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).

Thesis advisor: Solomon Friedberg / We study a new lifting of automorphic representations using the theta representation ϴ on the 4-fold cover of the symplectic group, $\overline{\Sp}_{2r}(\A)$. This lifting produces the first examples of CAP representations on higher degree metaplectic covering groups. Central to our analysis is the identification of the maximal nilpotent orbit associated to ϴ. We conjecture a natural extension of Arthur's parameterization of the discrete spectrum to $\overline{\Sp}_{2r}(\A)$. Assuming this, we compute the effect of our lift on Arthur parameters and show that the parameter of a representation in the image of the lift is non-tempered. We conclude by relating the lifting to the dimension equation of Ginzburg to predict the first non-trivial lift of a generic cuspidal representation of $\overline{\Sp}_{2r}(\A)$. / Thesis (PhD) — Boston College, 2018. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

Thesis advisor: Solomon Friedberg / We study the generalized theta lifting between the double covers of split special orthogonal groups, which uses the non-minimal theta representations constructed by Bump, Friedberg and Ginzburg. We focus on the theta liftings of non-generic representations and make a conjecture that gives an upper bound of the first non-zero occurrence of the liftings, depending only on the unipotent orbit. We prove both global and local results that support the conjecture. / Thesis (PhD) — Boston College, 2021. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第22968号 / 理博第4645号 / 新制||理||1668(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 市野 篤史, 教授 池田 保, 教授 加藤 周 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM