The Galois representations associated to a Drinfeld module in special characteristic—III: Image of the group ring

Let K be a finitely generated field of transcendence degree 1 over a finite field, and set G_K?Gal(K^sep/K). Let φ be a Drinfeld A-module over K in special characteristic. Set E?End_K(φ) and let Z be its center. We show that for almost all primes p of A, the image of the group ring A_p[G_K] in End_A(T_p(φ)) is the commutant of E. Thus, for almost all p it is a full matrix ring over Z⊗_AA_p. In the special case E=A it follows that the representation of G_K on the p-torsion points φ[p] is absolutely irreducible for almost all p.     

The Galois representations associated to a Drinfeld module in special characteristic—I: Zariski density

Let ? be a Drinfeld A-module of rank r over a finitely generated field K. Assume that ? has special characteristic p₀ and consider any prime p≠p₀ of A. If End_K^sep(?)=A, we prove that the image of Gal(K^sep/K) in its representation on the p-adic Tate module of ? is Zariski dense in GL_r.     

Image of the group ring of the Galois representation associated to Drinfeld modules

Let φ be a Drinfeld A-module of arbitrary rank and arbitrary characteristic over a finitely generated field K, and set GK=Gal(Ksep/K). Let E=EndK(φ). We show that for almost all primes p of A the image of the group ring A[GK] in EndA(Tp(φ)) is the commutant of E. In the special case E=A it follows that the representation of GK on the p-torsion points φ[p](Ksep) of φ is absolutely irreducible for almost all p.     

Adelic openness for Drinfeld modules in generic characteristic

Let φ be a Drinfeld A-module of arbitrary rank and generic characteristic over a finitely generated field K. If the endomorphism ring of φ over an algebraic closure of K is equal to A, we prove that the image of the adelic Galois representation associated to φ is open.     

The support problem for abelian varieties

Let A be an abelian variety over a number field K. If P and Q are K-rational points of A such that the order of the reduction of Q divides the order of the ) reduction of P for almost all prime ideals , then there exists a K-endomorphism φ of A and a positive integer k such that φ(P)=kQ.     

Linear equations with unknowns from a multiplicative group whose solutions lie in a small number of subspaces

Let K be a field of characteristic 0 and let (K^*)ⁿ denote the n-fold Cartesian product of K^*, endowed with coordinatewise multiplication. Let Γ be a subgroup of (K^*)ⁿ of finite rank. We consider equations (*) a₁x₁ + … + a_nx_n = 1 in x = (x₁x_n)Γ, where a = (a₁,a_n)(K^*)ⁿ. Two tuples a, b(K^*)ⁿ are called Γ-equivalent if there is a uΓ such that b = u · a. Gy?ry and the author [Compositio Math. 66 (1988) 329-354] showed that for all but finitely many Γ-equivalence classes of tuples a(K^*)ⁿ, the set of solutions of (*) is contained in the union of not more than 2⁽ⁿ⁺¹! proper linear subspaces of Kⁿ. Later, this was improved by the author [J. reine angew. Math. 432 (1992) 177-217] to (n!)²ⁿ⁺². In the present paper we will show that for all but finitely many Γ-equivalence classes of tuples of coefficients, the set of non-degenerate solutions of (*) (i.e., with non-vanishing subsums) is contained in the union of not more than 2ⁿ proper linear subspaces of Kⁿ. Further we give an example showing that 2ⁿ cannot be replaced by a quantity smaller than n.     

On p-quasi-hyponormal operators

A Hilbert space operator A ∈ B(H) is said to be p-quasi-hyponormal for some 0 < p ? 1, A ∈ p − QH, if A^∗(∣A∣^2p − ∣A^∗∣^2p)A ? 0. If H is infinite dimensional, then operators A ∈ p − QH are not supercyclic. Restricting ourselves to those A ∈ p − QH for which A⁻¹(0) ⊆ A^∗-1(0), A ∈ p_∗ − QH, a necessary and sufficient condition for the adjoint of a pure p_∗ − QH operator to be supercyclic is proved. Operators in p_∗ − QH satisfy Bishop’s property (β). Each A ∈ p_∗ − QH has the finite ascent property and the quasi-nilpotent part H₀(A − λI) of A equals (A − λI)^-1(0) for all complex numbers λ; hence f(A) satisfies Weyl’s theorem, and f(A^∗) satisfies a-Weyl’s theorem, for all non-constant functions f which are analytic on a neighborhood of σ(A). It is proved that a Putnam-Fuglede type commutativity theorem holds for operators in p_∗ − QH.     

Hyperelliptic jacobians with real multiplication

Let K be a field of characteristic p≠2, and let f(x) be a sextic polynomial irreducible over K with no repeated roots, whose Galois group is isomorphic to A₅. If the jacobian J(C) of the hyperelliptic curve C:y²=f(x) admits real multiplication over the ground field from an order of a real quadratic field D, then either its endomorphism algebra is isomorphic to D, or p>0 and J(C) is a supersingular abelian variety. The supersingular outcome cannot occur when p splits in D.     

On the independence of Heegner points in the function field case

Let ∞ be a fixed place of a global function field k. Let E be an elliptic curve defined over k which has split multiplicative reduction at ∞ and fix a modular parametrization ΦE:X₀(N)→E. Let be Heegner points associated to the rings of integers of distinct quadratic “imaginary” fields K₁,…,Kr over (k,∞). We prove that if the “prime-to-2p” part of the ideal class numbers of ring of integers of K₁,…,Kr are larger than a constant C=C(E,ΦE) depending only on E and ΦE, then the points P₁,…,Pr are independent in . Moreover, when k is rational, we show that there are infinitely many imaginary quadratic fields for which the prime-to-2p part of the class numbers are larger than C.     

Subcentrality of restrictions of boundary measures on state spaces of C1-algebras

Let F be a closed face of the weak¹ compact convex state space of a unital C¹-algebra A. The author has already shown that F is a Choquet simplex if and only if p_φ^Fπ_φ(A)″p_φ^F is abelian for any φ in F with associated cyclic representation ($\text{H}$_φ,π_φ,ξ_φ), where p_φ^F is the orthogonal projection of $\text{H}$_φ onto the subspace spanned by vectors η defining vector states a → 〈π_φ(a)η, η)〉 lying in F. It is shown here that if B is a C¹-subalgebra of A containing the unit and such that ξ_φ is cyclic in $\text{H}$_φ for π_φ(B) for any φ in F, then the boundary measures on F are subcentral as measures on the state space of B if and only if p_φ^F(π_φ(A), π_φ(B)′)″p_φ^F is abelian for all φ in F. If A is separable, this is equivalent to the condition that any state in F with (π_φ(A)′ ∩ π_φ(B)″) one-dimensional is pure. Taking A to be the crossed product of a discrete C¹-dynamical system (B, G, α), these results generalise known criteria for the system to be G-central.     

A generalization of amenability and inner amenability of groups

Let G be a locally compact group. We continue our work [A. Ghaffari: Γ-amenability of locally compact groups, Acta Math. Sinica, English Series, 26 (2010), 2313–2324] in the study of Γ-amenability of a locally compact group G defined with respect to a closed subgroup Γ of G × G. In this paper, among other things, we introduce and study a closed subspace A _Γ ^p (G) of L ^∞(Γ) and then characterize the Γ-amenability of G using A _Γ ^p (G). Various necessary and sufficient conditions are found for a locally compact group to possess a Γ-invariant mean.     

The 3-Sylow subgroup of the tame kernel of real number fields

Let F be a cubic cyclic field with exactly one ramified prime p,p>7, or , a real quadratic field with . In this paper, we study the 3-primary part of K₂O_F. If 3 does not divide the class number of F, we get some results about the 9-rank of K₂O_F. In particular, in the case of a cubic cyclic field F with only one ramified prime p>7, we prove that four conclusions concerning the 3-primary part of K₂O_F, obtained by J. Browkin by numerical computations for primes p, 7≤p≤5000, are true in general.     

Sharp bounds for the number of roots of univariate fewnomials

Let K be a field and t?0. Denote by Bm(t,K) the supremum of the number of roots in K^?, counted with multiplicities, that can have a non-zero polynomial in K[x] with at most t+1 monomial terms. We prove, using an unified approach based on Vandermonde determinants, that Bm(t,L)?t²Bm(t,K) for any local field L with a non-archimedean valuation v:L→R∪{∞} such that vZ≠0_|≡0 and residue field K, and that Bm(t,K)?(t²−t+1)(pf−1) for any finite extension K/Qp with residual class degree f and ramification index e, assuming that p>t+e. For any finite extension K/Qp, for p odd, we also show the lower bound Bm(t,K)?(2t−1)(pf−1), which gives the sharp estimation Bm(2,K)=3(pf−1) for trinomials when p>2+e.     

On the variation of Tate-Shafarevich groups of elliptic curves over hyperelliptic curves

Let E be an elliptic curve over F=F_q(t) having conductor (p)·∞, where (p) is a prime ideal in F_q[t]. Let d∈F_q[t] be an irreducible polynomial of odd degree, and let . Assume (p) remains prime in K. We prove the analogue of the formula of Gross for the special value L(E⊗_FK,1). As a consequence, we obtain a formula for the order of the Tate-Shafarevich group Ш(E/K) when L(E⊗_FK,1)≠0.     

On primitive roots for rank one Drinfeld modules

Let k be a global function field over a finite field and let A be the ring of the elements in k regular outside a fixed place ∞. Let K be a global A-field of finite A-characteristic and let ? be a rank one Drinfeld A-module over K. Given any α∈K, we show that the set of places P of K for which α is a primitive root modulo P under the action of ? possesses a Dirichlet density. We also give conditions for this density to be positive.     

Fourier transforms of smooth functions on certain nilpotent Lie groups

The authors consider irreducible representations $\text{π ?}\text{N}\text{?}$ of a nilpotent Lie group and define a Fourier transform for Schwartz class (and other) functions φ on N by forming the kernels K_φ(x, y) of the trace class operations π_φ = ∝_Nφ(n)π_ndn, regarding the π as modeled in L²(R^k) for all π in general position. For a special class of groups they show that the models, and parameters λ labeling the representations in general position, can be chosen so the joint behavior of the kernels K_φ(x, y, λ) can be interpreted in a useful way. The variables (x, y, λ) run through a Zariski open set in Rⁿ, n = dim N. The authors show there is a polynomial map u = A(x, y, λ) that is a birational isomorphism A: Rⁿ → Rⁿ with the following properties. The Fourier transforms F₁φ = K_φ(x, y, λ) all factor through A to give “rationalized” Fourier transforms Fφ(u) such that Fφ ° A = F₁φ. On the rationalized parameter space a function f(u) is of the form F_φ = f ? f is Schwartz class on Rⁿ. If polynomial operators T?P(N) are transferred to operators $\text{T}\text{?}$ on Rⁿ such that $\text{F(Tφ) =}\text{T}\text{?}\text{(Fφ), P(N)}$ is transformed isomorphically to P(Rⁿ).     

Equivariant cohomology of real flag manifolds

Let P=G/K be a semisimple non-compact Riemannian symmetric space, where G=I₀(P) and K=Gp is the stabilizer of p∈P. Let X be an orbit of the (isotropy) representation of K on Tp(P) (X is called a real flag manifold). Let K₀⊂K be the stabilizer of a maximal flat, totally geodesic submanifold of P which contains p. We show that if all the simple root multiplicities of G/K are at least 2 then K₀ is connected and the action of K₀ on X is equivariantly formal. In the case when the multiplicities are equal and at least 2, we will give a purely geometric proof of a formula of Hsiang, Palais and Terng concerning H^∗(X). In particular, this gives a conceptually new proof of Borel's formula for the cohomology ring of an adjoint orbit of a compact Lie group.     

The resolvent of a dilation-analytic three-particle system

If the potential in a three-particle system is the boundary value of an analytic function, the physical Hamiltonian H(0) has a dilation-analytic continuation H(φ). The continuous spectrum of H(φ) consists of half-lines Y(λ_p, φ) starting at the thresholds λ_p of scattering channels and making angles 2φ with the positive real axis. If the interaction is the sum of local two-body potentials in suitable $\text{L}$^p-spaces, each half-line Y(λ_p, φ) is associated with an operator P(λ_p, φ) that projects onto an invariant subspace of H(φ). Suppose Y(λ_p, φ) does not pass through any two- or three-particle eigenvalues λ ≠ λ_p when φ runs through some interval $\text{0 < α ? φ ? β <}\text{π}\text{2}$. For φ in [α, β], this paper shows that the resolvent R(λ, φ) has smoothness properties near Y(λ_p, φ) that are sufficient for P(λ_p, φ)[H(φ) ? λ_p] e^?2iφ to be spectral and to generate a strongly differentiable group. The projection, the group, and the spectral resolution operators are norm continuous in φ. These results are not affected by any spurious poles of the resolvent equation. At a spurious pole λ = λ_p + ze^2iφ, the resolvent R(λ_p + ze^2iφ,φ) is examined by a method that uses two resolvent equations in succession and shows that there is norm continuity in z, φ. The case of spurious poles on Y(λ_p, φ) is included.