2023 Theses Doctoral

CM congruences and anticyclotomic Euler systems

Lee, Yu-Sheng

We study congruences between theta series and other automorphic forms in the setting of adefinite unitary group. These theta series can be realized as the theta lifts of anticyclotomic characters. Following the Rallis inner product formula and Hida’s general philosophy, the ?-values of the characters should control the sizes of the congruences. The purpose of the thesis is to use the control and construct nontrivial anticyclotomic Euler systems for the characters. In order for the inner product on the definite unitary group to have arithmetic meaning, we show that the theta series in question are actually pull-backs of theta lifts to a quasi-split unitary group. Using the Fourier coefficients of which, we can make normalizations so that the theta series take ?-integral values.

We carry out this idea in families, and show that the congruence modules of Hida families of theta series are indeed annihilated by the ?-adic L-functions of the anticyclotomic characters. Then following Urban’s framework, we construct fundamental exact sequences that are extensions of Iwasawa algebras by the congruence modules, and nontrivial cocycles valued in which that are compatible under corestrictions when the tame conductor of the Iwasawa algebras varies. Since the ?-adic L-functions of two tame conductors are different by the Euler factors at primes dividing one conductor but not the other, after multiplying the cocycles by the ?-adic L-functions, we obtain cohomology classes valued in Iwasawa algebras that satisfies the desired Euler system relations.