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| Natura: | Preprint |
| Pubblicazione: |
2019
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| Accesso online: | https://arxiv.org/abs/1912.03657 |
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| _version_ | 1866908612699357184 |
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| author | Kings, Guido Sprang, Johannes |
| author_facet | Kings, Guido Sprang, Johannes |
| contents | We show that for an arbitrary totally complex number field $L$ the (regularized) critical $L$-values of algebraic Hecke characters of $L$ divided by certain periods are algebraic integers. This relies on a new construction of an equivariant coherent cohomology class with values in the completion of the Poincaré bundle on an abelian scheme $\cal{A}$. From this we obtain a cohomology class for the automorphism group of a CM abelian scheme $\cal{A}$ with values in some canonical bundles, which can be explicitly calculated in terms of Eisenstein-Kronecker series. As a further consequence, using an infinitesimal trivialization of the Poincaré bundle, we construct a $p$-adic measure interpolating the critical $L$-values in the ordinary case. This generalizes previous results for CM fields by Damerell, Shimura and Katz and settles the algebraicity and $p$-adic interpolation in the remaining open cases of critical values of Hecke $L$-functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1912_03657 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Eisenstein-Kronecker classes, integrality of critical values of Hecke $L$-functions and $p$-adic interpolation Kings, Guido Sprang, Johannes Number Theory Algebraic Geometry We show that for an arbitrary totally complex number field $L$ the (regularized) critical $L$-values of algebraic Hecke characters of $L$ divided by certain periods are algebraic integers. This relies on a new construction of an equivariant coherent cohomology class with values in the completion of the Poincaré bundle on an abelian scheme $\cal{A}$. From this we obtain a cohomology class for the automorphism group of a CM abelian scheme $\cal{A}$ with values in some canonical bundles, which can be explicitly calculated in terms of Eisenstein-Kronecker series. As a further consequence, using an infinitesimal trivialization of the Poincaré bundle, we construct a $p$-adic measure interpolating the critical $L$-values in the ordinary case. This generalizes previous results for CM fields by Damerell, Shimura and Katz and settles the algebraicity and $p$-adic interpolation in the remaining open cases of critical values of Hecke $L$-functions. |
| title | Eisenstein-Kronecker classes, integrality of critical values of Hecke $L$-functions and $p$-adic interpolation |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/1912.03657 |