A well-known conjecture, often attributed to Serre, asserts that any motive over any number field has infinitely many ordinary reductions (in the sense that the Newton polygon coincides with the Hodge polygon). In the case of Hilbert modular cuspforms $f$ of parallel weight $(2,\ldots ,2)$ , we show how to produce more ordinary primes by using the Sato–Tate equidistribution and combining it with the Galois theory of the Hecke field. Under the assumption of stronger forms of Sato–Tate equidistribution, we get stronger (but conditional) results. In the case of higher weights, we formulate the ordinariness conjecture for submotives of the intersection cohomology of proper algebraic varieties with motivic coefficients, and verify it for the motives whose $\ell$ -adic Galois realisations are abelian on a finite-index subgroup. We get some results for Hilbert cuspforms of weight $(3,\ldots ,3)$ , weaker than those for $(2,\ldots ,2)$ .

中文翻译：

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Hilbert 模簇中的普通素数

一个众所周知的猜想，通常归因于塞尔，断言任何数域上的任何动机都有无限多的普通归约（在牛顿多边形与霍奇多边形重合的意义上）。在 Hilbert 模尖点形成平行权重 $(2,\ldots,2)$ 的情况下，我们展示了如何通过使用 Sato-Tate 均衡分布并将其与 Hecke 场的伽罗瓦理论相结合来产生更多的普通素数. 在 Sato-Tate 等分布的更强形式的假设下，我们得到更强（但有条件）的结果。在更高权重的情况下，我们为具有动机系数的真代数簇的交上同调的子动机制定了平凡性猜想，并验证了其$\ell$ -adic Galois实现在有限指数子群上是阿贝尔的动机。