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In mathematics, Eichler cohomology (also called parabolic cohomology or cuspidal cohomology) is a cohomology theory for Fuchsian groups, introduced by Eichler, that is a variation of group cohomology analogous to the image of the cohomology with compact support in the ordinary cohomology group. The Eichler–Shimura isomorphism, introduced by Eichler for complex cohomology and by Shimura for real cohomology, is an isomorphism between an Eichler cohomology group and a space of cusp forms. There are several variations of the Eichler–Shimura isomorphism, because one can use either real or complex coefficients, and can also use either Eichler cohomology or ordinary group cohomology as in. There is also a variation of the Eichler–Shimura isomorphisms using l-adic cohomology instead of re

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  • Eichler–Shimura isomorphism (en)
  • アイヒラー・志村同型 (ja)
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  • 数学において、アイヒラーコホモロジー (Eichler cohomology) （また、放物型コホモロジー (parabolic cohomology) やカスプコホモロジー (cuspidal cohomology) とも呼ぶ）は、 (Fuchsian group) のコホモロジー論であり、により導入された。このコホモロジー論は、通常のコホモロジー群の中の(cohomology with compact support)の像に類似な群コホモロジーの変形である。アイヒラー・志村同型 (Eichler–Shimura isomorphism) は、複体のコホモロジーとしてアイヒラーにより導入され、実コホモロジーに対し Shimuraで導入され、アイヒラーコホモロジー群とカスプ形式の空間の間の同型写像である。に述べてあるように、係数として実数でも複素数でも使うことができ、アイヒラーコホモロジーでも通常の群コホモロジーでも使うことができるので、アイヒラー・志村同型はいくつかの変形がある。実コホモロジーの代わりに、l-進コホモロジーを使うアイヒラー・志村同型もあり、そこではカスプ形式の係数とこれらの群上に作用するフロベニウス写像の固有値の間を関連付ける。このことを使い、は、後に証明したヴェイユ予想へラマヌジャン予想を帰着させた。 (ja)
  • In mathematics, Eichler cohomology (also called parabolic cohomology or cuspidal cohomology) is a cohomology theory for Fuchsian groups, introduced by Eichler, that is a variation of group cohomology analogous to the image of the cohomology with compact support in the ordinary cohomology group. The Eichler–Shimura isomorphism, introduced by Eichler for complex cohomology and by Shimura for real cohomology, is an isomorphism between an Eichler cohomology group and a space of cusp forms. There are several variations of the Eichler–Shimura isomorphism, because one can use either real or complex coefficients, and can also use either Eichler cohomology or ordinary group cohomology as in. There is also a variation of the Eichler–Shimura isomorphisms using l-adic cohomology instead of re (en)
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  • Eichler cohomology (en)
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  • In mathematics, Eichler cohomology (also called parabolic cohomology or cuspidal cohomology) is a cohomology theory for Fuchsian groups, introduced by Eichler, that is a variation of group cohomology analogous to the image of the cohomology with compact support in the ordinary cohomology group. The Eichler–Shimura isomorphism, introduced by Eichler for complex cohomology and by Shimura for real cohomology, is an isomorphism between an Eichler cohomology group and a space of cusp forms. There are several variations of the Eichler–Shimura isomorphism, because one can use either real or complex coefficients, and can also use either Eichler cohomology or ordinary group cohomology as in. There is also a variation of the Eichler–Shimura isomorphisms using l-adic cohomology instead of real cohomology, which relates the coefficients of cusp forms to eigenvalues of Frobenius acting on these groups. used this to reduce the Ramanujan conjecture to the Weil conjectures that he later proved. (en)
  • 数学において、アイヒラーコホモロジー (Eichler cohomology) （また、放物型コホモロジー (parabolic cohomology) やカスプコホモロジー (cuspidal cohomology) とも呼ぶ）は、 (Fuchsian group) のコホモロジー論であり、により導入された。このコホモロジー論は、通常のコホモロジー群の中の(cohomology with compact support)の像に類似な群コホモロジーの変形である。アイヒラー・志村同型 (Eichler–Shimura isomorphism) は、複体のコホモロジーとしてアイヒラーにより導入され、実コホモロジーに対し Shimuraで導入され、アイヒラーコホモロジー群とカスプ形式の空間の間の同型写像である。に述べてあるように、係数として実数でも複素数でも使うことができ、アイヒラーコホモロジーでも通常の群コホモロジーでも使うことができるので、アイヒラー・志村同型はいくつかの変形がある。実コホモロジーの代わりに、l-進コホモロジーを使うアイヒラー・志村同型もあり、そこではカスプ形式の係数とこれらの群上に作用するフロベニウス写像の固有値の間を関連付ける。このことを使い、は、後に証明したヴェイユ予想へラマヌジャン予想を帰着させた。 (ja)
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