Siegel Eisenstein级数拉回的尖点性及其对Bloch-Kato猜想的应用
On the cuspidality of pullbacks of Siegel Eisenstein series and applications to the Bloch-Kato conjecture
摘要：
设$k > 3$为整数，$p$为素数，其中$p > 2k-2$为素数。设$f$是权$2k-2$的一种新形式，且等级为1，使得$f$在$p$是普通的，而$\barρ_{f}$是不可约的.在一些额外的假设下，我们证明了$ord_{p}(L_{alg}(k,f)) \leq ord_{p}(# S)$，其中$S$是与$ρ_{f} \otimes ε^{1-k}$相关联的Selmer群的Pontryagin对偶，$ε$是$p$ -adic分圆特征标。我们首先通过构造$f$的Saito-Kurokawa升力和非CAP Siegel尖点形式之间的同余来实现这一点。一旦这个同余成立，我们使用伽罗瓦表示，以获得上的Selmer群的下界。
英文摘要：
Let $k > 3$ be an integer and $p$ a prime with $p > 2k-2$. Let $f$ be a newform of weight $2k-2$ and level 1 so that $f$ is ordinary at $p$ and $\barρ_{f}$ is irreducible. Under some additional hypotheses we prove that $ord_{p}(L_{alg}(k,f)) \leq ord_{p}(# S)$ where $S$ is the Pontryagin dual of the Selmer group associated to $ρ_{f} \otimes ε^{1-k}$ with $ε$ the $p$-adic cyclotomic character. We accomplish this by first constructing a congruence between the Saito-Kurokawa lift of $f$ and a non-CAP Siegel cusp form. Once this congruence is established, we use Galois representations to obtain the lower bound on the Selmer group.
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