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 Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Given a Hecke eigenform $f$ of weight $2$ and square-free level $N$, by thework of Kohnen, there is a unique weight $3/2$ modular form of level $4N$mapping to $f$ under the Shimura correspondence. Furthermore, by the work ofWaldspurger the Fourier coefficients of such a form are related to thequadratic twists of the form $f$. Gross gave a construction of the halfintegral weight form when $N$ is prime, and such construction was latergeneralized to square-free levels. However, in the non-square free case, thesituation is more complicated since the natural construction is vacuous. Theproblem being that there are too many special points so that there iscancellation while trying to encode the information as a linear combination oftheta series.  In this paper, we concentrate in the case of level $p^2$, for $p>2$ a primenumber, and show how the set of special points can be split into subsets(indexed by bilateral ideals for an order of reduced discriminant $p^2$) whichgives two weight $3/2$ modular forms mapping to $f$ under the Shimuracorrespondence. Moreover, the splitting has a geometric interpretation whichallows to prove that the forms are indeed a linear combination of theta seriesassociated to ternary quadratic forms.  Once such interpretation is given, we extend the method of Gross-Zagier tothe case where the level and the discriminant are not prime to each other toprove a Gross-type formula in this situation.

Shimura correspondence for level $p^2$ and the central values of $L$-series II