Graded rings of Hermitian modular forms with singularities

We study graded rings of meromorphic Hermitian modular forms of degree two whose poles are supported on an arrangement of Heegner divisors. For the group $$\mathrm {SU}_{2,2}({\mathcal {O}}_K)$$ SU 2 , 2 ( O K ) where K is the imaginary-quadratic number field of discriminant $$-d$$ - d , $$d \in \{4, 7,8,11,15,19,20,24\}$$ d ∈ { 4 , 7 , 8 , 11 , 15 , 19 , 20 , 24 } we obtain a polynomial algebra without relations. In particular the Looijenga compactifications of the arrangement complements are weighted projective spaces.

Composition of binary quadratic forms over number fields

In this article, the standard correspondence between the ideal class group of a quadratic number field and the equivalence classes of binary quadratic forms of given discriminant is generalized to any base number field of narrow class number one. The article contains an explicit description of the correspondence. In the case of totally negative discriminants, equivalent conditions are given for a binary quadratic form to be totally positive definite.

Hermitian Theta Series and Maaß Spaces Under the Action of the Maximal Discrete Extension of the Hermitian Modular Group

Let $$\Gamma _n(\mathcal {\scriptstyle {O}}_{\mathbb {K}})$$ Γ n ( O K ) denote the Hermitian modular group of degree n over an imaginary quadratic number field $$\mathbb {K}$$ K and $$\Delta _{n,\mathbb {K}}^*$$ Δ n , K ∗ its maximal discrete extension in the special unitary group $$SU(n,n;\mathbb {C})$$ S U ( n , n ; C ) . In this paper we study the action of $$\Delta _{n,\mathbb {K}}^*$$ Δ n , K ∗ on Hermitian theta series and Maaß spaces. For $$n=2$$ n = 2 we will find theta lattices such that the corresponding theta series are modular forms with respect to $$\Delta _{2,\mathbb {K}}^*$$ Δ 2 , K ∗ as well as examples where this is not the case. Our second focus lies on studying two different Maaß spaces. We will see that the new found group $$\Delta _{2,\mathbb {K}}^*$$ Δ 2 , K ∗ consolidates the different definitions of the spaces.

Higher genus theory

In $1801$, Gauss found an explicit description, in the language of binary quadratic forms, for the $2$-torsion of the narrow class group and dual narrow class group of a quadratic number field. This is now known as Gauss’s genus theory. In this paper, we extend Gauss’s work to the setting of multi-quadratic number fields. To this end, we introduce and parametrize the categories of expansion groups and expansion Lie algebras, giving an explicit description for the universal objects of these categories. This description is inspired by the ideas of Smith [ 16] in his recent breakthrough on Goldfeld’s conjecture and the Cohen–Lenstra conjectures. Our main result shows that the maximal unramified multi-quadratic extension $L$ of a multi-quadratic number field $K$ can be reconstructed from the set of generalized governing expansions supported in the set of primes that ramify in $K$. This provides a recursive description for the group $\textrm{Gal}(L/\mathbb{Q})$ and a systematic procedure to construct the field $L$. A special case of our main result gives an upper bound for the size of $\textrm{Cl}^{+}(K)[2]$.

A Unit Norm Conjecture for Some Real Quadratic Number Fields: A Preliminary Heuristic Investigation

In this paper we make a conjecture about the norm of the fundamental unit, N(e), of some real quadratic number fields that have the form k = Q(√(p1.p2) where p1 and p2 are distinct primes such that pi = 2 or pi ≡ 1 mod 4, i = 1, 2. Our conjecture involves the case where the Kronecker symbol (p1/p2) = 1 and the biquadratic residue symbols (p1/p2)4 = (p2/p1)4 = 1, and is based upon Stevenhagen’s conjecture that if k = Q(√(p1.p2) is any real quadratic number field as above, then P(N(e) = -1)) = 2/3, i.e., the probability density that N(e) = -1 is 2/3. Given Stevenhagen’s conjecture and some theoretical assumptions about the probability density of the Kronecker symbols and biquadratic residue symbols, we establish that if k is as above with (p1/p2) = (p1/p2)4 = (p2/p1)4 = 1, then P(N(e) = -1)) = 1/3, and we support our conjecture with some preliminary heuristic data.

A simple criterion for the class number of a quadratic number field to be one

Let [Formula: see text] be a square-free positive integer, [Formula: see text] if [Formula: see text] and [Formula: see text] otherwise. Let [Formula: see text] and [Formula: see text] be integers, where [Formula: see text] is a prime. Suppose that [Formula: see text] for some integer [Formula: see text]. Suppose that there exist integers [Formula: see text] and [Formula: see text] such that [Formula: see text]. We prove that if [Formula: see text] is [Formula: see text] or a prime for all integers [Formula: see text] with [Formula: see text], then the class number of the field [Formula: see text] is [Formula: see text].

On the Hilbert 2-class field of some quadratic number fields

In this paper, we investigate the cyclicity of the [Formula: see text]-class group of the first Hilbert [Formula: see text]-class field of some quadratic number field whose discriminant is not a sum of two squares. For this, let [Formula: see text] be different prime integers. Put [Formula: see text], and denote by [Formula: see text] its [Formula: see text]-class group and by [Formula: see text] (respectively [Formula: see text]) its first (respectively second) Hilbert [Formula: see text]-class field. Then, we are interested in studying the metacyclicity of [Formula: see text] and the cyclicity of [Formula: see text] whenever the [Formula: see text]-rank of [Formula: see text] is [Formula: see text].

TWISTED TRIPLE PRODUCT -ADIC L-FUNCTIONS AND HIRZEBRUCH–ZAGIER CYCLES

Let $L/F$ be a quadratic extension of totally real number fields. For any prime $p$ unramified in $L$, we construct a $p$-adic $L$-function interpolating the central values of the twisted triple product $L$-functions attached to a $p$-nearly ordinary family of unitary cuspidal automorphic representations of $\text{Res}_{L\times F/F}(\text{GL}_{2})$. Furthermore, when $L/\mathbb{Q}$ is a real quadratic number field and $p$ is a split prime, we prove a $p$-adic Gross–Zagier formula relating the values of the $p$-adic $L$-function outside the range of interpolation to the syntomic Abel–Jacobi image of generalized Hirzebruch–Zagier cycles.