The $p$-adic Jacquet-Langlands correspondence and a question of Serre
The $p$-adic Jacquet-Langlands correspondence and a question of Serre
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$p$-adic Jacquet-Langlands 信件和 Serre 的问题
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发表时间：
2018
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通讯作者：
S. Howe
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作者：
S. Howe
我们证明，对于在 $p$ 和 $\infty$ 分支的四元数代数的单位，$p$-adic 模形式的完备赫克代数与连续 $p$-adic 自守形式的完备赫克代数是同构的。这对塞尔在 1987 年写给泰特的信中提出的问题给出了肯定的答案。证明是几何的，并由于 Serre 提出了 mod $p$ 论证：我们通过用 Hodge-Tate 周期图的纤维识别四元数双陪集来评估模形式，并使用假 Hasse 不变量将函数扩展到双陪集之外。假设局部-全局与 Knight 和 Scholze 的局部 $p$-adic Jacquet-Langlands 对应关系兼容，我们还表明，在许多情况下，局部代数向量在出现的四元数表示中并不稠密，这证实了 Knight 的期望。
We show that the completed Hecke algebra of $p$-adic modular forms is isomorphic to the completed Hecke algebra of continuous $p$-adic automorphic forms for the units of the quaternion algebra ramified at $p$ and $\infty$. This gives an affirmative answer to a question posed by Serre in a 1987 letter to Tate. The proof is geometric, and lifts a mod $p$ argument due to Serre: we evaluate modular forms by identifying a quaternionic double-coset with a fiber of the Hodge-Tate period map, and extend functions off of the double coset using fake Hasse invariants. Assuming local-global compatibility with the local $p$-adic Jacquet-Langlands correspondence of Knight and Scholze, we also show that in many cases the locally algebraic vectors are not dense in the quaternionic representations that appear, confirming an expectation of Knight.