Number Theory Seminar

Let $E$ be a CM field with maximal totally real subfield $F$. I will explain a Fourier spectral reciprocity formula that transforms toric periods over $E$ into a dual family of Hecke periods over $F$. Applied to canonical Hecke characters, it gives an explicit first moment for central $L$-values and derivatives, with power-saving errors uniform in the weight and quadratic twisting conductor. As the relative discriminant grows, we obtain quantitative nonvanishing in both root-number parities, including families of analytic rank one. When $F$ has odd degree over $\mathbb Q$, Zhang’s theorem turns the weight-one results into Mordell–Weil rank formulas and finiteness of Shafarevich–Tate groups for the associated dihedral abelian varieties over $F$ and $E$.