Hilbert class polynomials and traces of singular moduli

On the number of isomorphism classes of CM elliptic curves defined over a number field

In this article, we determine all positive definite integral binary quadratic forms that are represented by the sum of k integer squares in an essentially unique way for each integer $$k\ge 4$$.

A (positive definite integral) quadratic form is called diagonally 2-universal if it represents all positive definite integral binary diagonal quadratic forms. In this article, we show that, up to equivalence, there are exactly 18 (positive definite integral) quinary diagonal quadratic forms that are diagonally 2-universal. Furthermore, we provide a “diagonally 2-universal criterion” for diagonal quadratic forms, which is similar to “15-Theorem” proved by Conway and Schneeberger.

A collection $\mathcal S$ of equivalence classes of positive definite integral quadratic forms in $n$ variables is called an $n$-exceptional set if there exists a positive definite integral quadratic form, which represents all equivalence classes of positive definite integral quadratic forms in $n$ variables except those in $\mathcal S$. We show that, among other results, for any given positive integers $m$ and $n$, there is always an $n$-exceptional set of size $m$ and there are only finitely many of them.

We provide a characterization of integers represented by the positive definite binary quadratic form $$ax^2+bxy+cy^2$$ . Suppose that $$D=b^2-4ac$$ and $$d_K$$ is the discriminant of the imaginary quadratic field $$K={\mathbb {Q}}(\sqrt{D})$$ . We call $$f=\sqrt{D/d_K}$$ the conductor of $$ax^2+bxy+cy^2$$ . In order to prove the main results, we define the “relative conductor” of two orders in an imaginary quadratic field. We provide a characterization of decomposition of proper ideals of orders in imaginary quadratic fields. Next, we provide characterizations of prime powers $$l^h$$ , where l divides the conductor, represented by the positive definite binary quadratic form $$ax^2+bxy+cy^2$$ . Some interesting applications of the main results are also presented. For example, we provide an equivalent condition for when the equation $$m=4x^2+2xy+7y^2$$ has an integer solution. Note that its discriminant and conductor are $$-\,108$$ and 6 and we do not assume that m is prime to 2 or 3.

The j-function acts as a parametrization of the classical modular curve. Its values at complex multiplication (CM) points are called singular moduli and are algebraic integers. A Shimura curve is a generalization of the modular curve and, if the Shimura curve has genus 0, a rational parameterizing function exists and when evaluated at a CM point is again algebraic over Q. This paper shows that the coordinate maps given by N. Elkies for the Shimura curves associated to the quaternion algebras with discriminants 6 and 10 are Borcherds lifts of vector-valued modular forms. This property is then used to explicitly compute the rational norms of singular moduli on these curves. This method not only verifies conjectural values for the rational CM points, but also provides a way of algebraically calculating the norms of CM points with arbitrarily large negative discriminant.

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 [Formula: see text]. We also show that the same finiteness result holds for totally positive definite spinor regular ternary quadratic forms over [Formula: see text], and thus extends the corresponding finiteness results for spinor regular quadratic forms over ℤ obtained in [1,3].

For each positive integer n n , let g Z ( n ) g_\mathbb {Z}(n) be the smallest integer such that if an integral quadratic form in n n variables can be written as a sum of squares of integral linear forms, then it can be written as a sum of g Z ( n ) g_\mathbb {Z}(n) squares of integral linear forms. We show that every positive definite integral quadratic form is equivalent to what we call a balanced Hermite–Korkin–Zolotarev-reduced form and use it to show that the growth of g Z ( n ) g_\mathbb {Z}(n) is at most an exponential of n \sqrt {n} . Our result improves the best known upper bound on g Z ( n ) g_\mathbb {Z}(n) which is on the order of an exponential of n n . We also define an analogous number g O ∗ ( n ) g_{\mathcal O}^*(n) for writing Hermitian forms over the ring of integers O \mathcal O of an imaginary quadratic field as sums of norms of integral linear forms, and when the class number of the imaginary quadratic field is 1 1 , we show that the growth of g O ∗ ( n ) g_{\mathcal O}^*(n) is at most an exponential of n \sqrt {n} . We also improve on results of both Conway and Sloane and Kim and Oh on s s -integrable lattices.

Introduction. This research is concerned primarily with the construction of positive definite indecomposable hermitian forms over the integers of algebraic number fields. The interest in indecomposables stems from the fact that every definite hermitian form over the integers of a number field splits uniquely into the orthogonal sum of indecomposable components. Indecomposables have also been used to obtain a lower bound for the class number of a definite hermitian form in terms of the rank (see Gerstein [2]). The analogous problem for positive definite quadratic forms has been studied by O'Meara [13]. Combining his work with that of Erdos and Ko [1], O'Meara shows the existence of positive definite integral indecomposable quadratic forms of rank n and discriminant d over 7/ when n > 10 and d > 0, with exactly five exceptions. That no indecomposables exist in the exceptional cases had already been shown by Kneser [10]. In this paper we will first show that every indecomposable quadratic form over 7/ lifts (by tensor product) to an indecomposable hermitian form over the integers ? of any imaginary quadratic number field E. Hence the quadratic forms mentioned above provide indecomposable hermitian forms with the same invariants. The principle aim of this research is to give an explicit global construction of positive definite integral indecomposable hermitian forms over ? of arbitrary rank and discriminant. While this duplicates some of the results obtained by lifting indecomposable quadratic forms, many of the quadratic forms constructed by O'Meara were obtained by exhibiting their localizations. Further, our constructions will produce indecomposable hermitian forms which do not come from lifting quadratic forms. We will adopt the terminology of O'Meara's book [12] and refer to lattices instead of forms. Our results on indecomposable hermitian lattices are obtained by adapting some of the methods of O'Meara's paper [13] to the hermitian setting and by using a neighbor lattice technique over the integers of E. The neighbor lattice idea is due to Kneser [10], who developed it for quadratic lattices over Z. lyanaga [5] subsequently developed a neighbor lattice

Hodge structures of type (n, 0, . . . , 0, n) Burt Totaro Completing earlier work by Albert, Shimura found all the possible endomor- phism algebras (tensored with the rationals) for complex abelian varieties of a given dimension [12, Theorem 5]. In five exceptional cases, every abelian variety on which a certain algebra acts has “extra endomorphisms”, so that the full endomorphism algebra is bigger than expected. Complex abelian varieties X up to isogeny are equivalent to polarizable Q-Hodge structures of weight 1, with Hodge numbers (n, n) (where n is the dimension of X). In this paper, we generalize Shimura’s classification to determine all the possible endomorphism algebras for polarizable Q-Hodge structures with Hodge numbers (n, 0, . . . , 0, n). For Hodge structures of odd weight, the answer is the same as for abelian varieties. For Hodge structures of even weight, the answer is similar but not identical. The proof combines ideas from Shimura with Green-Griffiths-Kerr’s approach to computing Mumford-Tate groups [4, Proposition VI.A.5]. As with abelian varieties, the most interesting feature of the classification is that in certain cases, every Hodge structure on which a given algebra acts must have extra endomorphisms. (Throughout this discussion, we only consider polarizable Hodge structures.) One known case (pointed out to me by Beauville) is that every Q- Hodge structure with Hodge numbers (1, 0, 1) has endomorphisms by an imaginary quadratic field and hence is of complex multiplication (CM) type, meaning that its Mumford-Tate group is commutative. More generally, every Q-Hodge structure with Hodge numbers (n, 0, n) that has endomorphisms by a totally real field F of degree n has endomorphisms by a totally imaginary quadratic extension field of F , and hence is of CM type. Another case, which seems to be new, is that a Q-Hodge structure V with Hodge numbers (2, 0, 2) that has endomorphisms by an imaginary quadratic field F 0 must have endomorphisms by a quaternion algebra over Q. In this case, V need not be of CM type; there is a period space isomorphic to CP 1 of Hodge structures of this type, whereas there are only countably many Hodge structures of CM type. To motivate the results of this paper on endomorphism algebras, consider the geometric origin of Hodge structures. A Hodge structure comes from geometry if it is a summand of the cohomology of a smooth complex projective variety defined by an algebraic correspondence. Griffiths found (“Griffiths transversality”) that a family of Hodge structures coming from geometry can vary only in certain directions, expressed by the notion of a variation of Hodge structures [15, Theorem 10.2]. In particular, any variation of Hodge structures of weight at least 2 with Hodge numbers (n, 0, . . . , 0, n) (so there is at least one 0) is locally constant; more generally, this holds whenever there are no two adjacent nonzero Hodge numbers. This has the remarkable consequence that only countably many Hodge structures of weight at least 2 with Hodge numbers (n, 0, . . . , 0, n) come from geometry. Very little is

Modularity of Galois traces of Weber's resolvents

We describe deterministic and probabilistic algorithms to determine whether or not a given monic irreducible polynomial H ∈ Z [ X ] H\in \mathbb {Z}[X] is a Hilbert class polynomial, and if so, which one. These algorithms can be used to determine whether a given algebraic integer is the j j -invariant of an elliptic curve with complex multiplication (CM), and if so, the associated CM discriminant. More generally, given an elliptic curve E E over a number field, one can use them to compute the endomorphism ring of E E . Our algorithms admit simple implementations that are asymptotically and practically faster than previous approaches.

The modular trace of the normalized Hauptmodul has been extended to the Galois trace of a class invariant by Kaneko. It is an important issue to search for class invariants for which the Galois traces have modular properties. The crucial point in this paper is that we initiate a new notion, called Siegel resolvents; we define the Siegel resolvents as the quadratic polynomials of Siegel functions of level 3, so that they are modular functions of level 3 as well. We construct real-valued class invariants over imaginary quadratic fields by using the singular values of Siegel resolvents at imaginary quadratic irrationals. We also prove that the generating series of their Galois traces become a weakly holomorphic modular form with weight 3/2. This shows that the work of D. Zagier on traces of singular moduli can be extended to the modular functions of higher level.

After Zagier’s significant work (in: Bogomolov and Katzarkov (eds) Motives, polylogarithms and hodge theory, part I, International Press, Somerville, 2002) on traces of singular moduli, Bruinier and Funke (J Reine Angew Math 594:1–33, 2006) generalized his result to the traces of singular values of modular functions on modular curves of arbitrary genus. In class field theory, the extended ring class field is a generalization of the ray class field over an imaginary quadratic field. By using Shimura’s reciprocity law, we construct primitive generators of the extended ring class fields by using Siegel functions of arbitrary level $$N\ge 2$$ and identify their Galois traces with Fourier coefficients of weight 3/2 harmonic weak Maass forms. This would extend the results of Jeon et al. (Math Ann 353:37–63, 2012) and Jung et al. (Modularity of Galois traces of Weber’s resolvents, under revision).

The values of the elliptic modular $j$-invariant at imaginary quadratic arguments are algebraic integers, known as singular moduli of level one. If $d_1$ and $d_2$ are imaginary quadratic discriminants, then we may consider a generalized resultant of the class polynomials of the orders of discriminant $d_1$ and $d_2$; this is the norm of the differences of singular moduli of the corresponding orders, denoted here by $J(d_1,d_2)$. These resultants are highly factorizable; Gross-Zagier established a closed formula for $J(d_1,d_2)^2$ when $d_1$ and $d_2$ are fundamental discriminants, with $(d_1,d_2)=1$. In this paper we present a conjectural extension of the Gross-Zagier formula to the case when $d_1$ and $d_2$ are not necessarily fundamental, and $(d_1,d_2)=l^e$, where 1 is a prime not dividing the product of the conductors of $d_1$ and $d_2$.