arXiv math/96102281996-10-23math.NT

三重积L函数的p-adic测度与平方根

p-adic measures and square roots of triple product L-functions

Michael Harris, Jacques Tilouine

摘要：

设p为素数，f、g和h为$SL(2,\Bbb{Z})$的权重$κ$、$λ$和$μ$的三种模形式。我们假设$κ\geq λ+ μ$。在与Kudla的联合工作中，其中一位作者得到了三重积$L(s,f,g,h)$在$s = {1/2}(κ+ λ+ μ- 2)$（{ \it中心临界值}）处的值的归一化{ \it平方根}的公式。我们应用这个公式，让$f$（从而$κ$）在$p$ -进解析家庭${\bold f}$的普通模形式（一个Hida家庭）。通过修改Hida构造的$p$-adic Rankin-Selberg卷积，我们得到了一个广义$p$-adic测度，其相关的解析函数给出了$L(s,f,g,h)$的中心临界值的平方根的$p$-adic插值，并通过一定的通用校正因子进行归一化.阿基米德修正系数没有明确确定。这是$L$ -函数的正规化平方根沿着所谓的“反分圆超平面”的$p$ -adic插值的一个非常普遍的现象的例子。“我们注意到$p$ -adic三重积本身并没有在半空间$κ\geq λ+ μ$中构造。

英文摘要：

Let p be a prime number, and let f, g, and h be three modular forms of weights $κ$, $λ$, and $μ$ for $SL(2,\Bbb{Z})$. We suppose $κ\geq λ+ μ$. In joint work with Kudla, one of the authors obtained a formula for the normalized {\it square root} of the value at $s = {1/2}(κ+ λ+ μ- 2)$ (the {\it central critical value}) of the triple product $L(s,f,g,h)$. We apply this formula, letting $f$ (and thus $κ$) vary in a $p$-adic analytic family ${\bold f}$ of ordinary modular forms (a Hida family). By modifying Hida's construction of the $p$-adic Rankin-Selberg convolution, we obtain a generalized $p$-adic measure whose associated analytic function gives a $p$-adic interpolation of the square roots of the central critical values of $L(s,f,g,h)$, normalized by certain universal correction factors. The archimedean correction factor is not determined explicitly. This is an example of what appears to be a very general phenomenon of $p$-adic interpolation of normalized square roots of $L$-functions along the so-called "anti-cyclotomic hyperplane." We note that the $p$-adic triple product itself has not been constructed in the half-space $κ\geq λ+ μ$.

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