Proceedings of the London Mathematical Society, 2007
The main result of [Ash-Stevens 97] describes a framework for constructing padic analytic familie... more The main result of [Ash-Stevens 97] describes a framework for constructing padic analytic families of p-ordinary arithmetic Hecke eigenclasses in the cohomology of congruence subgroups of GL(n)/Q where the Hecke eigenvalues vary p-adic analytically as functions of the weight. Unanswered in that paper was the question of whether or not every p-ordinary arithmetic eigenclass can be "deformed" in a "positive dimensional family" of arithmetic eigenclasses. In this paper we make precise the notions of "deformation" and "rigidity" and investigate their properties. Rigidity corresponds to the non-existence of positive dimensional deformations other than those coming from twisting by the powers of the determinant. Formal definitions are given in §3. When n = 3, we will give a necessary and sufficient condition for a given p-ordinary arithmetic eigenclass to be p-adically rigid and we will use this criterion to give examples of p-adically rigid eigenclasses on GL(3). More precisely, we define a "Hecke eigenpacket" to be a map from the Hecke algebra to a coefficient ring which gives the Hecke eigenvalues attached to a Hecke eigenclass in the cohomology. We then consider p-adic analytic "deformations" of the Hecke eigenpacket. There are several advantages to this point of view, as opposed to deforming the cohomology classes themselves. For one thing, a surjective map of coefficient modules need not induce a surjective map on the cohomology. Standard tools of algebra and geometry can be used to study the Hecke algebra and the deformations of the eigenpackets. Moreover, it is the Hecke eigenvalues that carry the interesting arithmetic information attached to a cohomological eigenclass. Our main theoretical tool is the cohomology of congruence subgroups Γ ⊆ GL(n, Z) with coefficients in the space of bounded measures on the big cell of GL(n) (see §2). In §4 we define the ring R ord as a quotient of the Hecke algebra acting on the p-ordinary part of this cohomology. In Theorem 4.2 we prove that R ord is universal in the sense that every p-ordinary arithmetic Hecke eigenpacket factors through R ord. We show that R ord is a complete semilocal noetherian ring and study the geometry of its associated p-adic rigid analytic space X ord. This leads
Annales Scientifiques De L Ecole Normale Superieure, 2011
Cet article est exploration constructive des rapports entre les symboles modulaires classique et ... more Cet article est exploration constructive des rapports entre les symboles modulaires classique et les symboles modulaires p-adiques surconvergents. Plus précisément, on donne une preuve constructive d'un theorème de controle (Theoreme 1.1) du deuxiéme auteur [20]; ce theoréme preuve l'existence et l'unicité des "liftings propres" des symboles propres modulaires classiques de pente non-critique. En tant qu'application nous décrivons un algorithme en temps polynomial pour la calculation explicite des fonctions L p-adiques associées dans ce cas-là. Dans le cas de pente critique, le theorème de controleéchoue toujours de produire des "liftings propres" (voire Theoreme 5.14 et [17] pour une récupération), mais l'algorithme "réussit" néanmoins de produire des fonctions L p-adiques. Dans les deux dernières sections nous présentons des données numériques pour plusieurs exemples de pente critique et examinons le polygone de Newton des fonctions L p-adiques associées.
Journal of the Institute of Mathematics of Jussieu, 2014
Given a prime $p\gt 2$, an integer $h\geq 0$, and a wide open disk $U$ in the weight space $ \mat... more Given a prime $p\gt 2$, an integer $h\geq 0$, and a wide open disk $U$ in the weight space $ \mathcal{W} $ of ${\mathbf{GL} }_{2} $, we construct a Hecke–Galois-equivariant morphism ${ \Psi }_{U}^{(h)} $ from the space of analytic families of overconvergent modular symbols over $U$ with bounded slope $\leq h$, to the corresponding space of analytic families of overconvergent modular forms, all with ${ \mathbb{C} }_{p} $-coefficients. We show that there is a finite subset $Z$ of $U$ for which this morphism induces a $p$-adic analytic family of isomorphisms relating overconvergent modular symbols of weight $k$ and slope $\leq h$ to overconvergent modular forms of weight $k+ 2$ and slope $\leq h$.
Although human immunodeficiency virus type 1 (HIV-1) RNA is the acknowledged "gold standard" mark... more Although human immunodeficiency virus type 1 (HIV-1) RNA is the acknowledged "gold standard" marker for monitoring disease activity in patients receiving highly active antiretroviral therapy (HAART), it remains unaffordable in resource-constrained settings. The present study investigated two commercially available kits for the detection of HIV-1 viral load markers as more affordable alternatives to HIV-1 RNA quantitation. The greatly improved heat-denatured, signal-boosted HiSens HIV-1 p24 Ag Ultra kit (Perkin-Elmer) and the ExaVir Load Quantitative HIV-RT kit (Cavidi Tech AB) were compared with the Amplicor HIV-1 Monitor (version 1.5) assay (Roche Molecular Systems Inc.). A total of 117 samples containing HIV-1 subtype C were analyzed by all three methodologies. Eighty-nine of these samples represented serial measurements from 20 patients receiving HAART. The remaining samples analyzed were from a group of treatment-naïve patients. The association between the p24 antigen assay and the RNA assay was fairly strong (R 2 ؍ 0.686). The association between the reverse transcriptase (RT) quantitation assay and the RNA assay was strong (R 2 ؍ 0.810). Both alternative assays seemed most useful for the serial monitoring of patients receiving HAART (n ؍ 89 plasma samples from 20 patients), as all assays showed a statistically significant downward trend over time, with the trend being either linear or curvilinear. In addition, all three assays showed negative correlations with the CD4 count (CD4 count versus RNA load, r ؍ ؊0.336 and P ؍ 0.001; CD4 count versus p24 antigen level, r ؍ ؊0.541 and P < 0.0001; CD4 count versus RT level, r ؍ ؊0.358 and P ؍ 0.0006). Still of major concern are both the lack of sensitivity and the wide degrees of variability of both assays. However, both assays provide a less expensive alternative to the Roche viral load assay and demonstrate the same trends during treatment. on December 7, 2015 by guest http://jcm.asm.org/ Downloaded from 858 STEVENS ET AL. J. CLIN. MICROBIOL. on December 7, 2015 by guest http://jcm.asm.org/ Downloaded from FIG. 3. (a) Schematic box plots showing log 10 p24 antigen and log 10 RNA levels. Darker boxes, p24 antigen levels; lighter boxes, RNA load. (b) Schematic box plots showing log 10 RT level and log 10 RNA levels. Darker boxes, RT levels; lighter boxes, RNA level. The secondary axis versus the six visits over time was used (n ϭ 89 samples). A line has been drawn through the median of each visit. 860 STEVENS ET AL.
Proceedings of the London Mathematical Society, 2007
The main result of [Ash-Stevens 97] describes a framework for constructing padic analytic familie... more The main result of [Ash-Stevens 97] describes a framework for constructing padic analytic families of p-ordinary arithmetic Hecke eigenclasses in the cohomology of congruence subgroups of GL(n)/Q where the Hecke eigenvalues vary p-adic analytically as functions of the weight. Unanswered in that paper was the question of whether or not every p-ordinary arithmetic eigenclass can be "deformed" in a "positive dimensional family" of arithmetic eigenclasses. In this paper we make precise the notions of "deformation" and "rigidity" and investigate their properties. Rigidity corresponds to the non-existence of positive dimensional deformations other than those coming from twisting by the powers of the determinant. Formal definitions are given in §3. When n = 3, we will give a necessary and sufficient condition for a given p-ordinary arithmetic eigenclass to be p-adically rigid and we will use this criterion to give examples of p-adically rigid eigenclasses on GL(3). More precisely, we define a "Hecke eigenpacket" to be a map from the Hecke algebra to a coefficient ring which gives the Hecke eigenvalues attached to a Hecke eigenclass in the cohomology. We then consider p-adic analytic "deformations" of the Hecke eigenpacket. There are several advantages to this point of view, as opposed to deforming the cohomology classes themselves. For one thing, a surjective map of coefficient modules need not induce a surjective map on the cohomology. Standard tools of algebra and geometry can be used to study the Hecke algebra and the deformations of the eigenpackets. Moreover, it is the Hecke eigenvalues that carry the interesting arithmetic information attached to a cohomological eigenclass. Our main theoretical tool is the cohomology of congruence subgroups Γ ⊆ GL(n, Z) with coefficients in the space of bounded measures on the big cell of GL(n) (see §2). In §4 we define the ring R ord as a quotient of the Hecke algebra acting on the p-ordinary part of this cohomology. In Theorem 4.2 we prove that R ord is universal in the sense that every p-ordinary arithmetic Hecke eigenpacket factors through R ord. We show that R ord is a complete semilocal noetherian ring and study the geometry of its associated p-adic rigid analytic space X ord. This leads
Annales Scientifiques De L Ecole Normale Superieure, 2011
Cet article est exploration constructive des rapports entre les symboles modulaires classique et ... more Cet article est exploration constructive des rapports entre les symboles modulaires classique et les symboles modulaires p-adiques surconvergents. Plus précisément, on donne une preuve constructive d'un theorème de controle (Theoreme 1.1) du deuxiéme auteur [20]; ce theoréme preuve l'existence et l'unicité des "liftings propres" des symboles propres modulaires classiques de pente non-critique. En tant qu'application nous décrivons un algorithme en temps polynomial pour la calculation explicite des fonctions L p-adiques associées dans ce cas-là. Dans le cas de pente critique, le theorème de controleéchoue toujours de produire des "liftings propres" (voire Theoreme 5.14 et [17] pour une récupération), mais l'algorithme "réussit" néanmoins de produire des fonctions L p-adiques. Dans les deux dernières sections nous présentons des données numériques pour plusieurs exemples de pente critique et examinons le polygone de Newton des fonctions L p-adiques associées.
Journal of the Institute of Mathematics of Jussieu, 2014
Given a prime $p\gt 2$, an integer $h\geq 0$, and a wide open disk $U$ in the weight space $ \mat... more Given a prime $p\gt 2$, an integer $h\geq 0$, and a wide open disk $U$ in the weight space $ \mathcal{W} $ of ${\mathbf{GL} }_{2} $, we construct a Hecke–Galois-equivariant morphism ${ \Psi }_{U}^{(h)} $ from the space of analytic families of overconvergent modular symbols over $U$ with bounded slope $\leq h$, to the corresponding space of analytic families of overconvergent modular forms, all with ${ \mathbb{C} }_{p} $-coefficients. We show that there is a finite subset $Z$ of $U$ for which this morphism induces a $p$-adic analytic family of isomorphisms relating overconvergent modular symbols of weight $k$ and slope $\leq h$ to overconvergent modular forms of weight $k+ 2$ and slope $\leq h$.
Although human immunodeficiency virus type 1 (HIV-1) RNA is the acknowledged "gold standard" mark... more Although human immunodeficiency virus type 1 (HIV-1) RNA is the acknowledged "gold standard" marker for monitoring disease activity in patients receiving highly active antiretroviral therapy (HAART), it remains unaffordable in resource-constrained settings. The present study investigated two commercially available kits for the detection of HIV-1 viral load markers as more affordable alternatives to HIV-1 RNA quantitation. The greatly improved heat-denatured, signal-boosted HiSens HIV-1 p24 Ag Ultra kit (Perkin-Elmer) and the ExaVir Load Quantitative HIV-RT kit (Cavidi Tech AB) were compared with the Amplicor HIV-1 Monitor (version 1.5) assay (Roche Molecular Systems Inc.). A total of 117 samples containing HIV-1 subtype C were analyzed by all three methodologies. Eighty-nine of these samples represented serial measurements from 20 patients receiving HAART. The remaining samples analyzed were from a group of treatment-naïve patients. The association between the p24 antigen assay and the RNA assay was fairly strong (R 2 ؍ 0.686). The association between the reverse transcriptase (RT) quantitation assay and the RNA assay was strong (R 2 ؍ 0.810). Both alternative assays seemed most useful for the serial monitoring of patients receiving HAART (n ؍ 89 plasma samples from 20 patients), as all assays showed a statistically significant downward trend over time, with the trend being either linear or curvilinear. In addition, all three assays showed negative correlations with the CD4 count (CD4 count versus RNA load, r ؍ ؊0.336 and P ؍ 0.001; CD4 count versus p24 antigen level, r ؍ ؊0.541 and P < 0.0001; CD4 count versus RT level, r ؍ ؊0.358 and P ؍ 0.0006). Still of major concern are both the lack of sensitivity and the wide degrees of variability of both assays. However, both assays provide a less expensive alternative to the Roche viral load assay and demonstrate the same trends during treatment. on December 7, 2015 by guest http://jcm.asm.org/ Downloaded from 858 STEVENS ET AL. J. CLIN. MICROBIOL. on December 7, 2015 by guest http://jcm.asm.org/ Downloaded from FIG. 3. (a) Schematic box plots showing log 10 p24 antigen and log 10 RNA levels. Darker boxes, p24 antigen levels; lighter boxes, RNA load. (b) Schematic box plots showing log 10 RT level and log 10 RNA levels. Darker boxes, RT levels; lighter boxes, RNA level. The secondary axis versus the six visits over time was used (n ϭ 89 samples). A line has been drawn through the median of each visit. 860 STEVENS ET AL.
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