On the special values of certain L-series related to half-integral weight modular forms

Let h be a cuspidal Hecke eigenform of half-integral weight, and En/2+1/2 be Cohen’s Eisenstein series of weight n/2+1/2. For a Dirichlet character χ we define a certain linear combination R(χ)(s, h,En/+1/2) of the Rankin-Selberg convolution products of h and En/2+1/2 twisted by Dirichlet characters related with χ. We then prove a certain algebraicity result for R(χ)(l, h,En/2+1/2) with l integers

Role of liraglutide in Alzheimer's disease pathology

Background The described relationship between Alzheimer's disease (AD) and type 2 diabetes (T2D) and the fact that AD has no succesful treatment has led to the study of antidiabetic drugs that may limit or slow down AD pathology. Main body Although T2D treatment has evident limitations, options are increasing including glucagon-like peptide 1 analogs. Among these, liraglutide (LRGT) is commonly used by T2D patients to improve beta cell function and suppress glucagon to restore normoglycaemia. Interestingly, LRGT also counterbalances altered brain metabolism and has anti-inflammatory properties. Previous studies have reported its capacity to reduce AD pathology, including amyloid production and deposition, tau hyperphosphorylation, or neuronal and synaptic loss in animal models of AD, accompanied by cognitive improvement. Given the beneficial effects of LRGT at central level, studies in patients have been carried out, showing modest beneficial effects. At present, the ELAD trial (Evaluating Liraglutide in Alzheimer's Disease NCT01843075) is an ongoing phase IIb study in patients with mild AD. In this minireview, we resume the outcomes of LRGT treatment in preclinical models of AD as well as the available results in patients up to date. Conclusion The effects of LRGT on animal models show significant benefits in AD pathology and cognitive impairment. While studies in patients are limited, ongoing clinical trials will probably provide more definitive conclusions on the role of LRGT in AD patients