In this paper we give quantitative local test vectors for Waldspurger's period integral (i.e., a toric period on GL_2) in new cases with joint ramifications. The construction involves minimal vectors, rather than newforms and their variants. This paper gives a uniform treatment for the matrix algebra and division algebra cases under mild assumptions, and establishes an explicit relation between the size of the local integral and the finite conductor C(π×π_{χ^{−1}}). As an application, we combine the test vector results with the relative trace formula, and prove a hybrid type subconvexity bound which can be as strong as the Weyl bound in proper range.

We discuss Waldspurger’s local period integral for newforms in new cases. The main tool is the work of Hu and Nelson (2018) on Waldspurger’s period integral using minimal vectors, and the explicit relation between newforms and minimal vectors. We use a representation-theoretical trick to simplify computations for newforms. As an example, we compute the local integral coming from a special arithmetic setting which was used to study the 3-part full BSD conjecture by Hu et al. (2019).

Let $p\equiv 4,7\ \mathrm {mod}\ 9$ be a rational prime number such that $3\ \mathrm {mod}\ p$ is not a cube. In this paper, we prove the $3$-part of $|\textrm {III}(E_p)|\cdot |\textrm {III}(E_{3p^2})|$ is as predicted by the Birch and Swinnerton-Dyer conjecture, where $E_p: x^3+y^3=p$ and $E_{3p^2}: x^3+y^3=3p^2$ are the elliptic curves related to the Sylvester conjecture and cube sum problems.

In this short paper we give the sub-local upper bound for the sup norm of an automorphic form on PGL_n, whose associated automorphic representation has finite conductor C(π)=p^c with c→∞, and its local component at the place of ramification is a minimal vector belonging to an irreducible representation with generic induction datum.

In this paper we prove the equidistribution of the restriction of the mass of automorphic newforms to a nonsplit torus in the depth aspect. This result is better than the current known results on the similar problem in the eigenvalue aspect. The method is relatively elementary and makes use of the known effective QUE result in the depth aspect.