English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let $\pi$ be a cuspidal automorphic representation of $\operatorname{GL}_2$over a totally real number field $F$. Let $K$ be a totally imaginary quadraticextension of $F$. We estimate central values of the $\operatorname{GL}_2 \times\operatorname{GL}_2$ Rankin-Selberg $L$-functions associated to $\pi$ timesrepresentations induced from Hecke characters of $K$ which are ramified only ata given prime ideal $\mathfrak{p}$ of $F$. More specifically, we use spectraldecompositions of shifted convolution sums and relations to Fourier-Whittakercoefficients of genuine and non-genuine metaplectic forms to obtainnonvanishing estimates, averaging over primitive ring class characters of agiven exact order. When $\pi$ corresponds to a holomorphic Hilbert modular formof arithmetic weight $k \geq 2$, we then derive finer results from therationality theorems of Shimura, together with the existence of suitable$\mathfrak{p}$-adic $L$-functions. This allows us to generalize the theorems ofRohrlich, Vatsal, and Cornut-Vatsal to this setting. Finally, in aself-contained appendix, we explain how to use these results to deduce boundsfor Mordell-Weil ranks of the associated $\operatorname{GL}_2$-type abelianvarieties via existing Iwasawa main conjecture divisibilities.

Rankin-Selberg L-functions in cyclotomic towers, III