Around the support problem for Hilbert class polynomials

The Hilbert class polynomial has as roots the j-invariants of elliptic curves whose endomorphism ring is a given imaginary quadratic order. It can be used to compute elliptic curves over finite fields with a prescribed number of points. Since its coefficients are typically rather large, there has been continued interest in finding alternative modular functions whose corresponding class polynomials are smaller. Best known are Weber’s functions, which reduce the size by a factor of 72 for a positive density subset of imaginary quadratic discriminants. On the other hand, Bröker and Stevenhagen showed that no modular function will ever do better than a factor of 100.83. We introduce a generalization of class polynomials, with reduction factors that are not limited by the Bröker–Stevenhagen bound. We provide examples matching Weber’s reduction factor. For an infinite family of discriminants, their reduction factors surpass those of all previously known modular functions by a factor at least 2.

The purpose of this paper is to calculate the cyclic homology of rings of integers of global fields. We accomplish this by explicitly computing the homology of the simple complex associated to Tsygan’s double complex. To accomplish this, we first compute the cyclic homology of cyclic algebras, i.e., algebras of the form A = R[t]/(P (t)), where P is a monic polynomial with coefficients in R. More precisely, we give a complex (2.9) of finite free A-modules and R-linear maps, whose homology gives the cyclic homology HCr(A/R) when A is an integral domain. This complex enjoys a great advantage over the usual complex, (2.1), for cyclic homology, in that the rank of the modules grows linearly rather than exponentially in r. As a result, one can compute the homology groups explicitly in many interesting cases. If R is a Dedekind domain whose field of fractions K is a global field (i.e. a number field or the fraction field of a curve over a finite field), and S is the integral closure of R in a finite separable extension L/K, the cyclic homology of S/R can be obtained from the cyclic homology of certain cyclic algebras, by a kind of adelization procedure (3.2). To establish the relation between our cyclic complex and the “standard” cyclic complex of Tsygan, Loday, and Quillen [12], we describe a subcomplex of the latter, the small Tsygan complex, which is quasi-isomorphic to the full complex and which admits a quasiisomorphism to our cyclic complex. The construction of the latter quasi-isomorphism is the key technical feature of our paper. It is motivated by our attempts to solve the extension problem arising from the spectral sequence which computes the homology of the Tsygan complex. The differentials of this spectral sequence are implicitly worked out in the proof of Th. 2.10. The solution of the extension problem follows from the technical results Prop. 2.6 and Prop. 2.8. The first section discusses the Hochschild theory for cyclic algebras and Dedekind domains. For cyclic algebras, Hochschild homology can be computed from a periodic complex (1.6). This complex seems to be part of the folklore; but it seems to have been discovered independently, in varying degrees of generality, by a number of people, including Zack [18]; Cortinas, Guccione, and Villamayor [4]; Burghelea and ViguePoirrier [3]; Goodwillie (unpublished); Masuda and Natsume [14]; Wolffhardt [17]; and perhaps others) By adelizing, we obtain (Prop. 1.9) the periodicity of Hochschild homology for Dedekind domains. By constructing explicit quasi-isomorphisms between the Zack complex and the standard complex for Hochschild homology, we can study the maps HH∗(S/R) → HH∗(T/R) → HH∗(T/S), where R ⊂ S ⊂ T is a triple of Dedekind domains (Props. 1.11, 1.13), as well as the ring structure of HH∗(S/R) (1.16). We conclude with a construction of the small subcomplex of the standard Hochschild complex, which motivates the small subcomplex of the Tsygan complex.

Decomposition of integer-valued polynomial algebras

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.

On the joins of hensel rings

Let D be an integral domain with identity having quotient field K. This paper gives necessary and sufficient conditions on D in order that each integrally closed subring of D should belong to some subclass of the class of integrally closed domains; some of the subclasses considered are the completely integrally closed domains, Prufer domains, and Dedekind domains. 1. The class of integrally closed domains contains several classes of domains which are of fundamental importance in commutative algebra. Unique factorization domains, Krull domains, domains of finite character, Priifer domains, completely integrally closed domains, Dedekind domains, and principal ideal domains are examples of such subclasses of the class of integrally closed domains. This paper considers the problems of determining, conversely, necessary and sufficient conditions on an integral domain with identity in order that each of its integrally closed subrings should belong to some subclass of the class of integrally closed domains. An example of a typical result might be Theorem 2.3: If J is an integral domain with identity having quotient field K, then conditions (1) and (2) are equivalent. (1) Each integrally closed subring of J is completely integrally closed. (2) Either J has characteristic 0 and K is algebraic over the field of rational numbers or J has characteristic p # 0 and K has transcendence degree at most one over its prime subfield. If J is integrally closed, then conditions (1) and (2) are equivalent to: (3) Each integrally closed subring of J with quotient field K is completely integrally closed. In considering characterizations of integral domains with identity for which every integrally closed subring is Dedekind or almost Dedekind (?3), we are led to use some results of W. Krull to prove Theorem 4.1, which establishes the existence of, as well as a method for constructing, a field with certain specified valuations. We then use this theorem to construct an example of an infinite separable algebraic extension field K of FLp(X) such that the integral closure J of FLp[X] in K Received by the editors April 7, 1970. AMS 1969 subject classifications. Primary 1315, 1350; Secondary 1320.

We prove a new structural result for the spherical Tits building attached to SL_n(K) for many number fields K, and more generally for the fraction fields of many Dedekind domains O: the Steinberg module St_n(K) is generated by integral apartments if and only if the ideal class group cl(O) is trivial. We deduce this integrality by proving that the complex of partial bases of O^n is Cohen-Macaulay. We apply this to prove new vanishing and nonvanishing results for H^{vcd}(SL_n(O_K); Q), where O_K is the ring of integers in a number field and vcd is the virtual cohomological dimension of SL_n(O_K). The (non)vanishing depends on the (non)triviality of the class group of O_K. We also obtain a vanishing theorem for the cohomology H^{vcd}(SL_n(O_K); V) with twisted coefficients V.

The easiest way to start studying number fields is to consider them per se, as absolute extensions of √; this is, for example, what we have done in [Coh0]. In practice, however, number fields are frequently not given in this way. One of the most common other ways is to give a number field as a relative extension, in other words as an algebra L/K over some base field K that is not necessarily equal to √. necessarily equal to ℚ. In this case, the basic algebraic objects such as the ring of integers ℤ L and the ideals of ℤ L , are not only ℤ-modules, but also ℤ K- modules. The ℤ K -module structure is much richer and must be preserved. No matter what means are chosen to compute ℤ L , we have the problem of representing the result. Indeed, here we have a basic stumbling block: considered as ℤ-modules, ℤ L or ideals of ℤ L are free and hence may be represented by ℤ-bases, for instance using the Hermite normal form (HNF); see, for example, [Coh0, Chapter 2]. This theory can easily be generalized by replacing ℤ with any other explicitly computable Euclidean domain and, under certain additional conditions, to a principal ideal domain (PID). In general, ℤ K is not a PID, however, and hence there is no reason for ℤ L to be a free module over ℤ K- A simple example is given by K = ℚ(√-10) and L = K(√1) (see Exercise 22 of Chapter 2).KeywordsPrime IdealNumber FieldProjective ModuleIntegral IdealTorsion ModuleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This chapter develops tools for a more penetrating study of algebraic number theory than was possible in Chapter V and concludes by formulating two of the main three theorems of Chapter V in the modern setting of “adeles” and “ideles” commonly used in the subject.Sections 1–5 introduce discrete valuations, absolute values, and completions for fields, always paying attention to implications for number fields and for certain kinds of function fields. Section 1 contains a prototype for all these notions in the construction of the fieldQp of p-adic numbers formed out of the rationals. Discrete valuations in Section 2 are a generalization of the order-of-vanishing function about a point in the theory of one complex variable. Absolute values in Section 3 are real-valued multiplicative functions that give a metric on a field, and the pair consisting of a field and an absolute value is called a valued field. Inequivalent absolute values have a certain independence property that is captured by the Weak Approximation Theorem. Completions in Section 4 are functions mapping valued fields into their metric-space completions. Section 5 concerns Hensel’s Lemma, which in its simplest form allows one to lift roots of polynomials over finite prime fields Fp to roots of corresponding polynomials over p-adic fields Qp.Section 6 contains the main theorem for investigating the fundamental question of how prime ideals split in extensions. Let K be a finite separable extension of a field F, let R be a Dedekind domain with field of fractions F, and let T be the integral closure of R in K. The question concerns the factorization of an ideal pT in T when p is a nonzero prime ideal in R. If Fp denotes the completion of F with respect to p, the theorem explains how the tensor product K ⊗F Fp splits uniquely as a direct sum of completions of valued fields. The theorem in effect reduces the question of the splitting of pT in T to the splitting of Fp in a complete field in which only one of the prime factors of pT plays a role.Section 7 is a brief aside mentioning additional conclusions one can draw when the extension K/F is a Galois extension.Section 8 applies the main theorem of Section 6 to an analysis of the different of K/F and ultimately to the absolute discriminant of a number field. With the new sharp tools developed in the present chapter, including a Strong Approximation Theorem that is proved in Section 8, a complete proof is given for the Dedekind Discriminant Theorem; only a partial proof had been accessible in Chapter V.Sections 9–10 specialize to the case of number fields and to function fields that are finite separable extensions of Fq (X), where Fq is a finite field. The adele ring and the idele group are introduced for each of these kinds of fields, and it is shown how the original field embeds discretely in the adeles and how the multiplicative group embeds discretely in the ideles. The main theorems are compactness theorems about the quotient of the adeles by the embedded field and about the quotient of the normalized ideles by the embedded multiplicative group. Proofs are given only for number fields. In the first case the compactness encodes the Strong Approximation Theorem of Section 8 and the Artin product formula of Section 9. In the second case the compactness encodes both the finiteness of the class number and the Dirichlet Unit Theorem.KeywordsPrime IdealValuation RingBasic AlgebraDedekind DomainFractional IdealThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This chapter develops tools for a more penetrating study of algebraic number theory than was possible in Chapter V and concludes by formulating two of the main three theorems of Chapter V in the modern setting of “adeles” and “ideles” commonly used in the subject. Sections 1–5 introduce discrete valuations, absolute values, and completions for fields, always paying attention to implications for number fields and for certain kinds of function fields. Section 1 contains a prototype for all these notions in the construction of the field $\mathbb{Q}_p$ of $p$-adic numbers formed out of the rationals. Discrete valuations in Section 2 are a generalization of the order-of-vanishing function about a point in the theory of one complex variable. Absolute values in Section 3 are real-valued multiplicative functions that give a metric on a field, and the pair consisting of a field and an absolute value is called a valued field. Inequivalent absolute values have a certain independence property that is captured by the Weak Approximation Theorem. Completions in Section 4 are functions mapping valued fields into their metric-space completions. Section 5 concerns Hensel's Lemma, which in its simplest form allows one to lift roots of polynomials over finite prime fields $\mathbb{F}_p$ to roots of corresponding polynomials over $p$-adic fields $\mathbb{Q}_p$. Section 6 contains the main theorem for investigating the fundamental question of how prime ideals split in extensions. Let $K$ be a finite separable extension of a field $F$, let $R$ be a Dedekind domain with field of fractions $F$, and let $T$ be the integral closure of $R$ in $K$. The question concerns the factorization of an ideal $\mathfrak{p} T$ in $T$ when $\mathfrak{p}$ is a nonzero prime ideal in $R$. If $F_{\mathfrak{p}}$ denotes the completion of $F$ with respect to $\mathfrak{p}$, the theorem explains how the tensor product $K\otimes_FF_{\mathfrak{p}}$ splits uniquely as a direct sum of completions of valued fields. The theorem in effect reduces the question of the splitting of $\mathfrak{p} T$ in $T$ to the splitting of $F_{\mathfrak{p}}$ in a complete field in which only one of the prime factors of $\mathfrak{p} T$ plays a role. Section 7 is a brief aside mentioning additional conclusions one can draw when the extension $K/F$ is a Galois extension. Section 8 applies the main theorem of Section 6 to an analysis of the different of $K/F$ and ultimately to the absolute discriminant of a number field. With the new sharp tools developed in the present chapter, including a Strong Approximation Theorem that is proved in Section 8, a complete proof is given for the Dedekind Discriminant Theorem; only a partial proof had been accessible in Chapter V. Sections 9–10 specialize to the case of number fields and to function fields that are finite separable extensions of $\mathbb{F}_q(X)$, where $\mathbb{F}_q$ is a finite field. The adele ring and the idele group are introduced for each of these kinds of fields, and it is shown how the original field embeds discretely in the adeles and how the multiplicative group embeds discretely in the ideles. The main theorems are compactness theorems about the quotient of the adeles by the embedded field and about the quotient of the normalized ideles by the embedded multiplicative group. Proofs are given only for number fields. In the first case the compactness encodes the Strong Approximation Theorem of Section 8 and the Artin product formula of Section 9. In the second case the compactness encodes both the finiteness of the class number and the Dirichlet Unit Theorem.

We prove that logarithmic derivatives of certain twisted Hilbert class polynomials are holomorphic modular forms modulo p of filtration p + 1. We derive p-adic information about twisted Hecke traces and Hilbert class polynomials. In this framework, we formulate a precise criterion for p-divisibility of class numbers of imaginary quadratic fields in terms of the existence of certain cusp forms modulo p. We explain the existence of infinite classes of congruent twisted Hecke traces with fixed discriminant in terms of the factorization of the associated Hilbert class polynomial modulo p. Finally, we provide a new proof of a theorem of Ogg classifying those p for which all supersingular j-invariants modulo p lie in Fp.

In 2001, M. Bhargava stunned the mathematical world by extending Gauss's 200-year-old group law on integral binary quadratic forms, now familiar as the ideal class group of a quadratic ring, to yield group laws on a vast assortment of analogous objects. His method yields parametrizations of rings of degree up to 5 over the integers, as well as aspects of their ideal structure, and can be employed to yield statistical information about such rings and the associated number fields. In this paper, we extend a selection of Bhargava's most striking parametrizations to cases where the base ring is not Z but an arbitrary Dedekind domain R. We find that, once the ideal classes of R are properly included, we readily get bijections parametrizing quadratic, cubic, and quartic rings, as well as an analogue of the 2x2x2 cube law reinterpreting Gauss composition for which Bhargava is famous. We expect that our results will shed light on the analytic distribution of extensions of degree up to 4 of a fixed number field and their ideal structure.

Tracing back the Nile to its origin must be about as difficult as tracing back the origins of our interest in the theory of orders. At many junctions one has to choose in an almost arbitrary way which is the Nile and which is the other river joining it, wondering whether in such problems one should stick to the wider or to the deeper stream. Perhaps a convenient solution is to recognize that there are many sources and then to list just a few. Those inspired by number theory will certainly think first about the theory of maximal orders over Dedekind domains in number fields, the representation theory-based algebraist will refer to integral group rings, an algebraic geometer will perhaps point to orders over normal domains, and the ring theorist might view orders in central simple algebras as his favorite class of P.I. rings. In these topics graded orders and orders over graded rings appear not only as natural examples, but also as important basic ingredients: crossed products for finite groups, group rings considered as graded rings, orders over projective varieties, rings of generic matrices, trace rings, etc. On these observations we founded our belief that the application of methods from the theory of graded rings to the special case of orders may lead to some interesting topics for research, new points of view, and results. The formulation of this intent alone creates several problems of choice.

Let P and Q be two points on an elliptic curve defined over a number field K. For alpha in {text {End}}(E), define B_alpha to be the mathcal {O}_K-integral ideal generated by the denominator of x(alpha (P)+Q). Let mathcal {O} be a subring of {text {End}}(E), that is a Dedekind domain. We will study the sequence {B_alpha }_{alpha in mathcal {O}}. We will show that, for all but finitely many alpha in mathcal {O}, the ideal B_alpha has a primitive divisor when P is a non-torsion point and there exist two endomorphisms gne 0 and f so that f(P)= g(Q). This is a generalization of previous results on elliptic divisibility sequences.

Let K K be an algebraic number field and let O \mathfrak {O} be a ring of S S -integers in K K (where S S is a set of primes of K K containing all the archimedean primes); that is to say, O \mathfrak {O} is a Dedekind domain whose field of quotients is K K . In analogy with a theorem of T. Yamada in the case of a field of characteristic 0, it is shown that if S ( O ) S\left ( \mathfrak {O} \right ) is the Schur subgroup of the Brauer group B ( O ) B\left ( \mathfrak {O} \right ) and if o = O ∩ k \mathfrak {o} = \mathfrak {O} \cap k , where k k is any field containing the maximal abelian extension of Q \mathbb {Q} in K K , then S ( O ) = O ⊗ S ( o ) S\left ( \mathfrak {O} \right ) = \mathfrak {O} \otimes S\left ( \mathfrak {o} \right ) , i.e. the Brauer classes in S ( O ) S\left ( \mathfrak {O} \right ) are those obtained from S ( o ) S\left ( \mathfrak {o} \right ) by extension of the scalars to O \mathfrak {O} . A similar theorem is proved as well in the case of the Schur subgroup S ( O , ω ) S\left ( {\mathfrak {O},\omega } \right ) of the quadratic Brauer group B ( O , ω ) B\left ( {\mathfrak {O},\omega } \right ) , where ω \omega is an involution of O \mathfrak {O} .