On $p$-adic $L$-functions for $GL_{2n}$ in finite slope Shalika families
On $p$-adic $L$-functions for $GL_{2n}$ in finite slope Shalika families
复制标题
关于有限斜率 Shalika 族中 $GL_{2n}$ 的 $p$-adic $L$-函数
DOI：
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发表时间：
2021
期刊：
影响因子：
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通讯作者：
Chris Williams
中科院分区：
文献类型：
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作者：
Daniel Barrera Salazar;Mladen Dimitrov;Chris Williams
In this paper, we prove new results on the geometry of the cuspidal eigenvariety for $mathrm{GL}_{2n}$ over a totally real number field $F$ at classical points admitting Shalika models. We also construct $p$-adic $L$-functions over the eigenvariety around these points. Our proofs proceed in the opposite direction to established methods: rather than using the geometry of eigenvarieties to deduce results about $p$-adic $L$-functions, we instead show that -- via evaluation maps on parahoric overconvergent cohomology groups -- non-vanishing of a $p$-adic $L$-function implies smoothness of the eigenvariety at such points. More precisely, we attach a $p$-adic $L$-function to a non-critical refinement $ ildepi$ of a regular algebraic cuspidal automorphic representation $pi$ of $mathrm{GL}_{2n}/F$ which is spherical at $p$ and admits a Shalika model. This gives the first construction of $p$-adic $L$-functions in this generality beyond the $p$-ordinary setting. Further, when $pi$ has regular weight and the corresponding $p$-adic Galois representation is irreducible, we show that the parabolic eigenvariety for $mathrm{GL}_{2n}/F$ is 'etale at $ ildepi$ over an $([F:mathbb{Q}]+1)$-dimensional weight space and contains a dense set of classical points admitting Shalika models. Finally, under a hypothesis on the local Shalika models at bad places which is empty for $pi$ of level 1, we construct a $p$-adic $L$-function for the family.
In this paper, we prove new results on the geometry of the cuspidal eigenvariety for $mathrm{GL}_{2n}$ over a totally real number field $F$ at classical points admitting Shalika models. We also construct $p$-adic $L$-functions over the eigenvariety around these points. Our proofs proceed in the opposite direction to established methods: rather than using the geometry of eigenvarieties to deduce results about $p$-adic $L$-functions, we instead show that -- via evaluation maps on parahoric overconvergent cohomology groups -- non-vanishing of a $p$-adic $L$-function implies smoothness of the eigenvariety at such points. More precisely, we attach a $p$-adic $L$-function to a non-critical refinement $ ildepi$ of a regular algebraic cuspidal automorphic representation $pi$ of $mathrm{GL}_{2n}/F$ which is spherical at $p$ and admits a Shalika model. This gives the first construction of $p$-adic $L$-functions in this generality beyond the $p$-ordinary setting. Further, when $pi$ has regular weight and the corresponding $p$-adic Galois representation is irreducible, we show that the parabolic eigenvariety for $mathrm{GL}_{2n}/F$ is 'etale at $ ildepi$ over an $([F:mathbb{Q}]+1)$-dimensional weight space and contains a dense set of classical points admitting Shalika models. Finally, under a hypothesis on the local Shalika models at bad places which is empty for $pi$ of level 1, we construct a $p$-adic $L$-function for the family.