Quadratic forms over Z that represent all positive integers are called universal. Starting with Ramanujan, 54 universal quaternary quadratic forms without cross product terms were discovered. The form that is the sum of four squares was proved universal by Hurwitz using a special ring of quaternions. Here seven other quaternary quadratic forms are shown universal by investigation of appropriate rings of quaternions.
Recently the author used certain quaternion orders to demonstrate the universality of some quaternary quadratic forms. Here a further study is done on one of these orders analogous to Hurwitz's proof of the formula for the number of representations of a positive integer by sums of four squares using quaternions. The result is the formula for the number of representations of a positive integer by one of the above mentioned quaternary quadratic forms.
We study "quadratic periods" on quaternionic Shimura varieties and formulate an integral refinement of Shimura's conjecture regarding Petersson inner products of automorphic forms that are related by the Jacquet-Langlands correspondence. The main result is that this integral refinement is implied by another conjecture (Conjecture D below) regarding integrality of theta lifts between certain quaternionic unitary groups.
Rigid meromorphic cocycles were introduced by Darmon and Vonk as a conjectural $p$-adic extension of the theory of singular moduli to real quadratic base fields. They are certain cohomology classes of $\mathrm{SL}_2(\mathbb{Z}[1/p])$ which can be evaluated at real quadratic irrationalities and the values thus obtained are conjectured to lie in algebraic extensions of the base field. In this article we present a similar construction of cohomology casses in which $\mathrm{SL}_2(\mathbb{Z}[1/p])$ is replaced by an order in an indefinite quaternion algebra over a totally real number field $F$. The…
Darmon points on p-adic tori and Jacobians of Shimura curves over Q were introduced in previous joint works with Rotger as generalizations of Darmon's Stark-Heegner points. In this article we study the algebraicity over extensions of a real quadratic field K of the projections of Darmon points to elliptic curves. More precisely, we prove that linear combinations of Darmon points on elliptic curves weighted by certain genus characters of K are rational over the predicted genus fields of K. This extends to an arbitrary quaternionic setting the main theorem on the rationality of Stark-Heegner poi…
In a recent paper, Castella and Hsieh proved results for Selmer groups associated with Galois representations attached to newforms twisted by Hecke characters of an imaginary quadratic field. These results are obtained under the so-called Heegner hypothesis that the imaginary quadratic field satisfies with respect to the level of the modular form. In particular, Castella and Hsieh prove the rank 0 case of the Bloch-Kato conjecture for $L$-functions of modular forms in their setting. The key point of the work of Castella and Hsieh is a remarkable link between generalized Heegner cycles and $p$-…
We develop a theory of $p$-adic continued fractions for a quaternion algebra $B$ over $\mathbb Q$ ramified at a rational prime $p$. Many properties holding in the commutative case can be proven also in this setting. In particular, we focus our attention on the characterization of elements having a finite continued fraction expansion. By means of a suitable notion of quaternionic height, we prove a criterion for finiteness. Furthermore, we draw some consequences about the solutions of a family of quadratic polynomial equations with coefficients in $B$.
In a series of papers, published in Mathematische Annalen, Bianchi and Blumenthal introduced the notions of Bianchi orbifolds and Hilbert-Blumnethal surfaces as generalizations of modular curves associated to quadratic fields. In this paper, in the same spirit, and following a similar line of reasoning, we introduce the concept of Bianchi and Hilbert-Blumenthal quaternionic orbifolds as generalizations of the Lipschitz and Hurwitz quaternionic modular orbifolds defined recently by Díaz, Vlacci and the first author. In particular, we describe the cusp cross-sections of the Hilbert-Blumenthal qu…
For a set $S$ of (positive definite and integral) quadratic forms with bounded rank, a quadratic form $f$ is called $S$-universal if it represents all quadratic forms in $S$. A subset $S_0$ of $S$ is called an $S$-universality criterion set if any $S_0$-universal quadratic form is $S$-universal. We say $S_0$ is minimal if there does not exist a proper subset of $S_0$ that is an $S$-universality criterion set. In this article, we study various properties of minimal universality criterion sets. In particular, we show that for `most' binary quadratic forms $f$, minimal $S$-universality criterion …
Fix a quadratic order over the ring of integers. An embedding of the quadratic order into a quaternionic order naturally gives an integral binary hermitian form over the quadratic order. We show that, in certain cases, this correspondence is a discriminant preserving bijection between the isomorphism classes of embeddings and integral binary hermitian forms.
A positive definite integral quadratic form is said to be almost (primitively) universal if it (primitively) represents all but at most finitely many positive integers. In general, almost primitive universality is a stronger property than almost universality. The two main results of this paper are: 1) every primitively universal form nontrivially represents zero over every ring of p-adic integers, and 2) every almost universal form in five or more variables is almost primitively universal.
Using the methods developed for the proof that the 2-universality criterion is unique, we partially characterize criteria for the n-universality of positive-definite integer-matrix quadratic forms. We then obtain the uniqueness of Oh's 8-universality criterion as an application of our characterization results.
In this note, we give simple examples of sets S of quadratic forms that have minimal S-universality criteria of multiple cardinalities. This answers a question of Kim, Kim, and Oh in the negative.
Let $K$ be an imaginary quadratic field. Modular forms for GL(2) over $K$ are known as Bianchi modular forms. Standard modularity conjectures assert that every weight 2 rational Bianchi newform has either an associated elliptic curve over $K$ or an associated abelian surface with quaternionic multiplication over $K$. We give explicit evidence in the way of examples to support this conjecture in the latter case. Furthermore, the quaternionic surfaces given correspond to genuine Bianchi newforms, which answers a question posed by J. Cremona as to whether this phenomenon can happen.
A quadratic form over a non-archimedian local field of characteristic zero $F$ is called universal if it is integral and it represents all non-zero integers of $F$. Xu Fei and Zhang Yang determined all universal quadratic forms in the case when $F$ is non-dyadic. In the more complicated dyadic case, when $F$ is a finite extension of $\mathbb Q_2$, they solved the same problem only in the ternary case. In our paper we solve this problem in the general case. Our result is given in terms of BONGs (bases of norm generators) but in section 3 of the paper we translate (without a proof) our result in…
Centro de Matemática da Universidade do Porto, Faculdade de Ciências da Universidade do Porto（波尔图大学数学中心，波尔图大学理学院）；Centro de Matemática, Universidade do Minho — Polo CMAT-UTAD Universidade de Trás-os-Montes e Alto Douro（米尼奥大学数学中心 — UTAD波尔图CMAT-UTAD，特拉斯-奥蒙蒂斯和阿尔托杜罗大学）
Following Bhargava and Hanke's celebrated 290-theorem, we prove a universality theorem for all positive-definite integer-valued quadratic forms that represent all positive integers coprime to $3$. In particular, if a positive-definite quadratic form represents all positive integers coprime to $3$ and $\leq 290$, then it represents all positive integers coprime to $3$. We use similar methods to those used by Rouse to prove (assuming GRH) that a positive-definite quadratic form representing every odd integer between $1$ and $451$ represents all positive odd integers.
Let $a,b,c,d,e,f\in\mathbb N$ with $a\ge c\ge e>0$, $b\le a$ and $b\equiv a\pmod2$, $d\le c$ and $d\equiv c\pmod2$, $f\le e$ and $f\equiv e\pmod2$. If any nonnegative integer can be written as $x(ax+b)/2+y(cy+d)/2+z(ez+f)/2$ with $x,y,z\in\mathbb Z$, then the ordered tuple $(a,b,c,d,e,f)$ is said to be universal over $\mathbb Z$. Recently, Z.-W. Sun found all candidates for such universal tuples over $\mathbb Z$. In this paper, we use the theory of ternary quadratic forms to show that 44 concrete tuples $(a,b,c,d,e,f)$ in Sun's list of candidates are indeed universal over $\mathbb Z$. For exam…
Earnest and Khosravani, Iwabuchi, and Kim and Park recently gave a complete classification of the universal binary Hermitian forms. We give a unified proof of the universalities of these Hermitian forms, relying primarily on Ramanujan's list of universal quadratic forms and on the Bhargava-Hanke 290-Theorem. Our methods bypass nearly all of the ad hoc universality arguments required in the original classification.
A positive definite quadratic form is called perfect, if it is uniquely determined by its arithmetical minimum and the integral vectors attaining it. In this self-contained survey we explain how to enumerate perfect forms in $d$ variables up to arithmetical equivalence and scaling. We put an emphasis on practical issues concerning computer assisted enumerations. For the necessary theory of Voronoi we provide complete proofs based on Ryshkov polyhedra. This allows a very natural generalization to $T$-perfect forms, which are perfect with respect to a linear subspace $T$ in the space of quadrati…
