Schneider's p-adic L-function and the conjecture of Birch and Swinnerton-Dyer

Schneider 的 p 进 L 函数以及 Birch 和 Swinnerton-Dyer 的猜想

• 批准号: 11640048
• 负责人: KURIHARA Akira
• 金额: $ 0.64万
• 依托单位: Japan Women's University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640048/
• 关键词: automorphic form; elliptic curve; L-function; p-adic field; Mumford curve; the conjecture of Birch and Swinnerton-Dyer; Common Lisp

Let E be an elliptic curve over Q that is not of type CM.We choose a prime p that is a divisor of the conductor of E.Then we can consider an arithmetic group Γ.⊂ PGL_2(Q_p) and an automorphic form ψof weight 2 w.r.t. Γ on the p-adic upper half plane corresponding to E.Assume Γ is torsion-free for simplicity. Then ψ is realized as a p-adic Poincare series corresponding to an element 1≠γ∈Γ. Take another element 1≠δ∈Γ and choose the coordinate of P^1 so that 0 and ∞ are the fixed points of δ on P^1(Q_p). We define L(s, φ, δ)=Σ_<g∈<δ>＼Γ/<γ>>{χ(g・β)^<1-s>-χ(g・α)^<1-s>}. Here α, β are the fixed points of γ, and for the ratio q_δ of eigenvalues of δ we choose a suitable character χ : Q^X_p/q^Z_δ→1+p^<1+[1/(p-1)]>Z_p : (We have to modify this definition when δand γ are proportional in Γ/[Γ, Γ]) Then, we have (i) L (1, ψ, δ) = 0, (ii) d/(ds)L(s, φ, δ)|_<s=1>=Log_χ(<γ|δ>). Here Log_χ is defined by the relation d/(ds)χ(t)^s=Log_χ(t)χ(t)^s and <|> : Γ/[Γ, Γ]×Γ/[Γ, Γ]→Q^X_p is the Manin-Drinfeld pa … More iring which gives the p-adic periods of the jacobian variety of the Mumford curve uniformized by Γ. This L-function was introduced by P.Schneider and our purpose is to find a recipe to give an arithmetically nice δ and to establish the Birch-Swinnerton-Dyer conjecture for this L-function including some numerical evidences.At first we have to find a recipe to give δ. To do that we looked at an easier version of the above conjecture. Let (|) : Γ/[Γ, Γ]×Γ/[Γ, Γ]→Z be the pairing giving the length of intersection of chains on the quotient by Γ of the Bruhat-Tits building. Then, ord_p(<|>)=(|). Now the simplified conjecture is "∃δ, ∀γ, (γ|δ)=0⇔Q-rank of E is positive." In the elliptic modular case, the path on the upper half plane from 0 to √<-1>. ∞ plays the role for δ. However at present, we could not find a nice recipe for our δ. This means that although δ exists as a linar functional of the space of certain p-adic automorphic forms, we don't know whether δ exists as in a geometric sense as above (or as an element of a definite quaternion algebra). Less

设E是Q上非CM型椭圆曲线，我们选取一个素数p作为E的导子的因子，则我们可以考虑一个算术群Γ.在对应于E的p-adic上半平面上的Γ。为了简单起见，假设Γ是无挠的。然后将π实现为对应于元素1 <$γ∈Γ的p进Poincare级数。取另一个元素1 <$δ∈Γ，选择P^1的坐标，使得0和∞是δ在P^1（Q_p）上的不动点。定义L（s，φ，δ）= x_&lt;g∈&lt;δ&gt;\r/&lt;γ&gt;&gt;{x（g·β）^<1-s>-x（g·α）^<1-s>}.这里α，β是γ的不动点，对于δ的特征值之比q_δ，我们选择一个合适的特征标χ：Q^X_p/q^Z_δ→1+p^&lt;1+[1/（p-1）]&gt;Z_p：（当δ和γ在Γ/[Γ，Γ]中成比例时，我们必须修改这个定义）那么，我们有（i）L（1，φ，δ）= 0，（ii）d/（ds）L（s，φ，δ）|_&lt;s=1&gt;=Log_X（&lt;γ| δ&gt;）。这里，Log_x由关系d/（ds）x（t）^s=Log_x（t）x（t）^s定义，并且|&gt;：r/[r，r]× r/[r，r]→Q^X_p是Manin-Drinfeld路径 ...更多信息 环，给出了由Γ均匀化的Mumford曲线的雅可比簇的p-adic周期。这个L-函数是由P.Schneider引入的，我们的目的是找到一个给出算术量δ的方法，并建立关于这个L-函数的Birch-Swinnerton-Dyer猜想，包括一些数值证据。为了做到这一点，我们看了上述猜想的一个更简单的版本。让（|）：Γ/[Γ，Γ]×Γ/[Γ，Γ]→Z是给出Bruhat-Tits建筑的Γ商上的链的交叉长度的配对。然后，ord_p（&lt;|&gt;)=(|).现在简化的猜想是“&lt;$δ，&lt;$γ，（γ| δ）=0惠Q秩为正。“在椭圆模的情况下，上半平面上从0到0的路径<-1>。∞对δ起作用。然而，目前，我们无法为我们的δ找到一个好的配方。这意味着虽然δ作为某些p进自守形式空间的线性泛函存在，但我们不知道δ是否像上面那样在几何意义上存在（或者作为一个定四元数代数的元素）。少