The period polynomial r f (z) for an even weight k ≥ 4 newform f ∈ S k (Γ 0 (N)) is the generating function for the critical values of L(f , s). It has a functional equation relating r f (z) to r f (− 1 Nz ). We prove the Riemann hypothesis for these polynomials: that the zeros of r f (z) lie on the circle jzj = 1= ffiffiffiffi N p . We prove that these zeros are equidistributed when either k or N is large. Λðf , sÞ = N s=2

Stroke is the most common cause of disability globally. Neural stem cell (NSC) therapy, which can replace lost and damaged neurons, has been proposed as a potential treatment for stroke. The therapeutic efficacy of NSC therapy is hindered by the fact that only a small number of NSCs undergo neuronal differentiation. Neuron-specific miR-124, which promotes the differentiation of NSCs into mature neurons, can be combined with NSC therapy to cure ischemic stroke. However, the instability and poor internalization of miR-124 seriously hamper its broad clinical application. Herein, an innovative strategy involving delivery of miR-124 via a Ca-MOF@miR-124 nanodelivery system, which effectively prevents the degradation of miR-124 by nucleases and promotes the internalization of miR-124 by NSCs, is presented. The effect of accelerated neuronal directed differentiation of NSCs was assessed through in vitro cell experiments, and the clinical application potential of this nanodelivery system for the treatment of ischemic stroke was assessed through in vivo experiments involving the combination of NSC therapy and Ca-MOF@miR-124 nanoparticles. The results indicate that Ca-MOF@miR-124 nanoparticles can promote the differentiation of NSCs into mature neurons with electrophysiological function within 5 days. The differentiation rate of cells treated with Ca-MOF@miR-124 nanoparticles was at least 5 days faster than that of untreated cells. Moreover, Ca-MOF@miR-124 nanoparticles decreased the ischemic area to almost normal levels by day 7. The combination of Ca-MOF@miR-124 nanoparticles and NSC therapy will enhance the treatment of traumatic nerve injury and neurodegenerative diseases.

Osteoarthritis, a chronic degenerative cartilage disease, is the leading cause of movement disorders among humans. Although the specific pathogenesis and associated mechanisms remain unclear, oxidative stress-induced metabolic imbalance in chondrocytes plays a crucial role in the occurrence and development of osteoarthritis. In this study, a trimanganese tetroxide (Mn 3 O 4 ) nanozyme with superoxide dismutase (SOD)-like and catalase (CAT)-like activities is designed to reduce oxidative stress-induced damage and its therapeutic effect is investigated. In vitro, Mn 3 O 4 nanozymes are confirmed to reprogram both the imbalance of metabolism in chondrocytes and the uncontrolled inflammatory response stimulated by hydrogen peroxide. In vivo, a cross-linked chondroitin sulfate (CS) hydrogel is designed as a substrate for Mn 3 O 4 nanozymes to treat osteoarthritis in mouse models. As a result, even in the early stage of OA (4 weeks), the therapeutic effect of the Mn 3 O 4 @CS hydrogel is observed in both cartilage metabolism and inflammation. Moreover, the Mn 3 O 4 @CS hydrogel maintained its therapeutic effects for at least 7 days, thus revealing a broad scope for future clinical applications. In conclusion, these results suggest that the Mn 3 O 4 @CS hydrogel is a potentially effective therapeutic treatment for osteoarthritis, and a novel therapeutic strategy for osteoarthritis based on nanozymes is proposed.

Suppose that O L is the ring of integers of a number field L, and suppose that(note: q := e 2πiz ) is a normalized Hecke eigenform for SL 2 (Z). We say that f is non-ordinary at a prime p if there is a prime ideal p ⊂ O L above p for which a f (p) ≡ 0 (mod p).For any finite set of primes S, we prove that there are normalized Hecke eigenforms which are non-ordinary for each p ∈ S. The proof is elementary and follows from a generalization of work of Choie, Kohnen and the third author [1].

Let M be a pure motive over Q of odd weight w ≥ 3, even rank d ≥ 2, and global conductor N whose L-function L (s, M) coincides with the L-function of a self-dual algebraic tempered cuspidal symplectic representation of GL d (A Q ). We show that a certain polynomial which generates special values of L(s, M) (including all of the critical values) has all of its zeros equidistributed on the unit circle, provided that N or w are sufficiently large with respect to d. These special values have arithmetic significance in the context of the Bloch-Kato conjecture. We focus on applications to symmetric powers of semistable elliptic curves over Q. Using the Rodriguez-Villegas transform, we use these results to construct large classes of "zeta-polynomials" (in the sense of Manin) arising from symmetric powers of semistable elliptic curves; these polynomials have a functional equation relating s → 1 − s, and all of their zeros on the line (s) = 1/2.