Modular units and the surjectivity of a Galois representation

AbstractFor a prime p⩾7 the pth roots of certain modular units are shown to generate the second layer of the extension of function fields cut out by the universal Galois deformation of the representation on p-division points of a universal elliptic curve. It follows that certain Galois representations obtained by specializing the modular invariant to a rational number have large image

Quaternionic Artin representations and nontraditional arithmetic statistics

We classify and then attempt to count the real quadratic fields (ordered by the size of the totally positive fundamental unit, as in Sarnak [14], [15]) from which quaternionic Artin representations of minimal conductor can be induced. Some of our results can be interpreted as criteria for a real quadratic field to be contained in a Galois extension of Q with controlled ramification and Galois group isomorphic to a generalized quaternion group.Accepted manuscrip

Dihedral Artin representations and CM fields

For a fixed CM field K with maximal totally real subfield F, we consider dihedral Artin representations of F induced from K. We prove that a positive proportion of such representations have image D4.First author draf

Ruxolitinib for Glucocorticoid-Refractory Acute Graft-versus-Host Disease

BACKGROUND: Acute graft-versus-host disease (GVHD) remains a major limitation of allogeneic stem-cell transplantation; not all patients have a response to standard glucocorticoid treatment. In a phase 2 trial, ruxolitinib, a selective Janus kinase (JAK1 and JAK2) inhibitor, showed potential efficacy in patients with glucocorticoid-refractory acute GVHD. METHODS: We conducted a multicenter, randomized, open-label, phase 3 trial comparing the efficacy and safety of oral ruxolitinib (10 mg twice daily) with the investigator's choice of therapy from a list of nine commonly used options (control) in patients 12 years of age or older who had glucocorticoid-refractory acute GVHD after allogeneic stem-cell transplantation. The primary end point was overall response (complete response or partial response) at day 28. The key secondary end point was durable overall response at day 56. RESULTS: A total of 309 patients underwent randomization; 154 patients were assigned to the ruxolitinib group and 155 to the control group. Overall response at day 28 was higher in the ruxolitinib group than in the control group (62% [96 patients] vs. 39% [61]; odds ratio, 2.64; 95% confidence interval [CI], 1.65 to 4.22; P<0.001). Durable overall response at day 56 was higher in the ruxolitinib group than in the control group (40% [61 patients] vs. 22% [34]; odds ratio, 2.38; 95% CI, 1.43 to 3.94; P<0.001). The estimated cumulative incidence of loss of response at 6 months was 10% in the ruxolitinib group and 39% in the control group. The median failure-free survival was considerably longer with ruxolitinib than with control (5.0 months vs. 1.0 month; hazard ratio for relapse or progression of hematologic disease, non-relapse-related death, or addition of new systemic therapy for acute GVHD, 0.46; 95% CI, 0.35 to 0.60). The median overall survival was 11.1 months in the ruxolitinib group and 6.5 months in the control group (hazard ratio for death, 0.83; 95% CI, 0.60 to 1.15). The most common adverse events up to day 28 were thrombocytopenia (in 50 of 152 patients [33%] in the ruxolitinib group and 27 of 150 [18%] in the control group), anemia (in 46 [30%] and 42 [28%], respectively), and cytomegalovirus infection (in 39 [26%] and 31 [21%]). CONCLUSIONS: Ruxolitinib therapy led to significant improvements in efficacy outcomes, with a higher incidence of thrombocytopenia, the most frequent toxic effect, than that observed with control therapy

Galois theory, elliptic curves, and root numbers

The inverse problem of Galois theory asks whether an arbitrary finite group G can be real-ized as Gal(K/Q) for some Galois extension K of Q. When such a realization has been given for a particular G then a natural sequel is to find arithmetical realizations of the irreducible representations of G. One possibility is to ask for realizations in the Mordell-Weil groups of elliptic curves over Q: Given an irreducible complex representation τ of Gal(K/Q), does there exist an elliptic curve E over Q such that τ occurs in the natural representation of Gal(K/Q) on C ⊗Z E(K)? The present paper does not attempt to investigate this question directly. Instead we adopt Greenberg’s point of view in his remarks on nonabelian Iwasawa theory [5] and con-sider a related question about root numbers. Let ρE denote the representation of Gal(K/Q) on C ⊗Z E(K) and 〈τ, ρE 〉 the multiplicity of τ in ρE, and write L(E, τ, s) for the tensor product L-function associated to E and τ. The conjectures of Birch-Swinnerton-Dyer and Deligne-Gross imply that (0.1) ords=1L(E, τ, s) = 〈τ, ρE〉 (cf. [10], p. 127), whence for representations τ with real-valued character the root numberW (E, τ) should satisf