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Let L ( s , χ ) be the Dirichlet L-function associated to a non-principal primitive Dirichlet character χ defined mod q , where q ≥ 3 . We prove, under the assumption of the Generalised Riemann Hypothesis, the validity of estimates given by Lamzouri, Li, and Soundararajan on | L ( 1 , χ ) | . As a corollary, we have that similar estimates hold for the class number of the imaginary quadratic field Q ( − q ) , q ≥ 5 .

Let $E$ be an elliptic curve over an imaginary quadratic field $K$, and $p$ be an odd prime such that the residual representation $E[p]$ is reducible. The $\mu$-invariant of the fine Selmer group of $E$ over the anticyclotomic $\mathbb{Z}_p$-extension of $K$ is studied. We do not impose the Heegner hypothesis on $E$, thus allowing certain primes of bad reduction to decompose infinitely in the anticyclotomic $\mathbb{Z}_p$-extension. It is shown that the fine $\mu$-invariant vanishes if certain explicit conditions are satisfied. Further, a partial converse is proven., Comment: 16 pages, comments welcome

In this paper, we investigate the properties of well-rounded twists of a given ideal lattice of an imaginary quadratic field $K$. We show that every ideal lattice $I$ of $K$ has at least one well-rounded twist lattice. Moreover, we provide an explicit algorithm to compute all well-rounded twists of $I$., 24 pages

We compare the homology of a congruence subgroup Gamma of GL_2(Z) with coefficients in the Steinberg modules over Q and over E, where E is a real quadratic field. If R is any commutative base ring, the last connecting homomorphism psi_{Gamma,E} in the long exact sequence of homology stemming from this comparison has image in H_0(Gamma, St(Q^2;R)) generated by classes z_\beta indexed by beta in E \ Q. We investigate this image. When R=C, H_0(Gamma, St(Q^2;C)) is isomorphic to a space of classical modular forms of weight 2, and the image lies inside the cuspidal part. In this case, z_beta is closely related to periods of modular forms over the geodesic in the upper half plane from beta to its conjugate beta'. Assuming GRH we prove that the image of $\psi_{\Gamma,E}$ equals the entire cuspidal part. When R=Z, we have an integral version of the situation. We define the cuspidal part of the Steinberg homology, H_0^cusp(Gamma, St(Q^2;Z)). Assuming GRH we prove that for any congruence subgroup, psi_{Gamma,E} always has finite index in H_0^cusp(Gamma, St(Q^2;Z)), and if Gamma=Gamma_1(N)^pm or \Gamma_1(N), then the image is all of H_0^cusp(Gamma, St(Q^2;Z)). If Gamma=Gamma_0(N)^pm or Gamma_0(N), we prove (still assuming GRH) an upper bound for the size of H_0^cusp(Gamma, St(Q^2;Z))/image(psi_{Gamma,E}). We conjecture that the results in this paragraph are true unconditionally. We also report on extensive computations of the image of psi_{Gamma,E} that we made for Gamma=Gamma_0(N)^pm and Gamma=Gamma_0(N). Based on these computations, we believe that the image of psi_{Gamma,E} is not all of H_0^cusp(Gamma, St(Q^2;Z)) for these groups, for general N., Comment: 37 pages, 2 tables

Let $E(s, Q)$ be the Epstein zeta function attached to a positive definite quadratic form of discriminant $D, Comment: 13 pages. The proof has been changed to avoid an issue with a certain volume in $\mathbb{R}^{2J-1}$ that we could not compute. The main result remains essentially the same. The only difference is that the power of $\log\log T$ in the error term is replaced by a power of $\exp(\sqrt{\log\log T})$ (but the power of $\log T$ remains the same)

The exceptional zero phenomenon has been widely studied in the realm of $p$-adic $L$-functions, where the starting point lies in the foundational work of Mazur, Tate and Teitelbaum. This phenomenon also appears in the study of Euler systems, which comes as no surprise given the interaction between these two settings. When this occurs, one is led to study higher order derivatives of the Euler system in order to extract the arithmetic information which is usually encoded in the explicit reciprocity laws. In this work, we focus on the elliptic units of an imaginary quadratic field and study this exceptional zero phenomenon, proving an explicit formula relating the logarithm of a {\it derived} elliptic unit either to special values of Katz's two variable $p$-adic $L$-function or to its derivatives. Further, we interpret this fact in terms of an $\mathcal L$-invariant, and relate this result to other approaches to the exceptional zero phenomenon concerning Heegner points and Beilinson--Flach elements., Comment: To appear in Rev. Mat. Iberoam

We compute the diagonal restriction of the first derivative with respect to the weight of a p-adic family of Hilbert modular Eisenstein series attached to a general (odd) character of the narrow class group of a real quadratic field, and express the Fourier coefficients of its ordinary projection in terms of the values of a distinguished rigid analytic cocycle in the sense of Darmon and Vonk (Duke Math J, to appear, 2020) at appropriate real quadratic points of Drinfeld’s p-adic upper half-plane. This can be viewed as the p-adic counterpart of a seminal calculation of Gross and Zagier (J Reine Angew Math 355:191–220, 1985, §7) which arose in their “analytic proof” of the factorisation of differences of singular moduli, and whose inspiration can be traced to Siegel’s proof of the rationality of the values at negative integers of the Dedekind zeta function of a totally real field. Our main identity enriches the dictionary between the classical theory of complex multiplication and its extension to real quadratic fields based on RM values of rigid meromorphic cocycles, and leads to an expression for the p-adic logarithms of Gross–Stark units and Stark–Heegner points in terms of the first derivatives of certain twisted Rankin triple product p-adic L-functions.

The degeneracies of single-centered dyonic $\frac14$-BPS black holes (BH) in Type II string theory on K3$\times T^2$ are known to be coefficients of certain mock Jacobi forms arising from the Igusa cusp form $\Phi_{10}$. In this paper we present an exact analytic formula for these BH degeneracies purely in terms of the degeneracies of the perturbative $\frac12$-BPS states of the theory. We use the fact that the degeneracies are completely controlled by the polar coefficients of the mock Jacobi forms, using the Hardy-Ramanujan-Rademacher circle method. Here we present a simple formula for these polar coefficients as a quadratic function of the $\frac12$-BPS degeneracies. We arrive at the formula by using the physical interpretation of polar coefficients as negative discriminant states, and then making use of previous results in the literature to track the decay of such states into pairs of $\frac12$-BPS states in the moduli space. Although there are an infinite number of such decays, we show that only a finite number of them contribute to the formula. The phenomenon of BH bound state metamorphosis (BSM) plays a crucial role in our analysis. We show that the dyonic BSM orbits with $U$-duality invariant $\Delta, Comment: 58 pages, 9 figures, 3 tables

We study the $1$- or $2$-level density of families of $L$-functions for Hecke--Maass forms over an imaginary quadratic field $F$. For test functions whose Fourier transform is supported in $\left(-\frac 32, \frac 32\right)$, we prove that the $1$-level density for Hecke--Maass forms over $F$ of square-free level $\mathfrak{q}$, as $\mathrm{N}(\mathfrak{q})$ tends to infinity, agrees with that of the orthogonal random matrix ensembles. For Hecke--Maass forms over $F$ of full level, we prove similar statements for the $1$- and $2$-level densities, as the Laplace eigenvalues tends to infinity.

Let K be an imaginary quadratic field whose discriminant is congruent to one modulo 8 and O be the ring of integers of K. Let Γ denote the group S L ( 2 , O ) which acts discontinuously on the upper half space H. In this paper, we study a homomorphism φ : Γ → Z obtained from a branch of the logarithm of a theta function on H which is automorphic with respect to Γ and does not vanish on H. In particular, we determine explicitly the decomposition φ = φ c + φ e of φ into the cusp part φ c and the Eisenstein part φ e , and prove a congruence conjectured by Sczech [14] between φ and φ e modulo 8 under an assumption on the 2-divisibility of a certain L-value.

The investigation of arithmetic properties of the values of the generalized hypergeometric functions is often carried out by means of known in the theory of transcendental numbers Siegel’s method. The most general results in this field have been obtained precisely by this method. But the possibilities of Siegel’s method in case of hypergeometric functions with irrational parameters are restricted. This is connected with the fact that such hypergeometric functions are not E-functions and for that reason one is unable to construct linear approximating form with large order of zero by means of pigeonhole method. To consider problems connected with the investigation of arithmetic properties of the values of hypergeometric functions with irrational parameters it is possible in some cases to use the method based on the effective construction of linear approximating form but the possibilities of this method are also limited because of the absence of too general effective constructions. There are some difficulties also in the cases when such constructions are available. The peculiarities of these constructions often hinder the realization of arithmetic part of the method. For that reason of some interest are situations when one is able to realize the required investigation by means of specific properties of concrete functions. Sometimes it is possible to choose the parameters of the functions under consideration in such a way that one receives the possibility to overcome the difficulties of the general case. In this paper we consider hypergeometric function of a special kind and its derivatives. By means of effective construction it is possible not only to prove linear independence of the values of this function and its derivatives over some imaginary quadratic field but also to obtain corresponding quantitative result in the form of the estimation the modulus of the linear form in the aforesaid values.

Let p > 3 be an odd prime, p ≡ 3 mod 4 and let π + , π − be the pair of cuspidal representations of S L 2 ( F p ) . It is well known by Hecke that the difference m π + − m π − in the multiplicities of these two irreducible representations occurring in the space of weight 2 cusps forms with respect to the principal congruence subgroup Γ ( p ) , equals the class number h ( − p ) of the imaginary quadratic field Q ( − p ) . We extend this result to all fundamental discriminants −D of imaginary quadratic fields Q ( − D ) and prove that an alternating sum of multiplicities of certain irreducibles of S L 2 ( Z / D Z ) is an explicit multiple, up to a sign and a power of 2, of either the class number h ( − D ) or of the sums h ( − D ) + h ( − D / 2 ) , h ( − D ) + 2 h ( − D / 2 ) ; the last two possibilities occur in some of the cases when D ≡ 0 mod 8 . The proof uses the holomorphic Lefschetz number.

In this paper, we give an explicit lower bound for the class number of real quadratic field ℚd, where d=n2+4 is a square-free integer, using ωn which is the number of odd prime divisors of n.

The main difficulty one has to deal with while investigating arithmetic nature of the values of the generalized hypergeometric functions with irrational parameters consists in the fact that the least common denominator of several first coefficients of the corresponding power series increases too fast with the growth of their number. The last circumstance makes it impossible to apply known in the theory of transcendental numbers Siegel’s method for carrying out the above mentioned investigation. The application of this method implies usage of pigeon-hole principle for the construction of a functional linear approximating form. This construction is the first step in a long and complicated reasoning that leads ultimately to the required arithmetic result. The attempts to apply pigeon-hole principle in case of functions with irrational parameters encounters insurmountable obstacles because of the aforementioned fast growth of the least common denominator of the coefficients of the corresponding Taylor series. Owing to this difficulty one usually applies effective construction of the linear approximating form (or a system of such forms in case of simultaneous approximations) for the functions with irrational parameters. The effectively constructed form contains polynomials with algebraic coefficients and it is necessary for further reasoning to obtain a satisfactory upper estimate of the modulus of the least common denominator of these coefficients. The known estimates of this type should be in some cases improved. This improvement is carried out by means of the theory of divisibility in quadratic fields. Some facts concerning the distribution of the prime numbers in arithmetic progression are also made use of. In the present paper we consider one of the versions of effective construction of the simultaneous approximations for the hypergeometric function of the general type and its derivatives. The least common denominator of the coefficients of the polynomials included in these approximations is estimated subsequently by means of the improved variant of the corresponding lemma. All this makes it possible to obtain a new result concerning the arithmetic values of the aforesaid function at a nonzero point of small modulus from some imaginary quadratic field.

A Gaussian Pisot number is an algebraic integer with modulus greater than one whose other conjugates, over the quadratic field Q ( − 1 ) , are of modulus less than one. Given a nonreal algebraic number α , with modulus greater than one, we prove that there is a nonzero complex number λ such that the series ∑ n ∈ N ( Re ( λ α n ) 2 + Im ( λ α n ) 2 ) converges (resp. such that the sequence ( { Re ( λ α n ) } , { Im ( λ α n ) } ) n ∈ N has a unique limit point), if and only if α is a Gaussian Pisot number (resp. α is a Gaussian Pisot number satisfying P ( 1 ) ≥ 2 or deg ( P ) = 2 , where P is the minimal polynomial of α over Q ).

We confirm several conjectures of Sun involving quadratic residues modulo odd primes. For any prime $p\equiv 1\pmod 4$ and integer $a\not\equiv0\pmod p$, we prove that \begin{align*}&(-1)^{|\{1\le k\{ak^2\}_p\right\}\right| \\&+\left|\left\{(j,k):\ 1\le j\frac p2\right\}\right| \\\equiv&\left|\left\{1\le k\{\delta T_k\}_p\right\}\right|},$$ where $T_m$ denotes the triangular number $m(m+1)/2$., Comment: 10 pages. Accepted version for publication in Electron. Res. Arch

We give a new definition of a $p$-adic $L$-function for a mixed signature character of a real quadratic field and for a nontrivial ray class character of an imaginary quadratic field. We then state a $p$-adic Stark conjecture for this $p$-adic $L$-function. We prove our conjecture in the case when $p$ is split in the imaginary quadratic field by relating our construction to Katz's $p$-adic $L$-function. We also provide numerical evidence for our conjecture in three examples., Comment: In Section 6 confirmed earlier calculations to higher precision and included more numerics in two of the examples. Made minor changes to fix typos

Let p be an odd prime and K an imaginary quadratic field where p splits. Under appropriate hypotheses, Bertolini showed that the Selmer group of a p-ordinary elliptic curve over the anticyclotomic $${\mathbb {Z}}_p$$ -extension of K does not admit any proper $$\Lambda $$ -submodule of finite index, where $$\Lambda $$ is a suitable Iwasawa algebra. We generalize this result to the plus and minus Selmer groups (in the sense of Kobayashi) of p-supersingular elliptic curves. In particular, in our setting the plus/minus Selmer groups have $$\Lambda $$ -corank one, so they are not $$\Lambda $$ -cotorsion. As an application of our main theorem, we prove results in the vein of Greenberg–Vatsal on Iwasawa invariants of p-congruent elliptic curves, extending to the supersingular case results for p-ordinary elliptic curves due to Hatley–Lei.

Let $E/\mathbf{Q}$ be an elliptic curve of conductor $N$, let $p>3$ be a prime where $E$ has good ordinary reduction, and let $K$ be an imaginary quadratic field satisfying the Heegner hypothesis. In 1987, Perrin-Riou formulated an Iwasawa main conjecture for the Tate-Shafarevich group of $E$ over the anticyclotomic $\mathbf{Z}_p$-extension of $K$ in terms of Heegner points. In this paper, we give a proof of Perrin-Riou's conjecture under mild hypotheses. Our proof builds on Howard's theory of bipartite Euler systems and Wei Zhang's work on Kolyvagin's conjecture. In the case when $p$ splits in $K$, we also obtain a proof of the Iwasawa-Greenberg main conjecture for the $p$-adic $L$-functions of Bertolini-Darmon-Prasanna., 22 pages. Revised version, to appear in Algebra & Number Theory

In order to investigate arithmetic properties of the values of generalized hypergeometric functions with rational parameters one usually applies Siegel’s method. By means of this method have been achieved the most general results concerning the above mentioned properties. The main deficiency of Siegel’s method consists in the impossibility of its application for the hypergeometric functions with irrational parameters. In this situation the investigation is usually based on the effective construction of the functional approximating form (in Siegel’s method the existence of that form is proved by means of pigeon-hole principle). The construction and investigation of such a form is the first step in the complicated reasoning which leads to the achievement of arithmetic result. Applying effective method we encounter at least two problems which make extremely narrow the field of its employment. First, the more or less general effective construction of the approximating form for the products of hypergeometric functions is unknown. While using Siegel’s method one doesn’t deal with such a problem. Hence the investigator is compelled to consider only questions of linear independence of the values of hypergeometric functions over some algebraic field. Choosing this field is the second problem. The great majority of published results concerning corresponding questions deals with imaginary quadratic field (or the field of rational numbers). Only in exceptional situations it is possible to investigate the case of some other algebraic field. We consider here the case of a real quadratic field. By means of a special technique we establish linear independence of the values of some hypergeometric function with irrational parameter over such a field.

We will give a proof to the Prasad conjectures for $U_2$, $SO_4$ and $Sp_4$ over a quadratic field extension., Comment: 22 pages

Let $K$ be a subfield of $\mathbb{R}$. The theory of $\mathbb{R}$ viewed as an ordered $K$-vector space and expanded by a predicate for $\mathbb{Z}$ is decidable if and only if $K$ is a real quadratic field.

Let $K=\mathbb Q(\sqrt D)$ be a real quadratic field. We consider the additive semigroup $\mathcal O_K^+(+)$ of totally positive integers in $K$ and determine its generators (indecomposable integers) and relations; they can be nicely described in terms of the periodic continued fraction for $\sqrt D$. We also characterize all uniquely decomposable integers in $K$ and estimate their norms. Using these results, we prove that the semigroup $\mathcal O_K^+(+)$ completely determines the real quadratic field $K$., Comment: 15 pages, to appear in manuscripta mathematica

This article is a continuation of our previous work [7] on the Iwasawa theory of an elliptic modular form over an imaginary quadratic field $K$, where the modular form in question was assumed to be ordinary at a fixed odd prime $p$. We formulate integral Iwasawa main conjectures at non-ordinary primes $p$ for suitable twists of the base change of a newform $f$ to an imaginary quadratic field $K$ where $p$ splits, over the cyclotomic ${\mathbb{Z}}_p$-extension, the anticyclotomic ${\mathbb{Z}}_p$-extensions (in both the definite and the indefinite cases) as well as the ${\mathbb{Z}}_p^2$-extension of $K$. In order to do so, we define Kobayashi–Sprung-style signed Coleman maps, which we use to introduce doubly signed Selmer groups. In the same spirit, we construct signed (integral) Beilinson–Flach elements (out of the collection of unbounded Beilinson–Flach elements of Loeffler–Zerbes), which we use to define doubly signed $p$-adic $L$-functions. The main conjecture then relates these two sets of objects. Furthermore, we show that the integral Beilinson–Flach elements form a locally restricted Euler system, which in turn allow us to deduce (under certain technical assumptions) one inclusion in each one of the four main conjectures we formulate here (which may be turned into equalities in favorable circumstances).

We define the class of normalized Shintani L-functions of several variables. Unlike Shintani zeta functions, the normalized Shintani L-function is a holomorphic function. Moreover it satisfies a good functional equation. We show that Hecke L-function of a real quadratic field can be expressed as a diagonal part of some normalized Shintani L-function of several variables. This gives a good several variables generalization of a Hecke L-function of a real quadratic field.

In this paper, we give a nontrivial lower bound for the fundamental unit of norm $$-1$$ of a real quadratic field of class number 1.

Let F be the imaginary quadratic field of discriminant −3 and OF its ring of integers. Let Γ be the arithmetic group GL3(OF), and for any ideal n⊂OF let Γ0(n) be the congruence subgroup of level n ...

We study the question of when a congruence between two theta lifts on GSp 4 / Q descends to a congruence on modular forms on GL 2 over a quadratic field. In order to accomplish that, we use the theory of the local theta correspondence between similitude orthogonal groups and the similitude symplectic group GSp 4 , together with a classification for the degeneration modulo a prime of conductors for the L-parameters of irreducible admissible representations of GSp 4 over a non-archimedean local field. We explain that this is unlikely to be used in conjunction with existing results on congruences for GSp 4 / Q to deduce a theory of congruences over imaginary quadratic fields. On the other hand, we prove a result which does give some such congruence results by twisting.

We prove a functional equation for the three-variable $p$-adic $L$-function attached to the Rankin-Selberg convolution of a Coleman family and a CM Hida family, which was studied by Loeffler and Zerbes. Consequentially, we deduce that an anticyclotomic $p$-adic $L$-function attached to a $p$-non-ordinary modular form vanishes identically in the indefinite setting., Comment: Minor update. We have corrected some misprints and expanded some of the proofs

Stark-Heegner points, also known as Darmon points, were introduced by H. Darmon as certain local points on rational elliptic curves, conjecturally defined over abelian extensions of real quadratic fields. The rationality conjecture for these points is only known in the unramified case, namely, when these points are specializations of global points defined over the strict Hilbert class field $H^+_F$ of the real quadratic field $F$ and twisted by (unramified) quadratic characters of $Gal(H_c^+/F)$. We extend these results to the situation of ramified quadratic characters; more precisely, we show that Darmon points of conductor $c\geq 1$ twisted by quadratic characters of $G_c^+=Gal(H_c^+/F)$, where $H_c^+$ is the strict ring class field of $F$ of conductor $c$, come from rational points on the elliptic curve defined over $H_c^+$., 20 pages; final version

Inspired by the works of Zagier, we study the probability measures $\nu(t)$ with support on the flat tori which are the compact orbits of the maximal unipotent subgroup acting holomorphically on the positive orthonormal frame bundle $\mathcal{F}({M}_D)$ of 3-dimensional hyperbolic Bianchi orbifolds ${M}_D=\mathbb{H}^3/\widetilde{\Gamma}_D$, of finite volume and with only one cusp. Here $\Gamma_D=PSL(2, \mathcal{O})$, where $\mathcal{O}$ is the ring of integers of an imaginary quadratic field of class number one., Comment: We corrected a few typos and included a few details

Let L be an extended ring class field of an imaginary quadratic field K other than Q ( − 1 ) and Q ( − 3 ) . We show that there is a form class group induced from a congruence subgroup which describes the Galois group of L over K in a concrete way. We also construct a primitive generator of L over K as a real algebraic integer which can be applied to certain quadratic Diophantine equations.

Let $K$ be an imaginary quadratic field, and let $\mathfrak{f}$ be a nontrivial integral ideal of $K$. Hasse and Ramachandra asked whether the ray class field of $K$ modulo $\mathfrak{f}$ can be generated by a single value of the Weber function. We completely resolve this question when $\mathfrak{f}=(N)$ for an integer $N>1$., 13 pages

Let c be a square-free positive integer and p a prime satisfying $$p\not \mid c$$ . Let $$h(-c)$$ denote the class number of the imaginary quadratic field $$\mathbb {Q}(\sqrt{-c})$$ . In this paper, we consider the Diophantine equation $$\begin{aligned}&cx^2+p^{2m}=4y^n,~~x,y\ge 1, m\ge 0, n\ge 3,\\&\quad \gcd (x,y)=1, \gcd (n,2h(-c))=1, \end{aligned}$$ and we describe all its integer solutions. Our main tool here is the prominent result of Bilu, Hanrot and Voutier on existence of primitive divisors in Lehmer sequences.

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 d K . 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 ( χ d K , s ) . Using the properties of these functions we give new and computationally efficient formulas for computing some special values of L ( χ d K , s ) .

Let K be a quadratic number field and let \alpha \in K*. We present an explicit finite procedure to compute at once all Kummer degrees [K(\zeta_m,\sqrt[n]{\alpha}):K(\zeta_m)] for n,m \geq 1 with n|m, where \zeta_m denotes a primitive m-th root of unity. We can also replace \alpha by any finitely generated subgroup of K*.

Analogy between Anderson [Formula: see text]-motives and abelian varieties with multiplication by an imaginary quadratic field (MIQF) is a source of 2 results: (1) A description of abelian varieties with MIQF of dimension [Formula: see text] and signature [Formula: see text] in terms of ”lattices” of dimension [Formula: see text] in [Formula: see text]; (2) A construction of exterior powers of abelian varieties with MIQF having [Formula: see text].

In many ways, quaternion algebras are like “noncommutative quadratic field extensions”: this is apparent from their very definition, but also from their description as wannabe \(2\times 2\)-matrices. Just as the quadratic fields \(\mathbb Q (\sqrt{d})\) are wonderously rich, so too are their noncommutative analogues. In this part of the text, we explore these beginnings of noncommutative algebraic number theory.

Let $A$ be a quaternion algebra over a number field $F$, and $\mathcal{O}$ be an $O_F$-order of full rank in $A$. Let $K$ be a quadratic field extension of $F$ that embeds into $A$, and $B$ be an $O_F$-order in $K$. Suppose that $\mathcal{O}$ is a Bass order that is well-behaved at all the dyadic primes of $F$. We provide a necessary and sufficient condition for $B$ to be optimally spinor selective for the genus of $\mathcal{O}$. This partially generalizes previous results on optimal (spinor) selectivity by C. Maclachlan [Optimal embeddings in quaternion algebras. J. Number Theory, 128(10):2852-2860, 2008] for Eichler orders of square-free levels, and independently by M. Arenas et al. [On optimal embeddings and trees. J. Number Theory, 193:91-117, 2018] and by J. Voight [Chapter 31, Quaternion algebras, volume 288 of Graduate Texts in Mathematics. Springer-Verlag, 2021] for Eichler orders of arbitrary levels., 22 pages, made improvements and corrections, results unchanged