We show that the compactly supported cohomology of certain $\text{U}(n,n)$- or $\text{Sp}(2n)$-Shimura varieties with $\unicode[STIX]{x1D6E4}_{1}(p^{\infty })$-level vanishes above the middle degree. The only assumption is that we work over a CM field $F$ in which the prime $p$ splits completely. We also give an application to Galois representations for torsion in the cohomology of the locally symmetric spaces for $\text{GL}_{n}/F$. More precisely, we use the vanishing result for Shimura varieties to eliminate the nilpotent ideal in the construction of these Galois representations. This strengthens recent results of Scholze [On torsion in the cohomology of locally symmetric varieties, Ann. of Math. (2) 182 (2015), 945–1066; MR 3418533] and Newton–Thorne [Torsion Galois representations over CM fields and Hecke algebras in the derived category, Forum Math. Sigma 4 (2016), e21; MR 3528275].

Euler's identity equates the number of partitions of any non-negative integer n into odd parts and the number of partitions of n into distinct parts. Beck conjectured and Andrews proved the following companion to Euler's identity: the excess of the number of parts in all partitions of n into odd parts over the number of parts in all partitions of n into distinct parts equals the number of partitions of n with exactly one even part (possibly repeated). Beck's original conjecture was followed by generalizations and so-called "Beck-type" companions to other identities.In this paper, we establish a collection of Beck-type companion identities to the following result mentioned by Lehmer at the 1974 International Congress of Mathematicians: the excess of the number of partitions of n with an even number of even parts over the number of partitions of n with an odd number of even parts equals the number of partitions of n into distinct, odd parts. We also establish various generalizations of Lehmer's identity, and prove related Beck-type companion identities. We use both analytic and combinatorial methods in our proofs.

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Let f be a modular form with complex multiplication. If f has critical slope, then Coleman's classicality theorem implies that there is a padic overconvergent generalized Hecke eigenform with the same Hecke eigenvalues as f . We give a formula for the Fourier coefficiets of this generalized Hecke eigenform. We also investigate the dimension of the generalized Hecke eigenspace of p-adic overconvergent forms containing f .

The different polymorphisms of the transcription factor 7-like 2 (TCF7L2) gene promote variances in diabetes susceptibility in humans. We investigated whether these genotypes also promote differences in diabetic susceptibility in commercial pigs. Growing pigs (Landrace, both sex, 50–60 kg) with the C/C (n=4) and T/T (n=5) TCF7L2 genotypes were identified and intravenously injected with streptozotocin (STZ, 40 mg/kg) twice in weekly intervals, then a high-energy diet was offered. Oral glucose tolerance tests, blood analyses and the homeostasis model assessment-insulin resistance (HOMA-IR) index calculations were performed. The animals were sacrificed at the end of 12 weeks of treatment to reveal the pancreas histomorphometry. The results showed that all of the treated pigs grew normally despite exhibiting hyperglycemia at two weeks after the induction. The glycemic level of the fasting or postprandial pigs gradually returned to normal. The fasting insulin concentration was significantly decreased for the T/T carriers but not for the C/C carriers, and the resulting HOMA-IR index was significantly increased for the C/C genotype, indicating that the models of insulin dependence and resistance were respectively developed by T/T and C/C carriers. The histopathological results illustrated a significant reduction in the pancreas mass and insulin active sites, which suggested increased damage. The results obtained here could not be compared with previous studies because the TCF7L2 background has not been reported. Growing pigs may be an excellent model for diabetic in children if the animals are genetically pre-selected.