$，则$\boldsymbol f^{(p)}$中除有限个外的权为偶数且 Nebentypus 平凡的特殊化形式的解析秩均为$1$。该结果为 Greenberg 关于模形式族中解析秩的“极小性猜想”提供了证据。
英文摘要
Let $f$ be a non-CM newform of weight $k\geq4$, level $N$ and trivial Nebentypus. Let $p\nmid N$ be an odd prime number that is ordinary for $f$ and denote by $\boldsymbol f^{(p)}$ the $p$-adic Hida family passing through $f$. Assuming very specific instances of two general conjectures in arithmetic algebraic geometry (injectivity of $p$-adic Abel-Jacobi maps, positive definiteness of archimedean height pairings à la Gillet-Soulé), we prove that, for all but finitely many $p$ as above, if the analytic rank of $f$ is $1$, then all but finitely many specializations of $\boldsymbol f^{(p)}$ of even weight and trivial Nebentypus have analytic rank $1$. This result provides evidence for Greenberg's "minimality conjecture" on analytic ranks in families of modular forms.
