published_or_final_version / Mathematics / Master / Master of Arts

What if Light was used to emphasis the Lightness of a Lightweight structure. In this project I set out to explorer the way in which we visually and emotionally experience “Lightweight”. I feel that car companies are beginning to see previous mistakes and engineer for a future in which physically lighter cars are requisite. This leads me to the question; how do we as designers communicate this in a positive way and sell a future efficient, lightweight lifestyle? I began the project by considering “What is Lightweight” especially in its visual and emotional forms. This thought process lead me to the use of abstract photography of light itself to help me to create a new lightweight BMW design language. The final result is a mixture of lightweight, twisting, structural forms that flow around the user to create a floating interior architecture. This will allow for, in a future autonomous world, the exposure of a fully lightweight transportation experience.

As a generalization of the Riemann zeta function, L-function has become one of the central objects in Number Theory. The theory of L-functions, which produces a large family of consequences and conjectures in a unified way, concerns their zeros and poles, functional equations, special values and the connections between objects in different fields. Although most generalizations are largely conjectural, there are many existing results that provide us the evidence. In this thesis, we shall consider some L-functions and look into some problems mentioned above. More explicitly, for the L-functions associated to newforms of fixed square-free level, we will consider an average version of the fourth moments problem. The final bound is proven by considering definite rational quaternion algebras and divisor functions in them, generalizing Maass Correspondence Theorem and one of Duke's results and eventually applying the solution to Basis Problem. We then consider the problem of expressing the central value at 1/2 of the Rankin-Selberg L-function associated to two newforms in terms of the Pertersson inner product, where one of the newforms is twisted by the derivative of some Eisenstein series. Finally, we consider the Artin L-functions attached to irreducible $4$-dimensional $S_5$-Galois representations and deal with the modularity problem. One sufficient condition on the modularity is given, which may help to find an affirmative example for Strong Artin Conjecture in this case.

The study of periods of automorphic forms using the theta correspondence and the Weil representation was initiated by Waldspurger and his work relating Fourier coefficients of modular forms of half-integral weight, periods over tori of modular forms of integral weight and special values of L-functions attached to these modular forms. In this thesis, we show that there are general relations among periods of automorphic forms on groups related by the theta correspondence. For example, if G is a symplectic group and H is an orthogonal group over a number field k, these relations are identities equating Fourier coefficients of cuspidal automorphic forms on G (relative to the Siegel parabolic subgroup) and periods of cuspidal automorphic forms on H over orthogonal subgroups. These identities are quite formal and follow from the basic properties of theta functions and the Weil representation; further study is required to show how they compare to the results of Waldspurger. The second part of this thesis shows that, under some restrictions, the identities alluded to above are the result of a comparison of nonstandard relative traces formulas. In this comparison, the relative trace formula for H is standard however the relative trace formula for G is novel in that it involves the trace of an operator built from theta functions. The final part of this thesis explores some preliminary results on local height pairings of special cycles on the p-adic upper half plane following the work of Kudla and Rapoport. These calculations should appear as the local factors of arithmetic orbital integrals in an arithmetic relative trace formula built from arithmetic theta functions as in the work of Kudla, Rapoport and Yang. Further study is required to use this approach to relate Fourier coefficients of modular forms of half-integral weight and arithmetic degrees of cycles on Shimura curves (which are the analogues in the arithmetic situation of the periods of automorphic forms over orthogonal subgroups).

The study of periods of automorphic forms using the theta correspondence and the Weil representation was initiated by Waldspurger and his work relating Fourier coefficients of modular forms of half-integral weight, periods over tori of modular forms of integral weight and special values of L-functions attached to these modular forms. In this thesis, we show that there are general relations among periods of automorphic forms on groups related by the theta correspondence. For example, if G is a symplectic group and H is an orthogonal group over a number field k, these relations are identities equating Fourier coefficients of cuspidal automorphic forms on G (relative to the Siegel parabolic subgroup) and periods of cuspidal automorphic forms on H over orthogonal subgroups. These identities are quite formal and follow from the basic properties of theta functions and the Weil representation; further study is required to show how they compare to the results of Waldspurger. The second part of this thesis shows that, under some restrictions, the identities alluded to above are the result of a comparison of nonstandard relative traces formulas. In this comparison, the relative trace formula for H is standard however the relative trace formula for G is novel in that it involves the trace of an operator built from theta functions. The final part of this thesis explores some preliminary results on local height pairings of special cycles on the p-adic upper half plane following the work of Kudla and Rapoport. These calculations should appear as the local factors of arithmetic orbital integrals in an arithmetic relative trace formula built from arithmetic theta functions as in the work of Kudla, Rapoport and Yang. Further study is required to use this approach to relate Fourier coefficients of modular forms of half-integral weight and arithmetic degrees of cycles on Shimura curves (which are the analogues in the arithmetic situation of the periods of automorphic forms over orthogonal subgroups).

As a generalization of the Riemann zeta function, L-function has become one of the central objects in Number Theory. The theory of L-functions, which produces a large family of consequences and conjectures in a unified way, concerns their zeros and poles, functional equations, special values and the connections between objects in different fields. Although most generalizations are largely conjectural, there are many existing results that provide us the evidence. In this thesis, we shall consider some L-functions and look into some problems mentioned above. More explicitly, for the L-functions associated to newforms of fixed square-free level, we will consider an average version of the fourth moments problem. The final bound is proven by considering definite rational quaternion algebras and divisor functions in them, generalizing Maass Correspondence Theorem and one of Duke's results and eventually applying the solution to Basis Problem. We then consider the problem of expressing the central value at 1/2 of the Rankin-Selberg L-function associated to two newforms in terms of the Pertersson inner product, where one of the newforms is twisted by the derivative of some Eisenstein series. Finally, we consider the Artin L-functions attached to irreducible $4$-dimensional $S_5$-Galois representations and deal with the modularity problem. One sufficient condition on the modularity is given, which may help to find an affirmative example for Strong Artin Conjecture in this case.

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Thesis (Ph. D.)--University of Chicago, 1921. / Published also without thesis note.