A few uncaught universal Hermitian forms

Here we include some classical results from the theory of binary hermitian forms which originate from Hermite (1854). We discuss the reduction theory of binary hermitian forms as described for example in Bianchi (1892). Eventually our considerations lead to Humbert’s computation of the covolume of SL(2, Thuong) where Thuong is the ring of integers in an imaginary quadratic number field. The work of Humbert on hermitian forms is contained in his papers (1915), (1919a)—(1919e). It contains an interesting error, we correct it in Section 9.6. We also develop a theory of representation numbers of binary hermitian forms which is analogous to the theory of binary quadratic forms as in Landau (1927).KeywordsPrime NumberPrime IdealZeta FunctionDirichlet SeriesHermitian FormThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

For imaginary quadratic number fields F = <TEX>$\mathbb{Q}(\sqrt{{\varepsilon}p_1{\ldots}p_{t-1}})$</TEX>, where <TEX>${\varepsilon}{\in}$</TEX>{-1,-2} and distinct primes <TEX>$p_i{\equiv}1$</TEX> mod 4, we give condition of 8-ranks of class groups C(F) of F equal to 1 or 2 provided that 4-ranks of C(F) are at most equal to 2. Especially for F = <TEX>$\mathbb{Q}(\sqrt{{\varepsilon}p_1p_2)$</TEX>, we compute densities of 8-ranks of C(F) equal to 1 or 2 in all such imaginary quadratic fields F. The results are stated in terms of congruence relation of <TEX>$p_i$</TEX> modulo <TEX>$2^n$</TEX>, the quartic residue symbol <TEX>$(\frac{p_1}{p_2})4$</TEX> and binary quadratic forms such as <TEX>$p_2^{h+(2_{p_1})/4}=x^2-2p_1y^2$</TEX>, where <TEX>$h+(2p_1)$</TEX> is the narrow class number of <TEX>$\mathbb{Q}(\sqrt{2p_1})$</TEX>. The results are also very useful for numerical computations.

This paper classifies the even unimodular lattices that have a structure as a Hermitian [Formula: see text]-lattice of rank r ≤ 12 for rings of integers in imaginary quadratic number fields K of class number 1. The Hermitian theta series of such a lattice is a Hermitian modular form of weight r for the full modular group, therefore we call them theta lattices. For arbitrary imaginary quadratic fields we derive a mass formula for the principal genus of theta lattices which is applied to show completeness of the classifications.

We strongly develop the relationship between complex hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on complex hyperbolic spaces, especially in dimension $2$. We prove a Mertens' formula for the integer points over a quadratic imaginary number fields $K$ in the light cone of Hermitian forms, as well as an equidistribution theorem of the set of rational points over $K$ in Heisenberg groups. We give a counting formula for the cubic points over $K$ in the complex projective plane whose Galois conjugates are orthogonal and isotropic for a given Hermitian form over $K$, and a counting and equidistribution result for arithmetic chains in the Heisenberg group when their Cygan diameter tends to $0$.

We show that there are finitely many imaginary quadratic number fields for which the class group has exponent 5. Indeed there are finitely many with exponent at most 6. The proof is based on a method of Pierce [Pierce L. B.: A bound for the 3-part of class numbers of quadratic fields by means of the square sieve. Forum Math. 18 (2006), 677–698]. The problem is reduced to one of counting integral points on a certain affine surface. This is tackled using the author's “square-sieve”, in conjunction with estimates for exponential sums. The latter are derived using the q-analogue of van der Corput's method.

We all should learn in our abstract algebra classes that every Euclidean domain R has a minimal Euclidean function, ϕ R . Our short history starts with their introduction via Motzkin’s Lemma and moves onto Lenstra’s categorization of Euclidean functions in imaginary quadratic number fields. We examine computing minimal Euclidean functions in these fields, and apply them in short, easy proofs of standard results. Using the author’s simple formula for ϕ Z [ i ] , the only explicitly computable minimal Euclidean function the author knows for a number field other than Q , we apply the pre-images’ geometry to give a new elementary proof affirming Lenstra’s algebraic description of these sets. Figures illustrate the definitions and arguments.

We prove Manin's conjecture over imaginary quadratic number fields for a cubic surface with a singularity of type E6.

Positive definite binary hermitian forms with finitely many exceptions

Given an indefinite binary quaternionic Hermitian form $f$ with coefficients in a maximal order of a definite quaternion algebra over $\mathbb Q$, we give a precise asymptotic equivalent to the number of nonequivalent representations, satisfying some congruence properties, of the rational integers with absolute value at most $s$ by $f$, as $s$ tends to $+\infty$. We compute the volumes of hyperbolic 5-manifolds constructed by quaternions using Eisenstein series. In the Appendix, V. Emery computes these volumes using Prasad's general formula. We use hyperbolic geometry in dimension 5 to describe the reduction theory of both definite and indefinite binary quaternionic Hermitian forms.

The diophantine equation ax2+by2+cz2 = dxyz with a, b, c, d ∈ ℤ{0} and a, b, c ¦ d has been studied in connection with discrete subgroups of PSL(2, ℝ) ([R2, KR, K, Sch]). Rosenberger and Kern-Isberner have determined the complete set of integral solutions. Further Silverman ([S]) gave a description of the solutions of the equation x2 + y2 + z2 = dxyz, |d| ≥ 3, over orders in quadratic imaginary fields, whereas Bowditch, Maclachlan and Reid studied the equation x2 + y2 + z2 = xyz in order to describe the arithmetic once-punctered torus bundles ([BMR]). In this paper we give a survey of the set of solutions over the ring of integers Ok in quadratic imaginary fields, \(\mathbb{Q}\left( {\sqrt { - k} } \right)\), k > 0 squarefree, of the equation ax2 + by2 + cz2 = dxyz, where the coefficients are chosen as follows: $$\bullet a,b,c,d \in O_k \backslash \{ 0\} ,\left. {a,b,c} \right|d,\left| {\tfrac{d} {{\sqrt {abc} }}} \right| \geqslant 3$$ $$\bullet a,b,c,d \in \mathbb{Z}\backslash \{ 0\} ,\left. {a,b,c} \right|d,1 \leqslant \left| {\tfrac{d} {{\sqrt {abc} }}} \right| < 3, resp. a = b = c = 1, d \in O_k \backslash \{ 0\} ,\left| d \right| < 3.$$

In these lectures we give an introduction to the reduction theory of binary forms starting with quadratic forms with real coefficients, Hermitian forms, and then define the Julia quadratic for any degree n binary form. A survey of a reduction algorithm over ${\mathbb{Z}}$ is described based on recent work of Cremona and Stoll.

Kneser's method of constructing adjacent lattices will be used to determine class numbers of unimodular positive definite hermitian lattices of rank 2 and 3 over rings of integers in some imaginary quadratic fields. The same method will also be applied in order to construct indecomposable unimodular positive definite hermitian lattices of rank 2 and 3 over almost all orders in imaginary quadratic fields, even non-maximal ones. All exceptional cases will be determined explicitly.

V - HERMITIAN FORMS

We study a family of functions defined in a very simple way as sums of powers of binary Hermitian forms with coefficients in the ring of integers of an Euclidean imaginary quadratic field K with discriminant dK. Using these functions we construct a nontrivial cocycle belonging to the space of parabolic cocycles on Euclidean Bianchi groups. We also show that the average value of these functions is related to the special values of L(χdK,s). Using the properties of these functions we give new and computationally efficient formulas for computing some special values of L(χdK,s).

The families of simplest cubic, simplest quartic and simplest sextic fields and the related Thue equations are well known, see G. Lettl, A. Pethő and P. Voutier, Simple families of Thue inequalities, Trans. Amer. Math. Soc. 351 (1999) 1871–1894, On the arithmetic of simplest sextic fields and related Thue equations, in Number Theory: Diophantine, Computational and Algebraic Aspects, eds. K. Győry et al. (de Gruyter, Berlin, 1998), pp. 331–348. The family of simplest cubic Thue equations was already studied in the relative case, over imaginary quadratic fields. In the present paper, we give a similar extension of simplest quartic and simplest sextic Thue equations over imaginary quadratic fields. We explicitly give the solutions of these infinite parametric families of Thue equations over arbitrary imaginary quadratic fields.