On l-adic representations for a space of noncongruence cuspforms

This paper is concerned with a compatible family of 4-dimensional \ell-adic representations \rho_{\ell} of G_\Q:=\Gal(\bar \Q/\Q) attached to the space of weight 3 cuspforms S_3 (\Gamma) on a noncongruence subgroup \Gamma \subset \SL. For this representation we prove that: 1.)It is automorphic: the L-function L(s, \rho_{\ell}^{\vee}) agrees with the L-function for an automorphic form for \text{GL}_4(\mathbb A_{\Q}), where \rho_{\ell}^{\vee} is the dual of \rho_{\ell}. 2.) For each prime p \ge 5 there is a basis h_p = \{h_p ^+, h_p ^- \} of S_3 (\Gamma) whose expansion coefficients satisfy 3-term Atkin and Swinnerton-Dyer (ASD) relations, relative to the q-expansion coefficients of a newform f of level 432. The structure of this basis depends on the class of p modulo 12. The key point is that the representation $\rho_{\ell}$ admits a quaternion multiplication structure in the sense of a recent work of Atkin, Li, Liu and Long.Comment: Second revised version. To appear: Proceedings of the American Mathematical Societ

On ℓ-adic representations for a space of noncongruence cuspforms

This paper is concerned with a compatible family of 4-dimensional ℓ-adic representations ρℓ of GQ := Gal(Q/Q) attached to the space of weight-3 cuspforms S3(Γ) on a noncongruence subgroup Γ ⊂ SL2(Z). For this representation we prove that: 1. It is automorphic: the L-function L(s,ρℓ∨) agrees with the L-function for an automorphic form for GL4(AQ), where ρℓ∨ is the dual of ρℓ. 2. For each prime p≥5 there is a basis hp = {hp+, hp-} of S3(Γ) whose expansion coefficients satisfy 3-term Atkin and Swinnerton-Dyer (ASD) relations, relative to the q-expansion coefficients of a newform f of level 432. The structure of this basis depends on the class of p modulo 12. The key point is that the representation ρℓ admits a quaternion multiplication structure in the sense of Atkin, Li, Liu, and Long

Inadequate sensitivity of laboratory risk indicator to rule out necrotizing fasciitis in the emergency department

Introduction: Necrotizing fasciitis (NF) is a life-threatening illness, particularly when surgical debridement is delayed. The Laboratory Risk Indicator for Necrotizing Fasciitis (LRINEC) score was developed to identify patients at higher risk for NF. Despite limited information in this regard, the LRINEC score is often used to rule out NF if negative. We describe the sensitivity of the LRINEC score in emergency department (ED) patients for the diagnosis of NF. Methods: We conducted a chart review of ED patients in whom coding of hospital discharge diagnoses included NF. We employed standard methods to minimize bias. We used laboratory data to calculate the LRINEC score, and confirmed the diagnosis of NF via explicit chart review. We then calculated the sensitivity of a positive LRINEC score (standardly defined as six or greater) in our cohort. We examined the role of patient characteristics in the performance of the LRINEC score. Finally, we performed sensitivity analyses to estimate whether missing data for c-reactive protein (CRP) results were likely to impact our results. Results: Of 266 ED patients coded as having a discharge diagnosis of NF, we were able to confirm the diagnosis, by chart review, in 167. We were able to calculate a LRINEC score in only 80 patients (due to absence of an initial CRP value); an LRINEC score of 6 or greater had a sensitivity of 77%. Sensitivity analyses of missing data supported our finding of inadequate sensitivity to rule out NF. In sub-analysis, NF patients with concurrent diabetes were more likely to be accurately categorized by the LRINEC score. Conclusion: Used in isolation, the LRINEC score is not sufficiently sensitive to rule out NF in a general ED population. © 2016 Koenig et al

Nonconcordance between Clinical and Head CT Findings: The Specter of Overdiagnosis

Background:. It is unclear whether history and physical examination findings can predict abnormalities on head computed tomography (CT) believed to indicate increased risk of lumbar-puncture- (LP-) induced brain herniation. The objectives of this study were to (1) identify head CT findings felt to be associated with increased risk of brain herniation and (2) to assess the ability of history and physical examination to predict those findings. Methods:. Using a modified Delphi survey technique, an expert panel defined CT abnormalities felt to predict increased risk of LP-induced brain herniation. Presence of such findings on CT was compared with history and physical examination (H&P) variables in 47 patients. Results:. No H&P variable predicted “high-risk” CT; combining H&P variables to improve sensitivity led to extremely low specificity and still failed to identify all patients with high-risk CT. Conclusions:. “High-risk” CT is not uncommon in patients with clinical characteristics known to predict an absence of actual risk from LP, and thus it may not be clinically relevant. “Overdiagnosis” will be increasingly problematic as technological advances identify increasingly subtle deviations from “normal.