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—II: Openness

Let φ be a Drinfeld A-module in special characteristic p₀ over a finitely generated field K. For any finite set P of primes p≠p₀ of A let Γ_P denote the image of Gal(K^sep/K) in its representation on the product of the p-adic Tate modules of φ for all p∈P. We determine Γ_P up to commensurability.     

Partitions of natural numbers with the same representation functions

For a set A of nonnegative integers the representation functions R₂(A,n), R₃(A,n) are defined as the number of solutions of the equation n=a+a^′,a,a^′∈A with a<a^′, a?a^′, respectively. Let D(0)=0 and let D(a) denote the number of ones in the binary representation of a. Let A₀ be the set of all nonnegative integers a with even D(a) and A₁ be the set of all nonnegative integers a with odd D(a). In this paper we show that (a) if R₂(A,n)=R₂(N?A,n) for all n?2N−1, then R₂(A,n)=R₂(N?A,n)?1 for all n?12N²−10N−2 except for A=A₀ or A=A₁; (b) if R₃(A,n)=R₃(N?A,n) for all n?2N−1, then R₃(A,n)=R₃(N?A,n)?1 for all n?12N²+2N. Several problems are posed in this paper.     

Explicit constructions of infinite families of MSTD sets

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We explicitly construct infinite families of MSTD (more sums than differences) sets, i.e., sets where |A+A|>|A−A|. There are enough of these sets to prove that there exists a constant C such that at least C/r⁴ of the r² subsets of {1,…,r} are MSTD sets; thus our family is significantly denser than previous constructions (whose densities are at most f(r)/²r/2 for some polynomial f(r)). We conclude by generalizing our method to compare linear forms ?₁A+?+?nA with ?i∈{−1,1}.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=vIDDa1R2.     

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.     

Flat cyclotomic polynomials of order three

We say that a cyclotomic polynomial Φn has order three if n is the product of three distinct primes, p<q<r. Let A(n) be the largest absolute value of a coefficient of Φn. For each pair of primes p<q, we give an infinite family of r such that A(pqr)=1. We also prove that A(pqr)=A(pqs) whenever s>q is a prime congruent to .     

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.     

Intersections of q-ary perfect codes

The intersections of q-ary perfect codes are under study. We prove that there exist two q-ary perfect codes C ₁ and C ₂ of length N = qn + 1 such that |C ₁ ? C ₂| = k · |P _i |/p for each k ∈ {0,..., p · K ? 2, p · K}, where q = p ^r , p is prime, r ≥ 1, $n = \tfrac{{q^{m - 1} - 1}}{{q - 1}}$ , m ≥ 2, |P _i | = p ^nr(q?2)+n , and K = p ^{n(2r?1)?r(m?1)}. We show also that there exist two q-ary perfect codes of length N which are intersected by p ^nr(q?3)+n codewords.     

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 Selmer groups of abelian varieties over ℓ-adic Lie extensions of global function fields

Let F be a global function field of characteristic p > 0 and A/F an abelian variety. Let K/F be an ?-adic Lie extension (? ≠ p) unramified outside a finite set of primes S and such that Gal(K/F) has no elements of order ?. We shall prove that, under certain conditions, Sel _A (K) _? ^∨ has no nontrivial pseudo-null submodule.     

A formula for the central value of certain Hecke L-functions

Let be the negative of a prime, and O_K its ring of integers. Let D be a prime ideal in O_K of prime norm congruent to . Under these assumptions, there exists Hecke characters ψ_D of K with conductor (D) and infinite type (1,0). Their L-series L(ψ_D,s) are associated to a CM elliptic curve A(N,D) defined over the Hilbert class field of K. We will prove a Waldspurger-type formula for L(ψ_D,s) of the form L(ψ_D,1)=Ω∑_[A],Ir(D,[A],I)m_[A],I([D]) where the sum is over class ideal representatives I of a maximal order in the quaternion algebra ramified at |N| and infinity and [A] are class group representatives of K. An application of this formula for the case N=-7 will allow us to prove the non-vanishing of a family of L-series of level 7|D| over K.     

Irreducibility of associated matrices

For an n by n matrix A, let K(A) be the associated matrix corresponding to a permutation group (of degree m) and one of its characters. Let D_r(A) be the coefficient of x^m?r in K(A+xI). If A is reducible, then D_r(A) is reducible. If A is irreducible and the character is identically one, then D₁(A) is irreducible. If A is row stochastic and the character is identically one, then D_r(A) is essentially row stochastic. Finally, the results motivate the definition of group induced diagraphs.     

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.     

Upper bounds on n-dimensional Kloosterman sums

Let p^m be any prime power and K_n(a,p^m) be the Kloosterman sum , where the x_i are restricted to values not divisible by p. Let m,n be positive integers with m?2 and suppose that p^γ||(n+1). We obtain the upper bound , for odd p. For p=2 we obtain the same bound, with an extra factor of 2 inserted.     

On the prime power factorization of n!

For a fixed prime q, let eq(n) denote the order of q in the prime factorization of n!. For two fixed integers m?2 and r with 0?r?m−1, let A(x;m,q,r) denote the numbers of positive integers n?x for which . In this paper we shall prove a sharp asymptotic formula of A(x;m,q,r).     

On congruence relations between the fundamental units of biquadratic fields

Given any distinct prime numbers p,q, and r satisfying certain simple congruence conditions, we display a congruence relation between the fundamental units for the biquadratic field , modulo a certain prime ideal of OK. This congruence in particular implies the validity of the equivariant Tamagawa number conjecture formulated by Burns and Flach for the pair (h⁰(SpecK),Z[Gal(K/Q)]).     

A limiting absorption principle for Schrödinger operators with spherically symmetric exploding potentials

LetH=?Δ+V(r) be a Schrödinger operator with a spherically symmetric exploding potential, namely,V(r)=V _S(r)+V _L(r), whereV _S(r) is short-range and the exploding partV _L(r) satisfies the following assumptions: (a) Λ=lim sup _r→∞ V _L(r)<∞ (but Λ=?∞ is possible). Denote Λ⁺= max(Λ,0). (b)V _L(r)∈C ^2k (r ₀, ∞) and, with someδ>0 such that 2kδ>1: (d/dr) ^j V _L(r) · (Λ⁺?V _L(r))^?1=O(r ^jδ) asr → ∞,j=1, ..., 2k. (c) ∫ _r0 ^∞ dr|V _L(r|^1/2 dr|V _L(r)|^1/2=∞. (d) (d/dr)V _L(r)≦0. Under these assumptions a limiting absorption principle forR(z)=(H?z)^?1 is established. More specifically, ifK ?C ⁺={zImz≧0} is compact andK ∩ (?∞, Λ]=Ø thenR (z) can be extended as a continuous map ofK intoB (Y, Y*) (with the uniform operator topology), whereY ?L ²(R ⁿ) is a weighted-L ² space. To ensure uniqueness of solutions of (H?z)u=f, z ∈K, a suitable radiation condition is introduced.     

Maximum modulus sets in pseudoconvex boundaries

LedD be a strictly pseudoconvex domain in ? ⁿ withC ^∞ boundary. We denote byA ^∞(D) the set of holomorphic functions inD that have aC ^∞ extension to \(\bar D\) . A closed subsetE of ?D is locally a maximum modulus set forA ^∞(D) if for everyp∈E there exists a neighborhoodU ofp andf∈A ^∞(D∩U) such that |f|=1 onE∩U and |f|<1 on \(\bar D \cap U\backslash E\) . A submanifoldM of ?D is an interpolation manifold ifT _p (M)?T _p ^c (?D) for everyp∈M, whereT _p ^c (?D) is the maximal complex subspace of the tangent spaceT _p (?D). We prove that a local maximum modulus set forA ^∞ (D) is locally contained in totally realn-dimensional submanifolds of ?D that admit a unique foliation by (n?1)-dimensional interpolation submanifolds. LetD =D ₁ x ... xD _r ? ? ⁿ whereD _i is a strictly pseudoconvex domain withC ^∞ boundary in ? ⁿ _i ,i=1,…,r. A submanifoldM of ?D ₁×…×?D _r verifies the cone condition if \(II_p (T_p (M)) \cap \bar C[Jn_1 (p),...,Jn_r (p)] = \{ 0\} \) for everyp∈M, wheren _i (p) is the outer normal toD _i atp, J is the complex structure of ? ⁿ , \(\bar C[Jn_1 (p),...,Jn_r (p)]\) is the closed positive cone of the real spaceV _p generated byJ _{n 1}(p),…,J _{n r}(p), and II _p is the orthogonal projection ofT _p (?D) onV _p . We prove that a closed subsetE of ?D ₁×…×?D _r which is locally a maximum modulus set forA ^∞(D) is locally contained inn-dimensional totally real submanifolds of ?D ₁×…×?D _r that admit a foliation by (n?1)-dimensional submanifolds such that each leaf verifies the cone condition at every point ofE. A characterization of the local peak subsets of ?D ₁×…×?D _r is also given.     

A note on matricial minimality

Let A be a complex n×n matrix, θ a matricial norm and r(A) the spectral radius of A. Then, it is known [2,7], r(A?r(θ(A)). In our present note we investigate when r(A?αI_n)=r(θ(A?αI_n))?α?C.     

Density of sets of natural numbers and the Lévy group

Let N denote the set of positive integers. The asymptotic density of the set A⊆N is d(A)=limn→∞|A∩[1,n]|/n, if this limit exists. Let AD denote the set of all sets of positive integers that have asymptotic density, and let SN denote the set of all permutations of the positive integers N. The group L^? consists of all permutations f∈SN such that A∈AD if and only if f(A)∈AD, and the group L^* consists of all permutations f∈L^? such that d(f(A))=d(A) for all A∈AD. Let be a one-to-one function such that d(f(N))=1 and, if A∈AD, then f(A)∈AD. It is proved that f must also preserve density, that is, d(f(A))=d(A) for all A∈AD. Thus, the groups L^? and L^* coincide.