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MotivationStickelbergerThe fractional Galois idealRelation to Stark unitsCyclotomic example The higher K-group situation of Snaith Equivariant Birch–Swinnerton-Dyer A very early analytic-to-algebraic result is theanalytic class number formula: For a numberfield L, annZ(µ(L))J (L) annZ(Cl(L)) Question 1 If L/K is Galois with Galois groupG, can we find (non-trivial) elements α ∈ Q[G]such that annZ[G](µ(L))α ⊆ annZ[G](Cl(L))? We assume from now on that L/K is abelian.
A partial answer is provided by Stickelbergerelements: LL/K,S(0, χ)e¯χ ∈ C[G]. In fact, a known case (the rank zero case) ofStark’s conjecture, of which more later, showsthat θL/K,S ∈ Q[G].
In the case K = Q, Stickelberger’s Theoremplus work of Siegel (later generalized by Deligneand Ribet) gives: annZ[G](µ(L))θL/Q,S ⊆ annZ[G](Cl(L)). However, θL/K,S is often zero.
The analytic class number formula provides apossible hint: Take leading coefficients of L-functions and divide by regulators. But thenwe need a regulator for each character of theGalois group. This can be done (with a caveat).
The existence of an R[G]-module isomorphism(namely Dirichlet’s regulator map) implies the existence of a (non-canonical) Q[G]-module isomorphism L,S ⊗Z Q → X ⊗Z Q. Then for a representation V of G with charac-ter χ, Rfχ ∈ C× is defined to be the determinant(over C) of HomC[G](V ∗, X ⊗Z C) HomC[G](V ∗, X ⊗Z C) The fractional Galois ideal and Stark’s Stark’s conjecture (abelian case) says exactlythat Af ∈ Q[G]×. We assume this is the casefrom now on. This is known, for example, ifL/Q is abelian.
Define If to be the Z[G]-submodule of Q[G]generated by {detQ[G](α) | α ∈ EndQ[G](X⊗ZQ), α◦f( Definition 1 Define the fractional Galois idealto be J (L/K, S) = Af If ⊆ Q[G] for any choiceof Q[G]-module isomorphism f as above. (Thisis independent of the choice of f .) If L = K (and for simplicity S = S∞), thenJ (L/L, S) = J (L) as defined earlier.
J (L/K, S) always contains the Stickelbergerelement. Further, if for n ≥ 0 then e(0)J (L/K, S) = Z[G]θL/K,S.
What about characters whose L-functions havehigher orders of vanishing? For characters withr(χ) = 1, the Stark units come into the pic-ture.
Theorem 1 For simplicity of statement, sup-pose r(χ) = 1 for all χ ∈ G. Then under minorassumptions on the set S, where E is the group of Stark units. One can remove the assumption on the ordersof vanishing of the L-functions and still ob-tain a statement in a similar vein, concerninge(1)J (L/K, S).
Fix p an odd prime, let L/Q be a cyclotomicextension of p-power conductor, and let S ={∞, p}.
Proposition 1 With L/Q and S as above, 24annZ[G](µ(L))J (L/Q, S) annZ[G](Cl(L)). Proof: In this case, e(0) = e− and e(1) = e+.
e−J (L/Q, S) = Z[G]θL/Q,S e+J (L/Q, S) = Work of Rubin on cyclotomic units (which areexamples of Stark units), relates these units tothe class-group, finishing the proof.
From Stark-type elements to K-groups We assume that the Quillen–Lichtenbaum con-jecture holds, so that the Chern classes fromhigher K-groups of number fields to ´ Take p an odd prime again, and let m be aninteger prime to p and L = Q(ζmpn) for somen ≥ 1.
Theorem 2 (Snaith) For a negative integerr, annZp[G]((K12r(L)/E)p)er+ annZp[G](K−2r(OL,S)p) where E consists of the Stark elements in K12r(L)and er+ = 1(1 + (1)rc). If F• is a perfect chain complex of Zp[G]-modules(G any finite abelian group at the moment)which is acyclic outside degrees 0 and 1 andhas finite homology otherwise, then given t ∈annZp[G](H1(F•)), det(F•)tg ⊆ annZp[G](H0(F•)), where g is the minimal number of generators(over Zp[G]) for Hom(H1(F•), Qp/Zp) and det(F•)is the determinant of Knudsen and Mumford.
There exists a (cochain) complex to which theabove can be applied and which contains all thenecessary arithmetic information. It starts outas an ´ etale complex with cohomology K12r(L)p and K−2r(OL,S)p in degrees 1 and 2 (Q.–L.)and zero elsewhere. An appropriate modifica-tion gives rise to a complex C with  (K12r(L)/E)p if i = 1 Further, Hom((K12r(L)/E)p, Qp/Zp) is gener-ated by one element over Zp[G], so we can takeg = 1 in the previous slide.
det(C) is not found directly – Iwasawa the-oretic techniques are employed to deduce itfrom the determinant of a complex over anIwasawa algebra. Consider the tower of fieldsL0 ⊆ L1 ⊆ L2 ⊆ · · · where Ln = Q(ζmpn+1),with Iwasawa algebra Λ. Then indeed, there isa Λ-complex C∞ such that C ⊗Λ Zp[Gn] = Cnfor each n.
det(C∞)1 = er+ − er−θ∞ mod Λ× where θ∞ is a limit of Stickelberger elements.
A careful descent argument allows one to seethat er+det(Cn)1 = er+ from this.
Take a pair (L/Q, E) satisfying certain assump-tions, where E is an elliptic curve over Q andL/Q is a (finite) abelian extension.
A Stark conjecture exists for (L/Q, E) thanks to the Equivariant TamagawaNumber Conjecture of Burns and Flach.
J (L/Q, E) Q[G] can be defined as be-fore.
This elliptic curve Stark conjecture is integral in the sense of predicting spe-cial points on E(L), which we will callStark points.
E(L)/(Stark points) in a similar way tothe number field case.
From Stark points to the Selmer group There is a complex C of Zp[G]-modules, acyclic outside degrees 1, 2 and 3,whose cohomology groups are E(L)p,Sel(E/L)p and (E(L)tors)∨p in degrees 1,2 and 3.
This would be modified to introduce the A similar result to earlier deals with per- fect complexes acyclic outside three ad-jacent cohomology groups, relating theannihilator ideals of the outside two co-homology groups to that of the middleone.

Source: http://events.math.unipd.it/aag/slides/slides-buckingham.pdf

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