Let π be a cuspidal automorphic representation of PGL_2(AQ) of arithmetic conductor C and archimedean parameter T, and let ϕ be an L2-normalized automorphic form in the space of π. The sup-norm problem asks for bounds on ∥ϕ∥_∞ in terms of C and T. The quantum unique ergodicity (QUE) problem concerns the limiting behavior of the L2-mass |ϕ|^2(g)dg of ϕ. Previous work on these problems in the conductor-aspect has focused on the case that ϕ is a newform. In this work, we study these problems for a class of automorphic forms that are not newforms. Precisely, we assume that for each prime divisor p of C, the local component π_p is supercuspidal (and satisfies some additional technical hypotheses), and consider automorphic forms ϕ for which the local components ϕ_p∈π_p are "minimal" vectors. Such vectors may be understood as non-archimedean analogues of lowest weight vectors in holomorphic discrete series representations of PGL_2(R). For automorphic forms as above, we prove a sup-norm bound that is sharper than what is known in the newform case. In particular, if π_∞ is a holomorphic discrete series of lowest weight k, we obtain the optimal bound C^{1/8−ϵ} k^{1/4−ϵ} ≪_ϵ |ϕ|_∞ ≪_ϵ C^{1/8+ϵ} k^{1/4+ϵ}. We prove also that these forms give analytic test vectors for the QUE period, thereby demonstrating the equivalence between the strong QUE and the subconvexity problems for this class of vectors. This finding contrasts the known failure of this equivalence [31] for newforms of powerful level.

Let N be a fixed integer and f be a holomorphic newform of level q⁠, weight k and trivial nebentypus, where q is a multiple of N⁠. In this article, we prove that the pushforward to the modular curve of level N of the mass measure of f tends weakly to the Haar measure as qk→∞⁠. This generalizes the previous results for modular curve of level 1. The main innovation of this article is to obtain an upper bound for the local integral which cancels the convexity bound of the corresponding L-function in level aspect.

Let E be a quadratic algebra over a number field F. Let E(g, s) be an Eisenstein series on GL2(E), and let F be a cuspidal automorphic form on GL2(F). We will consider in this paper the following automorphic integral: ∫ZAGL2(F)∖GL2(AF)F(g)E(g,s)dg. This is in some sense the complementary case to the well-known Rankin–Selberg integral and the triple product formula. We will approach this integral by Waldspurger’s formula, giving a criterion about when the integral is automatically zero, and otherwise the L-functions it represents. We will also calculate the local integrals at some ramified places, where the level of the ramification can be arbitrarily large.

This paper is devoted to finding the highest possible focus order of planar polynomial differential equations. The results consist of two parts: (i) we explicitly construct a class of concrete systems of degree n, where n+1 is a prime p or a power of a prime p^k, and show that these systems can have a focus order n^2-n; (ii) we theoretically prove the existence of polynomial systems of degree n having a focus order n^2-1 for any even number n. Corresponding results for odd n and more concrete examples having higher focus orders are given too.

We investigate meromorphic quasi-modular forms and their L-functions. We study the space of meromorphic quasi-modular forms and Rankin–Cohen brackets of meromorphic modular forms. Then we define their L-functions by using multiple techniques of regularized integral. Explicit formulas for the L-functions are given.