On some questions related to the Gauss conjecture for function fields

We show that, for any finite field Fq, there exist infinitely many real quadratic function fields over Fq such that the numerator of their zeta function is a separable polynomial. As pointed out by Anglès, this is a necessary condition for the existence, for any finite field Fq, of infinitely many real function fields over Fq with ideal class number one (the so-called Gauss conjecture for function fields). We also show conditionally the existence of infinitely many real quadratic function fields over Fq such that the numerator of their zeta function is an irreducible polynomial.     

On the tensor rank of the multiplication in the finite fields

First, we prove the existence of certain types of non-special divisors of degree g−1 in the algebraic function fields of genus g defined over Fq. Then, it enables us to obtain upper bounds of the tensor rank of the multiplication in any extension of quadratic finite fields Fq by using Shimura and modular curves defined over Fq. From the preceding results, we obtain upper bounds of the tensor rank of the multiplication in any extension of certain non-quadratic finite fields Fq, notably in the case of F₂. These upper bounds attain the best asymptotic upper bounds of Shparlinski-Tsfasman-Vladut [I.E. Shparlinski, M.A. Tsfasman, S.G. Vladut, Curves with many points and multiplication in finite fields, in: Lecture Notes in Math., vol. 1518, Springer-Verlag, Berlin, 1992, pp. 145-169].     

Class number indivisibility for quadratic function fields

Let M?5. For any odd prime power q and any prime ??q, we show that there are at least pairwise coprime D∈Fq[T] which are square-free and of odd degree ?M, such that ? does not divide the class number of the complex quadratic functions fields .     

Reflection theorem for divisor class groups of relative quadratic function fields

We present the reflection theorem for divisor class groups of relative quadratic function fields. Let K be a global function field with constant field Fq. Let L₁ be a quadratic geometric extension of K and let L₂ be its twist by the quadratic constant field extension of K. We show that for every odd integer m that divides q+1 the divisor class groups of L₁ and L₂ have the same m-rank.     

Pesenti-Szpiro inequality for optimal elliptic curves

We study Pesenti-Szpiro inequality in the case of elliptic curves over F_q(t) which occur as subvarieties of Jacobian varieties of Drinfeld modular curves. In general, we obtain an upper-bound on the degrees of minimal discriminants of such elliptic curves in terms of the degrees of their conductors and q. In the special case when the level is prime, we bound the degrees of discriminants only in terms of the degrees of conductors. As a preliminary step in the proof of this latter result we generalize a construction (due to Gekeler and Reversat) of 1-dimensional optimal quotients of Drinfeld Jacobians.     

Using Stepanov's method for exponential sums involving rational functions

For any additive character ψ and multiplicative character χ on a finite field F_q, and rational functions f,g in F_q(x), we show that the elementary Stepanov-Schmidt method can be used to obtain the corresponding Weil bound for the sum ∑_x∈Fq?Sχ(g(x))ψ(f(x)) where S is the set of the poles of f and g. We also determine precisely the number of characteristic values ω_i of modulus q^1/2 and the number of modulus 1.     

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.     

Some formulas for the coefficients of Drinfeld modular forms

We obtain some formulas for t-expansion coefficients of meromorphic Drinfeld modular forms for GL₂(F_q[T]). Let j(z) be the Drinfeld modular invariant. As an application we show that the values of j(z) at points in the divisor of Drinfeld modular forms for GL₂(F_q[T]) are algebraic over F_q(T).     

Connection-graph and iteration-graph of monotone boolean functions

In this paper, we establish some relationships between the circuits of the connection-graph G_C(F), and the circuits of theiteration-graph G₁(F), of a monotone boolean function F. We first show that if G₁(F) contains an element circuit of length multiple of p? {2,3}, then G_C(F) contains an elementary circuit of length multiple of p; then we prove that if G_C(F) is a subgraph of a caterpillar, then G₁(F) is a subgraph of a tree; at last we exhibit an infinite family of monotone boolean functions {F_n; n = 2 × 5^q, q ≥ 1} such that any G_C(F_n) is a subgraph of a tree, and G₁(F_n) contains a circuit of length 2^q+1, i.e., of the order $\text{n}{}^{\mathrm{<mtext>log</mtext><mtext>2</mtext><mtext>log5</mtext>}}$.     

Maximal class numbers of CM number fields

Fix a totally real number field F of degree at least 2. Under the assumptions of the generalized Riemann hypothesis and Artin's conjecture on the entirety of Artin L-functions, we derive an upper bound (in terms of the discriminant) on the class number of any CM number field with maximal real subfield F. This bound is a refinement of a bound established by Duke in 2001. Under the same hypotheses, we go on to prove that there exist infinitely many CM-extensions of F whose class numbers essentially meet this improved bound and whose Galois groups are as large as possible.     

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 additive polynomials and certain maximal curves

We show that a maximal curve over F_q² given by an equation A(X)=F(Y), where A(X)∈F_q²[X] is additive and separable and where F(Y)∈F_q²[Y] has degree m prime to the characteristic p, is such that all roots of A(X) belong to F_q². In the particular case where F(Y)=Y^m, we show that the degree m is a divisor of q+1.     

On subword decomposition and balanced polynomials

Let H(x) be a monic polynomial over a finite field F=GF(q). Denote by Na(n) the number of coefficients in Hn which are equal to an element a∈F, and by G the set of elements a∈F^× such that Na(n)>0 for some n. We study the relationship between the numbers (Na(n))_a∈G and the patterns in the base q representation of n. This enables us to prove that for “most” n's we have Na(n)≈Nb(n), a,b∈G. Considering the case H=x+1, we provide new results on Pascal's triangle modulo a prime. We also provide analogous results for the triangle of Stirling numbers of the first kind.     

Surfaces and Curves Corresponding to the Solutions Generated from Periodic“Seed”of NLS Equation

The solutions q [n] generated from a periodic "seed" q = cei(as+bt) of the nonlinear Schrdinger(NLS) by n-fold Darboux transformation is represented by determinant.Furthermore,the s-periodic solution and t-periodic solution are given explicitly by using q [1].The curves and surfaces(F1,F2,F3) associated with q [n] are given by means of Sym formula.Meanwhile,we show periodic and asymptotic properties of these curves.     

On the restriction of cross characteristic representations of 2 F 4(q) to proper subgroups

We prove that the restriction of any nontrivial representation of the Ree groups ² F ₄(q), q = 2²ⁿ⁺¹ ≥ 8 in odd characteristic to any proper subgroup is reducible. We also determine all triples (K, V, H) such that ${K \in \{^2F_4(2), ^2F_4(2)'\} }We prove that the restriction of any nontrivial representation of the Ree groups ² F ₄(q), q = 2²ⁿ⁺¹ ≥ 8 in odd characteristic to any proper subgroup is reducible. We also determine all triples (K, V, H) such that K ? {²F₄(2), ²F₄(2)￠}{K \in \{^2F_4(2), ^2F_4(2)'\} } , H is a proper subgroup of K, and V is a representation of K in odd characteristic restricting absolutely irreducibly to H.     

Maximum IPP Codes of Length 3

Let Q be an alphabet with q elements. For any code C over Q of length n and for any two codewords a = (a ₁, . . . , a _n ) and b = (b ₁, . . . , b _n ) in C, let ${D({\bf a, b}) = \{(x_1, . . . , x_n) \in {Q^n} : {x_i} \in \{a_i, b_i\}\,{\rm for}\,1 \leq i \leq n\}}Let Q be an alphabet with q elements. For any code C over Q of length n and for any two codewords a = (a ₁, . . . , a _n ) and b = (b ₁, . . . , b _n ) in C, let D(a, b) = {(x₁, . . . , x_n) ? Qⁿ : x_i ? {a_i, b_i} for 1 ￡ i ￡ n}{D({\bf a, b}) = \{(x_1, . . . , x_n) \in {Q^n} : {x_i} \in \{a_i, b_i\}\,{\rm for}\,1 \leq i \leq n\}}. Let C^* = è_{a, b ? C}D(a, b){C^* = {{\bigcup}_{\rm {a,\,b}\in{C}}}D({\bf a, b})}. The code C is said to have the identifiable parent property (IPP) if, for any x ? C^*{{\rm {\bf x}} \in C^*}, ?_{x ? D(a, b)}{a, b} 1 ?{{\bigcap}_{{\rm x}{\in}D({\rm a,\,b})}\{{\bf a, b}\}\neq \emptyset} . Codes with the IPP were introduced by Hollmann et al [J. Combin. Theory Ser. A 82 (1998) 21–133]. Let F(n, q) = max{|C|: C is a q-ary code of length n with the IPP}.T? and Safavi-Naini [SIAM J. Discrete Math. 17 (2004) 548–570] showed that 3q + 6 - 6 é?{q+1}ù ￡ F(3, q) ￡ 3q + 6 - é6 ?{q+1}ù{3q + 6 - 6 \lceil\sqrt{q+1}\rceil \leq F(3, q) \leq 3q + 6 - \lceil 6 \sqrt{q+1}\rceil}, and determined F (3, q) precisely when q ≤ 48 or when q can be expressed as r ² + 2r or r ² + 3r +2 for r ≥ 2. In this paper, we establish a precise formula of F(3, q) for q ≥ 24. Moreover, we construct IPP codes of size F(3, q) for q ≥ 24 and show that, for any such code C and any x ? C^*{{\rm {\bf x}} \in C^*}, one can find, in constant time, a ? C{{\rm {\bf a}} \in C} such that if x ? D (c, d){{\rm {\bf x}} \in D ({\bf c, d})} then a ? {c, d}{{\rm {\bf a}} \in \{{\rm {\bf c, d}}\}}.     

Arithmetical properties of linear recurrent sequences

Let F(z)∈R[z] be a polynomial with positive leading coefficient, and let α>1 be an algebraic number. For r=degF>0, assuming that at least one coefficient of F lies outside the field Q(α) if α is a Pisot number, we prove that the difference between the largest and the smallest limit points of the sequence of fractional parts {F(n)αn}_n=1,2,3,… is at least 1/?(Pr+1), where ? stands for the so-called reduced length of a polynomial.     

Path-fan Ramsey numbers

For two given graphs F and H, the Ramsey number R(F,H) is the smallest positive integer p such that for every graph G on p vertices the following holds: either G contains F as a subgraph or the complement of G contains H as a subgraph. In this paper, we study the Ramsey numbers R(P_n,F_m), where P_n is a path on n vertices and F_m is the graph obtained from m disjoint triangles by identifying precisely one vertex of every triangle (F_m is the join of K₁ and mK₂). We determine the exact values of R(P_n,F_m) for the following values of n and m: 1?n?5 and m?2; n?6 and 2?m?(n+1)/2; 6?n?7 and m?n-1; n?8 and n-1?m?n or ((q·n-2q+1)/2?m?(q·n-q+2)/2 with 3?q?n-5) or m?(n-3)²/2; odd n?9 and ((q·n-3q+1)/2?m?(q·n-2q)/2 with 3?q?(n-3)/2) or ((q·n-q-n+4)/2?m?(q·n-2q)/2 with (n-1)/2?q?n-5). Moreover, we give nontrivial lower bounds and upper bounds for R(P_n,F_m) for the other values of m and n.     

Further examples of maximal curves

It is shown that the curve over F_q²ⁿ with n≥3 odd, that generalizes Serre’s curve y⁴+y=x³ over F₆₄, is also maximal. We also investigate a family of maximal curves over F_q²ⁿ and provide isomorphisms between these curves.     

Counting hyperelliptic curves

We find a closed formula for the number hyp(g) of hyperelliptic curves of genus g over a finite field k=Fq of odd characteristic. These numbers hyp(g) are expressed as a polynomial in q with integer coefficients that depend on g and the set of divisors of q−1 and q+1. As a by-product we obtain a closed formula for the number of self-dual curves of genus g. A hyperelliptic curve is defined to be self-dual if it is k-isomorphic to its own hyperelliptic twist.