臺灣博碩士論文加值系統

論文的主結果是給出了近全純 齋藤-黑川 模型式之限制公式。我們推廣了市 野 篤史的公式，涵蓋了具有更高模群之模型式，並且解決了對於模式型的權的 限制。限制公式給出了 Gross-Prasad 猜想在 (SO(5),SO(4)) 以及非緩增加表現的 情形下的例子。作為應用，我們證明了某些 GL(3) × GL(2) 上的自守L-函數的 Deligne 猜想。

We give explicit pullback formulas for nearly holomorphic Saito-Kurokawa lifts restrict to product of upper half-plane against with product of elliptic modular forms. We generalize the formula of Ichino to modular forms of higher level and free the restriction on weights. The explicit formulas provide non-trivial examples for the refined Gross-Prasad conjecture for (SO(5), SO(4)) in the non-tempered cases. As an application, we obtain new cases for Deligne’s conjecture for central critical values of certain automorphic L-functions for GL(3) × GL(2).

口試委員會審定書 ............................................................................#
謝辭 ................................................................................................ i
中文摘要 ......................................................................................... ii
Abstract .................................................................... ....................iii
Contents ........................................................................................iv
Chapter 1 Introduction ....................................................................1
1.1 Main results .............................................................................1
1.2 An outline of the proof ............................................................3
1.3 Structure of the thesis ............................................................4
1.4 Saito-Kurokawa packets ..........................................................5
1.5 Notation and conventions ........................................................6
Chapter 2 Whittaker functions ........................................................9
2.1 The metaplectic groups ............................................................9
2.2 Whittaker functions on widetilde{SL(2)} .................................10
2.3 Whittaker functions on GL(2) ...................................................19
Chapter 3 Weil representations and theta correspondence .............26
3.1 Local Weil representations ........................................................26
3.2 Global Weil representations and theta functions .......................33
Chapter 4 Shimura-Shintani-Waldspurger correspondence ..............34
4.1 Setting .......................................................................................34
4.2 Local theta lifts ..........................................................................36
4.3 Global theta lifts ........................................................................46
Chapter 5 Saito-Kurokawa lifts ..........................................................49
5.1 Setting .......................................................................................49
5.2 Local theta lifts ..........................................................................51
5.3 Global theta lifts ........................................................................57
Chapter 6 Jacquet-Langlands-Shimizu correspondence ..................59
6.1 Setting .......................................................................................59
6.2 Local theta lifts .........................................................................59
6.3 Global theta lifts .......................................................................61
Chapter 7 Base change for GL(2) .....................................................63
7.1 Setting .......................................................................................63
7.2 Local theta lifts ..........................................................................64
7.3 Global theta lifts ........................................................................66
Chapter 8 Triple product L-functions ................................................68
8.1 Setting .......................................................................................68
8.2 Local trilinear period integrals ...................................................70
8.3 Explicit central value formula ....................................................75
Chapter 9 Pullback formula ...............................................................76
9.1 Setting and main theorem ..........................................................76
9.2 Seesaw identities and explicit central value formula ..................79
9.3 Proof of the main theorem .........................................................82
Chapter 10 Local trilinear period integral in the C × R case ...............83
10.1 Setting ......................................................................................83
10.2 Comparison of invariant pairings ..............................................84
10.3 Local trilinear period integral and local zeta integral ................86
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