L-functions of Symmetric Products of the Kloosterman Sheaf over Z

The classical $n$-variable Kloosterman sums over the finite field ${\bf F}_p$ give rise to a lisse $\bar {\bf Q}_l$-sheaf ${\rm Kl}_{n+1}$ on ${\bf G}_{m, {\bf F}_p}={\bf P}^1_{{\bf F}_p}-\{0,\infty\}$, which we call the Kloosterman sheaf. Let $L_p({\bf G}_{m,{\bf F}_p}, {\rm Sym}^k{\rm Kl}_{n+1}, s)$ be the $L$-function of the $k$-fold symmetric product of ${\rm Kl}_{n+1}$. We construct an explicit virtual scheme $X$ of finite type over ${\rm Spec} {\bf Z}$ such that the $p$-Euler factor of the zeta function of $X$ coincides with $L_p({\bf G}_{m,{\bf F}_p}, {\rm Sym}^k{\rm Kl}_{n+1}, s)$. We also prove similar results for $\otimes^k {\rm Kl}_{n+1}$ and $\bigwedge^k {\rm Kl}_{n+1}$.Comment: 16 page

Frequency of seizures and epilepsy in neurological HIV-infected patients

SummaryBackgroundInfection with the human immunodeficiency virus (HIV) is associated both with infections of the central nervous system and with neurological deficits due to direct effects of the neurotropic virus. Seizures and epilepsy are not rare among HIV-infected patients. We investigated the frequency of acute seizures and epilepsy of patients in different stages of HIV infection. In addition, we compared the characteristics of patients who experienced provoked seizures only with those of patients who developed epilepsy.MethodsThe database of the Department of Neurology, University of Münster, was searched for patients with HIV infection admitted between 1992 and 2004. Their charts were reviewed regarding all available sociodemographic, clinical, neurophysiological, imaging and laboratory data, therapy and outcome. Stage of infection according to the CDC classification and the epileptogenic zone were determined.ResultsOf 831 HIV-infected patients treated in our department, 51 (6.1%) had seizures or epilepsy. Three of the 51 patients (6%) were diagnosed with epilepsy before the onset of the HIV infection. Fourteen patients (27%) only had single or few provoked seizures in the setting of acute cerebral disorders (eight patients), drug withdrawal or sleep withdrawal (two patients), or of unknown cause (four patients). Thirty-four patients (67%) developed epilepsy in the course of their HIV infection. Toxoplasmosis (seven patients), progressive multifocal leukencephalopathy (seven patients) and other acute or subacute cerebral infections (five patients) were the most frequent causes of seizures. EEG data of 38 patients were available. EEG showed generalized and diffuse slowing only in 9 patients, regional slowing in 14 patients and regional slowing and epileptiform discharges in 1 patient. Only 14 of the patients had normal EEG. At the last contact, the majority of the patients (46 patients=90%) were on highly active antiretroviral therapy (HAART). Twenty-seven patients (53%) were on anticonvulsant therapy (gabapentin: 14 patients, carbamazepine: 9 patients, valproate: 2 patients, phenytoin: 1 patient, lamotrigine: 1 patient). Patients with only provoked seizures had no epilepsy risk factors except HIV infection, and were less likely to be infected via intravenous drug abuse.ConclusionsSeizures are a relevant neurological symptom during the course of HIV infection. Although in some patients seizures only occur provoked by acute disease processes, the majority of patients with new onset seizures eventually develops epilepsy and require anticonvulsant therapy. Intravenous drug abuse and the presence of non-HIV-associated risk factors for epilepsy seem to be associated with the development of chronic seizures in this patient group