On cyclic caps in 4-dimensional projective spaces

For any divisor k of q ⁴−1, the elements of a group of k th-roots of unity can be viewed as a cyclic point set C _k in PG(4,q). An interesting problem, connected to the theory of BCH codes, is to determine the spectrum A(q) of maximal divisors k of q ⁴−1 for which C _k is a cap. Recently, Bierbrauer and Edel [Edel and Bierbrauer (2004) Finite Fields Appl 10:168–182] have proved that 3(q ² + 1)∈A(q) provided that q is an even non-square. In this paper, the odd order case is investigated. It is proved that the only integer m for which m(q ² + 1)∈A(q) is m = 2 for q ≡ 3 (mod 4), m = 1 for q ≡ 1 (mod 4). It is also shown that when q ≡ 3 (mod 4), the cap is complete.      

Small solutions of polynomial congruences

Let p be prime and q|p − 1. Suppose x ^q ≡ a(mod p) has a solution. We estimate the size of the smallest solution x ₀ with 0 < x ₀ < p. We prove that |x ₀| ≪ p ³/² q ⁻¹ log p. By applying the Burgess character sum estimates, and estimates of certain exponential sums due to Bourgain, Glibichuk and Konyagin, we derive refinements of our result.     

On the ternary goldbach problem with primes in independent arithmetic progressions

We show that for every fixed A > 0 and θ > 0 there is a ϑ = ϑ(A, θ) > 0 with the following property. Let n be odd and sufficiently large, and let Q ₁ = Q ₂:= n ^1/2(log n)^−ϑ and Q ₃:= (log n) ^θ . Then for all q ₃ ≦ Q ₃, all reduced residues a ₃ mod q ₃, almost all q ₂ ≦ Q ₂, all admissible residues a ₂ mod q ₂, almost all q ₁ ≦ Q ₁ and all admissible residues a ₁ mod q ₁, there exists a representation n = p ₁ + p ₂ + p ₃ with primes p _i ≡ a _i (q _i ), i = 1, 2, 3.      

Mordell-Weil groups and Selmer groups of twin- prime elliptic curves

Let E = Eσ : y2 = x(x + σp)(x + σq) be elliptic curves, where σ = ±1, p and q are primenumbers with p+2 = q. (i) Selmer groups S(2)(E/Q), S(φ)(E/Q), and S(φ)(E/Q) are explicitly determined,e.g. S(2)(E+1/Q)= (Z/2Z)2, (Z/2Z)3, and (Z/2Z)4 when p ≡ 5, 1 (or 3), and 7(mod 8), respectively. (ii)When p ≡ 5 (3, 5 for σ = -1) (mod 8), it is proved that the Mordell-Weil group E(Q) ≌ Z/2Z Z/2Z,symbol, the torsion subgroup E(K)tors for any number field K, etc. are also obtained.     

On a Criterion for Catalan's Conjecture

We give a new proof of a theorem of P. Mihailescu which states that the equation x ^p – y ^q = 1 is unsolvable with x, y integral and p, q odd primes, unless the congruences p ^q p (mod q ²) and q ^p q (mod p ²) hold.     

Distribution of a certain partition function modulo powers of primes

In this paper, we study a certain partition function a(n) defined by Σ _n≥0 a(n)q ⁿ := Π _n=1(1 − q ⁿ )⁻¹(1 − q ²ⁿ )⁻¹. We prove that given a positive integer j ≥ 1 and a prime m ≥ 5, there are infinitely many congruences of the type a(An + B) ≡ 0 (mod m ^j ). This work is inspired by Ono’s ground breaking result in the study of the distribution of the partition function p(n).     

Variation of local systems and parabolic cohomology

Given a family of local systems on a punctured Riemann sphere, with moving singularities, its first parabolic cohomology is a local system on the base space. We study this situation from different points of view. For instance, we derive universal formulas for the monodromy of the resulting local system. We use a particular example of our construction to prove that the simple groups PSL₂(p ²) admit regular realizations over the field ℚ(t) for primesp≢ 1, 4, 16 mod 21.     

The intersection problem for m-cycle systems

Let I_m(v) denote the set of integers k for which a pair of m-cycle systems of K_v, exist, on the same vertex set, having k common cycles. Let J_m(v) = {0, 1, 2,…, t_v ?2, t_v} where t_v = v(v ? 1)/2m. In this article, if 2mn + x is an admissible order of an m-cycle system, we investigate when I_m(2mn + x) = J_m(2mn + x), for both m even and m odd. Results include J^m(2mn + 1) = I^m(2mn + 1) for all n > 1 if m is even, and for all n > 2 if n is odd. Moreover, the intersection problem for even cycle systems is completely solved for an equivalence class x (mod 2m) once it is solved for the smallest in that equivalence class and for K_2m+1. For odd cycle systems, results are similar, although generally the two smallest values in each equivalence class need to be solved. We also completely solve the intersection problem for m = 4, 6, 7, 8, and 9. (The cased m = 5 was done by C-M. K. Fu in 1987.) © 1993 John Wiley & Sons, Inc.     

On the iteration of holomorphic self-maps of ℂ*

Letf be a holomorphic self-map of the punctured plane ℂ^*=ℂ\{0} with essentially singular points 0 and ∞. In this note, we discuss the setsI ₀(f)={z ∈ ℂ^*:f ⁿ (z) → 0,n → ∞} andI _∞(f)={z ∈ ℂ^*:f ⁿ (z) → 0,n → ∞}. We try to find the relation betweenI ₀(f),I _∞(t) andJ(f). It is proved that both the boundary ofI ₀(f) and the boundary ofI _∞)f) equal toJ(f),I ₀(f) ∩J(f) ≠ θ andI _∞(f) ∩J(f) ≠ θ. As a consequence of these results, we find bothI ₀(f) andI _∞(f) are not doubly-bounded. Supported by the National Natural Science Foundation of China     

On the cohomology of orbit space of free ℤ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-actions on lens spaces

Let G = ℤ _p , p an odd prime, act freely on a finite-dimensional CW-complex X with mod p cohomology isomorphic to that of a lens space L ^2m−1(p; q ₁, …, q _m ). In this paper, we determine the mod p cohomology ring of the orbit space X/G, when p ² ∤ m.     

Flag transitive planes of orderq n with a long cyclel ∞ as a collineation

Baker and Ebert [1] presented a method for constructing all flag transitive affine planes of orderq ² havingGF(q) in their kernels for any odd prime powerq. Kantor [6; 7; 8] constructed many classes of nondesarguesian flag transitive affine planes of even order, each admitting a collineation, transitively permuting the points at infinity. In this paper, two classes of non-desarguesian flag transitive affine planes of odd order are constructed. One is a class of planes of orderq ⁿ , whereq is an odd prime power andn 3 such thatq ⁿ 1 (mod 4), havingGF(q) in their kernels. The other is a class of planes of orderq ⁿ , whereq is an odd prime power andn 2 such thatq ⁿ 1 (mod 4), havingGF(q) in their kernels. Since each plane of the former class is of odd dimension over its kernel, it is not isomorphic to any plane constructed by Baker and Ebert [1]. The former class contains a flag transitive affine plane of order 27 constructed by Kuppuswamy Rao and Narayana Rao [9]. Any plane of the latter class of orderq ⁿ such thatn 1 (mod 2), is not isomorphic to any plane constructed by Baker ad Ebert [1].The author is grateful to the referee for many helpful comments.     

Graphs on which a group of order <Emphasis Type="Italic">pq</Emphasis> acts edge-transitively

Let F be a finite simple undirected graph with no isolated vertices. Let p, q be prime numbers with p≥q. We complete the classification of the graphs on which a group of order pq acts edge-transitively. The results are the following. If Aut（Г） contains a subgroup G of order pq that acts edge-transitively on F, then F is one of the following graphs： （1） pK1,1; （2） pqK1,1; （3） pgq,1; （4） qKp,1 （p 〉 q）; （5） pCq （q 〉 2）; （6） qCp （p 〉 q）; （7） Cp （p 〉 q = 2）; （8） Cpq; （9） （Zp, C） whereC={±r^μ ｜μ∈Zq} withq〉2, q｜（p-1） and r≠1≡r^q （modp）; （10） Kp,1 （p 〉 q）; （11） a double Cayley graph B（G,C） with C = {1-r^μ ｜ μ ∈ Zq} and r≠1≡r^q （modp）; （12） Kpq,1;or （13） Kp,q.     

The Kernel of the Eisenstein Ideal

Let N be a prime number, and let J₀(N) be the Jacobian of the modular curve X₀(N). Let T denote the endomorphism ring of J₀(N). In a seminal 1977 article, B. Mazur introduced and studied an important ideal I⊆T, the Eisenstein ideal. In this paper we give an explicit construction of the kernel J₀(N)[I] of this ideal (the set of points in J₀(N) that are annihilated by all elements of I). We use this construction to determine the action of the group Gal(Q/Q) on J₀(N)[I]. Our results were previously known in the special case where N−1 is not divisible by 16.     

Small complete caps in Galois affine spaces

Some new families of caps in Galois affine spaces AG(N, q) of dimension N≡ 0(mod 4) and odd order q are constructed. Such caps are proven to be complete by using some new ideas depending on the concept of a regular point with respect to a complete plane arc. As a corollary, an improvement on the currently known upper bounds on the size of the smallest complete caps in AG(N, q) is obtained. This research was performed within the activity of GNSAGA of the Italian INDAM, with the financial support of the Italian Ministry MIUR project “Strutture geometriche, combinatorica e loro applicazioni”, PRIN 2004–2005.     

A congruence for the Fourier coefficients of a modular form and its application to quadratic forms

Let F(z)=∑ _n=1^∞ A(n)q ⁿ denote the unique weight 6 normalized cuspidal eigenform on Γ₀(4). We prove that A(p)≡0,2,−1(mod 11) when p≠11 is a prime. We then use this congruence to give an application to the number of representations of an integer by quadratic form of level 4.      

Multi-parameter singular radon transforms I: The <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> theory

The purpose of this paper is to study the L ² boundedness of operators of the form f ↦ ψ(x) ∫ f (γ _t (x))K(t)dt, where γ _t (x) is a C ^∞ function defined on a neighborhood of the origin in (t, x) ∈ ℝ ^N × ℝ ⁿ , satisfying γ ₀(x) ≡ x, ψ is a C ^∞ cut-off function supported on a small neighborhood of 0 ∈ ℝ ⁿ , and K is a “multi-parameter singular kernel” supported on a small neighborhood of 0 ∈ ℝ ^N . The goal is, given an appropriate class of kernels K, to give conditions on γ such that every operator of the above form is bounded on L ². The case when K is a Calderón-Zygmund kernel was studied by Christ, Nagel, Stein, and Wainger; we generalize their conditions to the case when K has a “multi-parameter” structure. For example, when K is given by a “product kernel.” Even when K is a Calderón- Zygmund kernel, our methods yield some new results. This is the first paper in a three part series, the later two of which are joint with E. M. Stein. The second paper deals with the related question of L ^p boundedness, while the third paper deals with the special case when γ is real analytic.     

On a Problem of D. H. Lehmer and General Kloosterman Sums

Abstract Let q ≥ 3 be an odd number, a be any fixed positive integer with (a, q) = 1. For each integer b with 1 ≤ b < q and (b, q) = 1, it is clear that there exists one and only one c with 0 < c < q such that bc ≡ a (mod q). Let N(a, q) denote the number of all solutions of the congruent equation bc ≡ a (mod q) for 1 ≤ b, c < q in which b and c are of opposite parity, and let . The main purpose of this paper is to study the distribution properties of E(a, q), and give a sharper hybrid mean-value formula involving E(a, q) and general Kloosterman sums. This work is supported by the NSF and the PSF of P. R. China     

A Note on the Mean Value of Numbersof the Solutions of x^α≡ 1（modn）

Let T = T(p, q, α) be the number of solutions of the congruence x^α ≡ 1 (mod p^ηq^θ). Let A and B be sets of primes satisfying x₁ < p ≤ x₂ and y₁ < q ≤ y₂, respectively. A mean value estimation of is given. Supported by National Natural Science Foundation of China (No. 19971024) and Zhejiang Provincial Natural Science Foundation of China (No. 199047)     

Complete ( q 2?+ q +?8)/2-caps in the spaces PG (3, q ), q ≡ 2 (mod 3) an odd prime, and a complete 20-cap in PG (3, 5)

An infinite family of complete (q ² + q + 8)/2-caps is constructed in PG(3, q) where q is an odd prime ≡ 2 (mod 3), q ≥ 11. This yields a new lower bound on the second largest size of complete caps. A variant of our construction also produces one of the two previously known complete 20-caps in PG(3, 5). The associated code weight distribution and other combinatorial properties of the new (q ² + q + 8)/2-caps and the 20-cap in PG(3, 5) are investigated. The updated table of the known sizes of the complete caps in PG(3, q) is given. As a byproduct, we have found that the unique complete 14-arc in PG(2, 17) contains 10 points on a conic. Actually, this shows that an earlier general result dating back to the Seventies fails for q = 17.      

Codes Over the p-adic Integers

We study codes over the p-adic integers and correct errors in the existing literature. We show that MDS codes exist over the p-adics for all lengths, ranks and p. We show that self-dual codes exist over the 2-adics if and only if the length is a multiple of 8 and that self-dual codes exist over the p-adics with p odd if and only if the length is 0 (mod 4) for p ≡ 3 (mod 4) and 0 (mod 2) for p ≡ 1 (mod 4).