International Journal of Number Theory

Let Hⁿ be the upper half-space model of the n-dimensional hyperbolic space. For n=3, Hermitian points in the Markov spectrum of the extended Bianchi group B_d are introduced for any d. If ν is a Hermitian point in the spectrum, then there is a set of extremal geodesics in H³ with diameter 1/ν, which depends on one continuous parameter. It is shown that ν² ≤ |D|/24 for any imaginary quadratic field with discriminant D, whose ideal-class group contains no cyclic subgroup of order 4, and in many other cases. Similarly, in the case of n = 4, if ν is a Hermitian point in the Markov spectrum for SV(Z⁴), some discrete group of isometries of H⁴, then the corresponding set of extremal geodesics depends on two continuous parameters.

Let H³ be the upper half-space model of the three-dimensional hyperbolic space. For certain cocompact Fuchsian subgroups Γ of an extended Bianchi group B_d, the extremality of the axis of hyperbolic F ∈ Γ in H₃ with respect to Γ implies its extremality with respect to B_d. This reduction is used to obtain sharp lower bounds for the Hurwitz constants and lower bounds for the highest limit points in the Markov spectra of B_d for some d < 1000. In particular, such bounds are found for all non-Euclidean class one imaginary quadratic fields. The Hurwitz constants for the imaginary quadratic fields with discriminants -120 and -132 are given. The second minima are also indicated for these fields.

For a prime q ≡ 3 (mod 4) and the character , we consider the subspace of the space of holomorphic cusp forms of weight one, level q and character χ that is spanned by forms that correspond to Galois representations of octahedral type. We prove that this subspace has dimension bounded by upto multiplication by a constant that depends only on ε.

We introduce a new combinatorial proof of the Lebesgue identity which allows us to find a new finite form of the identity. Using this new finite form we are able to make new observations about special cases of the Lebesgue identity, namely the "little" Göllnitz theorems and Sylvester's identity.

Let G be a finite, non-trivial Abelian group of exponent m, and suppose that B₁,…, B_k are generating subsets of G. We prove that if k > 2m ln log₂ |G|, then the multiset union B₁ ∪ B_k forms an additive basis of G; that is, for every g ∈ G, there exist A₁ ⊆ B₁,…, A_k ⊆ B_k such that . This generalizes a result of Alon, Linial and Meshulam on the additive bases conjecture.

As another step towards proving the conjecture, in the case where B₁,…, B_k are finite subsets of a vector space, we obtain lower-bound estimates for the number of distinct values, attained by the sums of the form , where A_i vary over all subsets of B_i for each i = 1,…, k.

Finally, we establish a surprising relation between the additive bases conjecture and the problem of covering the vertices of a unit cube by translates of a lattice, and present a reformulation of (the strong form of) the conjecture in terms of coverings.

We show the existence of abelian surfaces over ℚ_p having good reduction with supersingular special fiber whose associated p-adic Galois module is not semisimple.

In this paper, we study the divisibility of the function a(n) defined by . In particular, we prove certain "Ramanujan type congruences" for a(n) modulo powers of 3.

We consider a class of generating functions analogous to the generating function of the partition function and establish a bound on the primes ℓ for which their coefficients c(n) obey congruences of the form c(ℓn + a) ≡ 0 (mod ℓ). We apply this result to obtain a complete characterization of the congruences of the same form that the sequences c_N(n) satisfy, where c_N(n) is defined by . This last result answers a question of H.-C. Chan.

Zhao established the following harmonic congruence for prime p > 3:

In this short paper, we extend results of Kohnen and Sengupta on the sign of eigenvalues of Siegel cuspforms. We show that their bound for the first negative Hecke eigenvalue of a genus 2 Siegel cuspform of level 1 extends to the case of level N > 1. We also discuss the signs of Hecke eigenvalues of CAP forms.

The n × n circulant matrix associated with the polynomial (with d < n) is the one with first row (a₀ ⋯ a_d 0 ⋯ 0). The problem as to when such circulants are unimodular arises in the theory of cyclically presented groups and leads to the following question, previously studied by Odoni and Cremona: when is Res(f(t), tⁿ-1) = ±1? We give a complete answer to this question for trinomials f(t) = t^m ± t^k ± 1. Our main result was conjectured by the author in an earlier paper and (with two exceptions) implies the classification of the finite Cavicchioli–Hegenbarth–Repovš generalized Fibonacci groups, thus giving an almost complete answer to a question of Bardakov and Vesnin.

We obtain a lower bound on the multiplicative order of Gauss periods which generate normal bases over finite fields. This bound improves the previous bound of von zur Gathen and Shparlinski.

Define e_n(t) = {t/n}. Let d_N denote the distance in L²(0, ∞; t^-2dt) between the indicator function of [1, ∞[ and the vector space generated by e₁,…, e_N. A theorem of Báez-Duarte states that the Riemann hypothesis (RH) holds if and only if d_N → 0 when N → ∞. Assuming RH, we prove the estimate

We specialize a problem studied by Elliott, the behavior of arbitrary sequences a_p of complex numbers on residue classes to prime moduli to the case a_p = e(αp). For these special cases, we obtain under certain additional conditions improvements on Elliott's results.

We generalize Gel'fond's criterion for algebraic independence to the context of a sequence of polynomials whose first derivatives take small values on large subsets of a fixed subgroup of ℂ, instead of just one point (one extension deals with a subgroup of ℂ^×).