Citation
Sorensen, Claus Mazanti (2006) LevelRaising for GSp(4). Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/83EQJ244. https://resolver.caltech.edu/CaltechETD:etd05152006143522
Abstract
This thesis provides congruences between unstable and stable automorphic forms for the symplectic similitude group $GSp(4)$. More precisely, we raise the level of certain CAP representations $Pi$ of SaitoKurokawa type, arising from classical modular forms $f in S_4(Gamma_0(N))$ of squarefree level and root number $epsilon_f=1$. We first transfer $Pi$ to a suitable inner form $G$ such that $G(R)$ is compact modulo its center. This is achieved by viewing $G$ as a similitude spin group of a definite quadratic form in five variables, and then $ heta$lifting the whole Waldspurger packet for $widetilde{SL}(2)$ determined by $f$. Thereby we obtain an automorphic representation $pi$ of $G$. For the inner form we prove a precise levelraising result, inspired by the work of Bellaiche and Clozel, and relying on computations of Schmidt. Thus we obtain a $ ilde{pi}$ congruent to $pi$, with a local component that is irreducibly induced from an unramified twist of the Steinberg representation of the Klingen Levi subgroup. To transfer $ ilde{pi}$ back to $GSp(4)$, we use Arthur's stable trace formula and the exhaustive work of Hales on Shalika germs and the fundamental lemma in this case. Since $ ilde{pi}$ has a local component of the above type, all endoscopic error terms vanish. Indeed, by Weissauer, we only need to show that such a component does not participate in the $ heta$correspondence with any $GO(4)$. This is an exercise in using Kudla's filtration of the Jacquet modules of the Weil representation. Thus we get a cuspidal automorphic representation $ tilde{Pi}$ of $GSp(4)$ congruent to $Pi$, which is neither CAP nor endoscopic. In particular, its Galois representations are irreducible by work of Ramakrishnan. It is crucial for our application that we can arrange for $ ilde{Pi}$ to have vectors fixed by the nonspecial maximal compact subgroups at all primes dividing $N$. Since $G$ is necessarily ramified at some prime $r$, we have to show a nonspecial analogue of the fundamental lemma at $r$. Fortunately, by work of Kottwitz we can compare the involved orbital integrals to twisted orbital integrals over the unramified quadratic extension of $Q_r$. The inner form $G$ splits over this extension, and the comparison of the twisted orbital integrals can be done by hand. Finally we give an application of our main result to the BlochKato conjecture. Assuming a conjecture of Skinner and Urban on the rank of the monodromy operators at the primes dividing $N$, we construct a torsion class in the Selmer group of the motive $M_f(2)$.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  automorphic forms; Selmer groups; trace formula 
Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Mathematics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  4 May 2006 
Record Number:  CaltechETD:etd05152006143522 
Persistent URL:  https://resolver.caltech.edu/CaltechETD:etd05152006143522 
DOI:  10.7907/83EQJ244 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  1817 
Collection:  CaltechTHESIS 
Deposited By:  Imported from ETDdb 
Deposited On:  16 May 2006 
Last Modified:  08 Apr 2020 19:11 
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