Transfer of Siegel Cusp Forms of Degree 2.
Pitale, Ameya.
Transfer of Siegel Cusp Forms of Degree 2. - 1st ed. - 1 online resource (120 pages) - Memoirs of the American Mathematical Society Series ; v.232 . - Memoirs of the American Mathematical Society Series .
Cover -- Title page -- Introduction -- Notation -- Chapter 1. Distinguished vectors in local representations -- 1.1. Parabolic induction to (2,2) -- 1.2. Distinguished vectors: non-archimedean case -- 1.3. Distinguished vectors: archimedean case -- 1.4. Intertwining operator: non-archimedean case -- 1.5. Intertwining operator: archimedean case -- Chapter 2. Global -functions for ₄× ₂ -- 2.1. Bessel models for ₄ -- 2.2. Local zeta integrals -- 2.3. The global integral representation -- 2.4. The functional equation -- Chapter 3. The pullback formula -- 3.1. Local sections: non-archimedean case -- 3.2. The local pullback formula: non-archimedean case -- 3.3. Local sections: archimedean case -- 3.4. The local pullback formula: archimedean case -- 3.5. The global pullback formula -- 3.6. The second global integral representation -- Chapter 4. Holomorphy of global -functions for ₄× ₂ -- 4.1. Preliminary considerations -- 4.2. Eisenstein series and Weil representations -- 4.3. The Siegel-Weil formula and the proof of entireness -- Chapter 5. Applications -- 5.1. The transfer theorems -- 5.2. Analytic properties of -functions -- 5.3. Critical values of -functions -- Bibliography -- Back Cover.
Let \pi be the automorphic representation of \textrm_4(\mathbb) generated by a full level cuspidal Siegel eigenform that is not a Saito-Kurokawa lift, and \tau be an arbitrary cuspidal, automorphic representation of \textrm_2(\mathbb). Using Furusawa's integral representation for \textrm_4\times\textrm_2 combined with a pullback formula involving the unitary group \textrm(3,3), the authors prove that the L-functions L(s,\pi\times\tau) are "nice". The converse theorem of Cogdell and Piatetski-Shapiro then implies that such representations \pi have a functorial lifting to a cuspidal representation of \textrm_4(\mathbb). Combined with the exterior-square lifting of Kim, this also leads to a functorial lifting of \pi to a cuspidal representation of \textrm_5(\mathbb). As an application, the authors obtain analytic properties of various L-functions related to full level Siegel cusp forms. They also obtain special value results for \textrm_4\times\textrm_1 and \textrm_4\times\textrm_2.
9781470418939
Cusp forms (Mathematics).
Siegel domains.
Modular groups.
Electronic books.
QA243 .P58 2014
512.7
Transfer of Siegel Cusp Forms of Degree 2. - 1st ed. - 1 online resource (120 pages) - Memoirs of the American Mathematical Society Series ; v.232 . - Memoirs of the American Mathematical Society Series .
Cover -- Title page -- Introduction -- Notation -- Chapter 1. Distinguished vectors in local representations -- 1.1. Parabolic induction to (2,2) -- 1.2. Distinguished vectors: non-archimedean case -- 1.3. Distinguished vectors: archimedean case -- 1.4. Intertwining operator: non-archimedean case -- 1.5. Intertwining operator: archimedean case -- Chapter 2. Global -functions for ₄× ₂ -- 2.1. Bessel models for ₄ -- 2.2. Local zeta integrals -- 2.3. The global integral representation -- 2.4. The functional equation -- Chapter 3. The pullback formula -- 3.1. Local sections: non-archimedean case -- 3.2. The local pullback formula: non-archimedean case -- 3.3. Local sections: archimedean case -- 3.4. The local pullback formula: archimedean case -- 3.5. The global pullback formula -- 3.6. The second global integral representation -- Chapter 4. Holomorphy of global -functions for ₄× ₂ -- 4.1. Preliminary considerations -- 4.2. Eisenstein series and Weil representations -- 4.3. The Siegel-Weil formula and the proof of entireness -- Chapter 5. Applications -- 5.1. The transfer theorems -- 5.2. Analytic properties of -functions -- 5.3. Critical values of -functions -- Bibliography -- Back Cover.
Let \pi be the automorphic representation of \textrm_4(\mathbb) generated by a full level cuspidal Siegel eigenform that is not a Saito-Kurokawa lift, and \tau be an arbitrary cuspidal, automorphic representation of \textrm_2(\mathbb). Using Furusawa's integral representation for \textrm_4\times\textrm_2 combined with a pullback formula involving the unitary group \textrm(3,3), the authors prove that the L-functions L(s,\pi\times\tau) are "nice". The converse theorem of Cogdell and Piatetski-Shapiro then implies that such representations \pi have a functorial lifting to a cuspidal representation of \textrm_4(\mathbb). Combined with the exterior-square lifting of Kim, this also leads to a functorial lifting of \pi to a cuspidal representation of \textrm_5(\mathbb). As an application, the authors obtain analytic properties of various L-functions related to full level Siegel cusp forms. They also obtain special value results for \textrm_4\times\textrm_1 and \textrm_4\times\textrm_2.
9781470418939
Cusp forms (Mathematics).
Siegel domains.
Modular groups.
Electronic books.
QA243 .P58 2014
512.7