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In Mathematische Werke, Hecke defines the operator [Formula: see text] and describes their utility in conjunction with theta series of quadratic forms. In particular, he shows that the image of theta series associated to classes of binary quadratic forms in CL[Formula: see text] is again a theta series associated to a collection of forms in CL[Formula: see text]. We state and prove an explicit formula for the action of [Formula: see text] on a binary quadratic form of negative discriminant.

The modular properties of generalized theta-functions with characteristics are used to obtain the formulas for the number of representations of positive integers by the diagonal quadratic forms in nine variables with coefficients that are ones and twos.

Let $K$ be a number field, $\mathfrak{q}$ be an integral ideal, and $\mathrm{Cl}(\mathfrak{q})$ be the associated ray class group. Suppose $\mathrm{Cl}(\mathfrak{q})$ possesses a real exceptional character $\psi$, possibly principal, with a Siegel zero $\beta$. For $\mathcal{C} \in \mathrm{Cl}(\mathfrak{q})$ satisfying $\psi(\mathcal{C}) = 1$, we establish an effective $K$-uniform Linnik-type bound with explicit constants for the least norm of a prime ideal $\mathfrak{p} \in \mathcal{C}$. A special case of this result is related to rational primes represented by certain binary quadratic forms., Comment: 25 pages

In this paper, we find many new parity results for broken 7-diamond partitions by using a result of Berkovich and Patane. For example, we prove that [Formula: see text] for all [Formula: see text] and [Formula: see text].

An explicit formula is given for the representation number of each of the 75 reduced, positive-definite, integral, primitive, quaternary quadratic forms [Formula: see text], which belong to a genus with discriminant [Formula: see text] containing one and only one form class.

Let [Formula: see text] denote the number of overpartitions of [Formula: see text]. In this paper, we will give a complete determination of [Formula: see text] modulo [Formula: see text] by relating it to some binary quadratic forms; this will generalize the result of Kim on [Formula: see text] modulo [Formula: see text]. Moreover, we give infinitely many new Ramanujan-like congruences for overpartition function modulo [Formula: see text].

Diophantine equations [Formula: see text] and [Formula: see text] are connected with the arithmetic once punctured torus groups having Fricke coordinates [Formula: see text] and [Formula: see text]. We describe and analyze the structure of these equations by a graph, which we call a two-color Markoff graph. The faces of the graph are labeled by matrices in the corresponding once punctured torus group, the word representation of which is easily described owing to arithmetic properties of the graph. As an application, we show that these matrices define Markoff minimal forms attaining the values of the Markoff spectrum given by the positive integer solutions of these equations.

We investigate the relationship between the numbers of representations of certain integers by a primitive integral binary quadratic form [Formula: see text] of discriminant [Formula: see text] and the order of the class of [Formula: see text] in the form class group of discriminant [Formula: see text], in the case when this order is even. The explicit form of the solutions obtained is used to give a partial answer to a question regarding which multiples of [Formula: see text] can be parameterized in a particular way.

In this paper, we investigate the nonvanishing coefficients of half-integral weight modular forms modulo a prime. We also consider the indivisibility of class numbers of imaginary quadratic fields.

Let [Formula: see text] be a prime greater than [Formula: see text] and let [Formula: see text] be a rational [Formula: see text]-adic integer. In this paper, we try to determine [Formula: see text], and reveal the connection between cubic congruences and the sum [Formula: see text], where [Formula: see text] is the greatest integer not exceeding [Formula: see text]. Suppose that [Formula: see text] are rational [Formula: see text]-adic integers, [Formula: see text], [Formula: see text] and [Formula: see text]. In this paper, we show that the number of solutions of the congruence [Formula: see text] depends only on [Formula: see text]. Let [Formula: see text] be a prime of the form [Formula: see text] and so [Formula: see text] with [Formula: see text]. When [Formula: see text] and [Formula: see text], we establish congruences for [Formula: see text] and [Formula: see text] modulo p. As a consequence, when [Formula: see text] we show that [Formula: see text] has three solutions if and only if [Formula: see text] is a cubic residue of [Formula: see text].

We prove a general theorem that evaluates the Legendre symbol [Formula: see text] under certain conditions on the integers A, B, m and the prime p. The evaluation is in terms of parameters appearing in a binary quadratic form representing p. The theorem has applications to quartic residuacity.

For any square-free polynomial D over a finite field of characteristic at least 5, we present an algorithm for generating all cubic function fields of discriminant D. We also provide a count of all these fields according to their splitting at infinity. When D′ = D/(-3) has even degree and a leading coefficient that is a square, i.e. D′ is the discriminant of a real quadratic function field, this method makes use of the infrastructures of this field. This infrastructure method was first proposed by Shanks for cubic number fields in an unpublished manuscript from the late 1980s. While the mathematical ingredients of our construction are largely classical, our algorithm has the major computational advantage of finding very small minimal polynomials for the fields in question.

In this work, we establish formulas for the representations of binary quadratic forms of discriminants -44, -92, -108, -135 and -140 as a sum of Lambert series and eta-quotients.

Let ℤ be the set of integers, and let p be a prime of the form 4k + 1 and so p = c2 + d2 with c, d ∈ ℤ. Let q be an integer of the form 4k + 3. Assume that 4n2p = x2 + qy2 with c, d, n, x, y ∈ ℤ and (q, n) = (x, y) = 1, where (a, b) is the greatest common divisor of integers a and b. In this paper, we establish congruences for (-q)[p/8] ( mod p) in terms of c, d, n, x and y, where [⋅] is the greatest integer function. In particular, we establish a reciprocity law and give an explicit criterion for (-11)[p/8] ( mod p).

Let k ≥ 1 be an integer and ped k(n) be the number of k-tuple partitions of an integer n wherein the even parts are distinct (and the odd parts are unrestricted). In this paper, we examine the behavior of ped 4(n) modulo 8 in detail. As a result, we obtain infinite families of congruences for ped 4(n) modulo 8.

We prove that the dimension of the Hecke module generated by a certain eta-quotient is equal to the class number of an imaginary quadratic field. To do this, we relate the eta-quotient to the Hecke theta series attached to a ray class character of the imaginary quadratic field.

Let ℕ denote the set of positive integers and ℤ the set of all integers. Let ℕ0 = ℕ ∪{0}. Let a1x2 + b1xy + c1y2 and a2z2 + b2zt + c2t2 be two positive-definite, integral, binary quadratic forms. The number of representations of n ∈ ℕ0 as a sum of these two binary quadratic forms is [Formula: see text] When (b1, b2) ≠ (0, 0) we prove under certain conditions on a1, b1, c1, a2, b2 and c2 that N(a1, b1, c1, a2, b2, c2; n) can be expressed as a finite linear combination of quantities of the type N(a, 0, b, c, 0, d; n) with a, b, c and d positive integers. Thus, when the quantities N(a, 0, b, c, 0, d; n) are known, we can determine N(a1, b1, c1, a2, b2, c2; n). This determination is carried out explicitly for a number of quaternary quadratic forms a1x2 + b1xy + c1y2 + a2z2 + b2zt + c2t2. For example, in Theorem 1.2 we show for n ∈ ℕ that [Formula: see text] where N is the largest odd integer dividing n and [Formula: see text]

The main purpose of this paper is to give an intrinsic interpretation of the space Θ(D) generated by the binary theta series ϑf attached to the positive binary quadratic forms f whose discriminant has the form D(f) = D/t2, for some integer t. It turns out that [Formula: see text], the space of modular forms of weight 1 and of level |D| which have complex multiplication (CM) by their Nebentypus character [Formula: see text]. As an application, we obtain a structure theorem of the space [Formula: see text]. The proof of this theorem rests on the results of [The space of binary theta series, Ann. Sci. Math. Québec36 (2012) 501–534] together with a characterization of the newforms f which have CM by their Nebentypus character in terms of properties of the associated Deligne–Serre Galois representationρf.

Let p be a prime greater than 3. In this paper, by using expansions and congruences for Lucas sequences and the theory of cubic residues and cubic congruences, we establish some congruences for [Formula: see text] and [Formula: see text] modulo p, where [x] is the greatest integer not exceeding x, and m is a rational p-adic integer with m ≢ 0 ( mod p).

We study the number of divisors in residue classes modulo m and prove, for example, that the exact equidistribution holds for almost all natural numbers coprime to m in the sense of natural density if and only if m = 2kp1p2…ps, where k and s are non-negative integers and pj are distinct Fermat primes. We also provide a general and exact lower bound for the proportion of divisors in the residue class 1 mod m. The same combinatorial technique using Davenport's constant leads to exact lower bounds for the number of representations of a natural number by a given binary quadratic form with a negative discriminant.

In this study we derive formulae for the number of representations of n by the octonary quadratic forms [Formula: see text] for any n ∈ ℕ0.

Using known formulas for R(a, b)(n), the number of representations of n as a times a square plus b times a square, we prove 21 conjectured formulas of Melham on the number of representations of n as sums of triangular, pentagonal and heptagonal numbers. We also demonstrate how our technique can be used to prove the other 277 conjectured formulas of Melham concerning representations by other figurate numbers.

In this paper, we prove a generalization of a correspondence between holomorphic Jacobi cusp forms of higher degree (matrix index) and elliptic cusp forms obtained by K. Bringmann [Lifting maps from a vector space of Jacobi cusp forms to a subspace of elliptic modular forms, Math. Z.253 (2006) 735–752], for forms of higher levels (for congruence subgroups). To achieve this, we make use of the method adopted by M. Manickam and the first author in Sec. 3 of [On Shimura, Shintani and Eichler–Zagier correspondences, Trans. Amer. Math. Soc.352 (2000) 2601–2617], who obtained similar correspondence in the degree one case. We also derive a similar correspondence in the case of skew-holomorphic Jacobi forms (matrix index and for congruence subgroups). Such results in the degree one case (for the full group) were obtained by N.-P. Skoruppa [Developments in the theory of Jacobi forms, in Automorphic Functions and Their Applications, Khabarovsk, 1988 (Acad. Sci. USSR, Inst. Appl. Math., Khabarovsk, 1990), pp. 168–185; Binary quadratic forms and the Fourier coefficients of elliptic and Jacobi modular forms, J. Reine Angew. Math.411 (1990) 66–95] and by M. Manickam [Newforms of half-integral weight and some problems on modular forms, Ph.D. thesis, University of Madras (1989)].

Suppose that p is an odd prime and d is a positive integer. Let x and y be integers given by p = x2+dy2 or 4p = x2+dy2. In this paper we determine x( mod p) for many values of d. For example, [Formula: see text] where x is chosen so that x ≡ 1 ( mod 3). We also pose some conjectures on supercongruences modulo p2 concerning binary quadratic forms.

It is known that there is a one-to-one correspondence between equivalence classes of primitive indefinite binary quadratic forms and primitive hyperbolic conjugacy classes of the modular group. Due to such a correspondence, Sarnak obtained the asymptotic formula for the class number sum in order of the fundamental unit by using the prime geodesic theorem for the modular group. In the present paper, we propose asymptotic formulas of the class number sums over discriminants in arithmetic progressions. Since there are relations between the arithmetic properties of the discriminants and the conjugacy classes in the finite groups given by the modular group and its congruence subgroups, we can get the desired asymptotic formulas by arranging the Tchebotarev-type prime geodesic theorem. While such asymptotic formulas were already given by Raulf, the approaches are quite different, the expressions of the leading terms of our asymptotic formulas are simpler and the estimates of the reminder terms are sharper., 22 pages

We consider a variation of the unit sum number problem for quadratic fields and prove various results.

We study annihilating polynomials and annihilating ideals for elements of Witt rings for groups of exponent 2. With the help of these results and certain calculations involving the Clifford invariant, we are able to give full sets of generators for the annihilating ideal of both the isometry class and the equivalence class of an arbitrary quadratic form over a local field. By applying the Hasse–Minkowski theorem, we can then achieve the same for an arbitrary quadratic form over a global field.

We use the recent evaluation of certain convolution sums involving the sum of divisors function to determine the number of representations of a positive integer by certain diagonal octonary quadratic forms whose coefficients are 1, 2 or 4.

We present a reduction theory for certain binary quadratic forms with coefficients in ℤ[λ], where λ is the minimal translation in a Hecke group. We generalize from the modular group Γ(1) = PSL(2,ℤ) to the Hecke groups and make extensive use of modified negative continued fractions. We also define and characterize "reduced" and "simple" hyperbolic fixed points of the Hecke groups.

We shall obtain the formulae for the number of representations of positive integers by a direct sum of k binary quadratic forms of the kind [Formula: see text], when k = 3, 4, 5.

Some theta function identities are proved and used to give formulae for the number of representations of a positive integer by certain quaternary forms x2 + ey2 + fz2 + gt2 with e, f, g ∈ {1, 2, 4, 8}.

In this paper, we show that the number of real quadratic fields 𝕂 of discriminant Δ𝕂 < x whose class group has an element of order g (with g even) is ≥ x1/g/5 if x > x0, uniformly for positive integers g ≤ ( log log x)/(8 log log log x). We also apply the result to find real quadratic number fields whose class numbers have many prime factors.

Some new theta function identities are proved and used to determine the number of representations of a positive integer n by certain quaternary quadratic forms.

Let q be a positive squarefree integer. A prime p is said to be q-admissible if the equation p = u2+ qv2has rational solutions u, v. Equivalently, p is q-admissible if there is a positive integer k such that [Formula: see text], where [Formula: see text] is the set of norms of algebraic integers in [Formula: see text]. Let k(q) denote the smallest positive integer k such that [Formula: see text] for all q-admissible primes p. It is shown that k(q) has subexponential but suprapolynomial growth in q, as q → ∞.

Let 𝔬 be the ring of integers in a number field. An integral quadratic form over 𝔬 is called regular if it represents all integers in 𝔬 that are represented by its genus. In [13,14] Watson proved that there are only finitely many inequivalent positive definite primitive integral regular ternary quadratic forms over ℤ. In this paper, we generalize Watson's result to totally positive regular ternary quadratic forms over ${\mathbb Z}[\frac{1 + \sqrt{5}}{2}]$. We also show that the same finiteness result holds for totally positive definite spinor regular ternary quadratic forms over ${\mathbb Z}[\frac{1 + \sqrt{5}}{2}]$, and thus extends the corresponding finiteness results for spinor regular quadratic forms over ℤ obtained in [1,3].