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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenTue, 16 Apr 2024 15:12:04 +0000The Twisted Weighted Fundamental Lemma for the Transfer of Automorphic Forms from GSp(4) to GL(4)
https://resolver.caltech.edu/CaltechETD:etd-05242005-104909
Authors: {'items': [{'email': 'whitehouse_david@hotmail.com', 'id': 'Whitehouse-David', 'name': {'family': 'Whitehouse', 'given': 'David'}, 'show_email': 'NO'}]}
Year: 2005
DOI: 10.7907/26W7-8M37
We prove the twisted weighted fundamental lemma for the group GL(4) x GL(1) relative to a certain outer automorphism α, which yields GSp(4) as a twisted endoscopic group. This version of the fundamental lemma is needed to stabilize the twisted trace formula for the pair (GL(4) xGL(1),α). This stabilized twisted trace formula is required for Arthur's classification of the discrete spectrum of $GSp(4)$ in terms of automorphic representations of GL(4).
https://thesis.library.caltech.edu/id/eprint/1995Artin L-Functions for Abelian Extensions of Imaginary Quadratic Fields
https://resolver.caltech.edu/CaltechETD:etd-06062005-134908
Authors: {'items': [{'email': 'jenfns@gmail.com', 'id': 'Johnson-Jennifer-Michelle', 'name': {'family': 'Johnson', 'given': 'Jennifer Michelle'}, 'show_email': 'NO'}]}
Year: 2005
DOI: 10.7907/8T84-BQ83
Let F be an abelian extension of an imaginary quadratic field K with Galois group G. We form the Galois-equivariant L-function of the motive h(Spec F)(j) where the Tate twists j are negative integers. The leading term in the Taylor expansion at s=0 decomposes over the group algebra Q[G] into a product of Artin L-functions indexed by the characters of G. We construct a motivic element via the Eisenstein symbol and relate the L-value to periods via regulator maps. Working toward the equivariant Tamagawa number conjecture, we prove that the L-value gives a basis in etale cohomology which coincides with the basis given by the p-adic L-function according to the main conjecture of Iwasawa theory.https://thesis.library.caltech.edu/id/eprint/2475Level-Raising for GSp(4)
https://resolver.caltech.edu/CaltechETD:etd-05152006-143522
Authors: {'items': [{'email': 'csorensen@ucsd.edu', 'id': 'Sorensen-Claus-Mazanti', 'name': {'family': 'Sorensen', 'given': 'Claus Mazanti'}, 'show_email': 'NO'}]}
Year: 2006
DOI: 10.7907/83EQ-J244
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 Saito-Kurokawa type, arising from classical modular forms $f in S_4(Gamma_0(N))$ of square-free 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 level-raising 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 non-special maximal compact subgroups at all primes dividing $N$. Since $G$ is necessarily ramified at some prime $r$, we have to show a non-special 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 Bloch-Kato 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 $M_f(2)$.https://thesis.library.caltech.edu/id/eprint/1817On the Equivariant Tamagawa Number Conjecture
https://resolver.caltech.edu/CaltechETD:etd-05242006-225912
Authors: {'items': [{'email': 'navilarekallu@gmail.com', 'id': 'Navilarekallu-Tejaswi', 'name': {'family': 'Navilarekallu', 'given': 'Tejaswi'}, 'show_email': 'NO'}]}
Year: 2006
DOI: 10.7907/7HZ0-F068
For a finite Galois extension K/Q of number fields with Galois group G and a motive M = M' ⊗ h⁰(Spec(K))(0) with coefficients in Q[G], the equivariant Tamagawa number conjecture relates the special value L*(M,0) of the motivic L-function to an element of K₀(Z[G];R) constructed via complexes associated to M. The conjecture for nonabelian groups G is very much unexplored. In this thesis, we will develop some techniques to verify the conjecture for Artin motives and motives attached to elliptic curves. In particular, we consider motives h⁰(Spec(K))(0) for an A₄-extension K/Q and, h¹ (E x Spec(L))(1) for an S₃-extension L/Q and an elliptic curve E/Q.
https://thesis.library.caltech.edu/id/eprint/2017