Bimodules and Abelian Surfaces
Bimodules and Abelian Surfaces
复制标题
双模和阿贝尔曲面
DOI：
10.2969/aspm/01710359
复制
发表时间：
1989
期刊：
影响因子：
--
通讯作者：
K. Ribet
中科院分区：
文献类型：
--
作者：
K. Ribet
在一篇关于模形式的模表示的手稿[26]中，作者介绍了一个关于某些Shimura曲线的mod p约化和相应的经典模曲线的mod q约化的精确序列。这里p和q是不同的素数。更精确地说，在Q上判别式pq的四元数代数中确定一个最大阶O。设M是与pq互素的正整数。设C是Shimura曲线，它将具有作用量O的交换曲面分类，并具有“Γ o（M）-结构.设X为标准模曲线Xo（Mpq）。根据定义，这两条曲线是粗模量方案，并且最熟悉的是Q上的曲线（例如，参见[28]，Th. 9.6）。然而，它们作为Z上的概型存在：对于C参见[4，6]，对于X参见[5，13]。特别地，已知C和X分别在特征p和q中的约化C Fp和X Fq是完全曲线，其唯一奇点是普通的双点。在这两种情况下，奇点集可以根据在q和∞处精确分歧的有理四元数代数的算术来计算。（有一个这样的四元数代数直到同构。）在[26]中，作者观察到这些计算导致“相同的答案”，并得出结论，两组奇点之间存在1-1对应关系。他接着讲述了两条曲线X和C的雅可比行列式的算术（参见图1）。[14][10，11]）。[26]的对应关系取决于几个任意选择。更准确地说，[26]使用Drinfeld定理[6]将Zp上的Shimura曲线C视为适当的“p-adic上半平面“与PGL 2（Qp）的NSF子群Γ部分支持的离散 * 的商。这个群是通过选择：（1）判别式为q的有理四元数代数H，（2）H中水平M的Eichler阶，（3）同构H <$Q p <$M（2，Q p）而得到的。PGL 2（Q p）中的Γ的共轭类与这些选择无关，但没有规范的方式在两个不同的Γ之间移动。这种软弱使得[26]的对应关系与[27]的对应关系兼容的验证变得尴尬。
Introduction In a manuscript on mod representations attached to modular forms [26], the author introduced an exact sequence relating the mod p reduction of certain Shimura curves and the mod q reduction of corresponding classical modular curves. Here p and q are distinct primes. More precisely, fix a maximal order O in a quaternion algebra of discriminant pq over Q. Let M be a positive integer prime to pq. Let C be the Shimura curve which classifies abelian surfaces with an action of O, together with a " Γ o (M)-structure. " Let X be the standard modular curve X o (M pq). These two curves are, by definition, coarse moduli schemes and are most familiar as curves over Q (see, for example, [28], Th. 9.6). However, they exist as schemes over Z: see [4, 6] for C and [5, 13] for X. In particular, the reductions C Fp and X Fq of C and X , in characteristics p and q respectively, are known to be complete curves whose only singular points are ordinary double points. In both cases, the sets of singular points may be calculated in terms of the arithmetic of " the " rational quaternion algebra which is ramified precisely at q and ∞. (There is one such quater-nion algebra up to isomorphism.) In [26], the author observed that these calculations lead to the " same answer " and concluded that there is a 1-1 correspondence between the two sets of singular points. He went on to relate the arithmetic of the Jacobians of the two curves X and C (cf. [14] and [10, 11]). The correspondence of [26] depends on several arbitrary choices. More precisely, [26] used Drinfeld's theorem [6] to view the Shimura curve C over Z p as the quotient of the appropriate " p-adic upper half-plane " by a discrete * Partially supported by the NSF subgroup Γ of PGL 2 (Q p). This group is obtained by choosing: (1) a rational quaternion algebra H of discriminant q, (2) an Eichler order in H of level M , and (3) an isomorphism H ⊗ Q p ≈ M(2, Q p). The conjugacy class of Γ in PGL 2 (Q p) is independent of these choices, but there is no canonical way to move between two different Γ's. This flabbiness makes awkward the verification that the correspondence of [26] is compatible with the …
DOI：
--
发表时间：
1984
期刊：
--
影响因子：
--
作者：
M. Wodzicki
通讯作者：
M. Wodzicki