Drinfeld modular functions

By a change of variables we obtain new <TEX>$y$</TEX>-coordinates of elliptic curves. Utilizing these <TEX>$y$</TEX>-coordinates as meromorphic modular functions, together with the elliptic modular function, we generate the fields of meromorphic modular functions. Furthermore, by means of the special values of the <TEX>$y$</TEX>-coordinates, we construct the ray class fields over imaginary quadratic fields as well as normal bases of these ray class fields.

We know that the elliptic modular function j(z) gives the birational invariant of the elliptic curve with the analytic modulus z, and that the elliptic modular functions belonging to the congruence-subgroups are obtained from the values of elliptic functions at the points of finite order on elliptic curves. This is not only the origin of those functions, but one of the most essential points to which we may ascribe the significance of elliptic modular functions in number-theory. It is important to generalize these facts, namely, to investigate in a more general case the relation between automorphic functions with one or more variables and systems of algebraic varieties, especially, systems of abelian varieties. The object of this paper is to give some results on this subject. We shall deal with certain systems of polarized abelian varieties parametrized by holomorphic functions and show that there exist meromorphic functions whose values are considered as of the members of the systems (Theorem 1). The theory developed here will be chiefly concerned with systems of abelian varieties with non-trivial endomorphisms, since I have already given elsewhere a theory for Siegel's modular functions [11]. I am particularly interested, in the present paper, in the determination of the fields of definition for fields of automorphic functions. This is the first problem which confronts us when we proceed beyond a formal treatment in the arithmetic theory of automorphic functions. We obtain a criterion (Theorem 2) applicable even in the case of compact fundamental domain where one can not employ Fourier expansions. The last part of the paper is devoted to the theory of a certain type of automorphic functions of one variable known in the literature as functions belonging to indefinite ternary quadratic forms [8], [5]; they occur as moduli of abelian varieties of dimension 2 whose endomorphismrings are isomorphic to an order of an indefinite quaternion algebra. We get in this case the field of rational numbers as a field of definition for the function-field. We can also prove the congruence formulae for the modular correspondences, which give a generalization of what is obtained in [3], [10]. We shall treat these formulae in a subsequent paper together with the theory of the functions belonging to congruence-subgroups. Our method can be applied to other types of automorphic functions, for example, to Hilbert's modular functions, by considering a system of

FROM FERMAT TO GAUSS. Fermat, Euler and Quadratic Reciprocity. Lagrange, Legendre and Quadratic Forms. Gauss, Composition and Genera. Cubic and Biquadratic Reciprocity. CLASS FIELD THEORY. The Hilbert Class Field and p = x 2 + ny 2 . The Hilbert Class Field and Genus Theory. Orders in Imaginary Quadratic Fields. Class Fields Theory and the Cebotarev Density Theorem. Ring Class Field and p = x 2 + ny 2 . COMPLEX MULTIPLICATION. Elliptic Functions and Complex Multiplication. Modular Functions and Ring Class Fields. Modular Functions and Singular j--Invariants. The Class Equation. Ellpitic Curves. References. Index.

This note points out a new aspect of the well-known relationship between the subjects mentioned in the title. The following result and its generalization in totally real algebraic number fields is central to the discussion. Let denote the Legendre symbol for relatively prime numbers a and b ℇ ℤ and a substitution of the modular subgroup Γ0(4). Then, if γ&gt;0 and b≡1 mod 2,withandAccording to (1), the Legendre symbol behaves somewhat like a modular function ﹙apart from the known behaviour under and ﹚. (1) follows (see below) from the functional equationwithprovided thatHere we used and always will use the abbreviationand ℇδ means the absolutely least residue of δ mod 4. In the proof, Hecke [4] assumed γ&gt;0 (see also Shimura [5]).

This chapter is meant as another motivation (cf. introduction) for the construction of the ascending pseudodifferential analysis, rather than as an introduction to a new point of view in modular form theory: we hope to come back to possible developments in this direction at some later occasion.We here wish to show that the parallel treatments of anaplectic analysis and associated alternative pseudodifferential analysis on one hand, of usual analysis and pseudodifferential analysis on the other hand, extend, up to some point, as parallel sources of holomorphic modular forms on one hand, nonholomorphic modular forms on the other hand. In particular, let us direct the interested reader to Remark 5.1.1 at the end of this section, which puts on an absolutely equal footing the Rankin–Cohen brackets of holomorphic modular forms, well known to modular form theorists, and their analogues in nonholomorphic modular form theory, which do not seem to have kept their attention. As explained in the introduction, making this possible was one of the initial aims of the present work.

Modularity of Galois traces of Weber's resolvents

Olga Taussky’s interest in class field theory began when she was a student with Ph. Furtwangler at the University of Vienna (she received her doctorate under his supervision in 1930). At this time, E. Artin, using his general reciprocity law, reformulated the “Principal Ideal Theorem” as a purely grouptheoretic question regarding the triviality of the “Verlagerung” homomorphism from a finite group to its commutator subgroup. Since Furtwangler had been developing group theoretic methods in his own number theoretic investigations, Artin communicated this new idea to him. This was considered to be one of the central problems in the subject at the time, and so there was a good deal of excitement when Furtwangler eventually succeeded in proving the result using this new approach. Olga Taussky had been looking for a thesis topic and Furtwangler suggested several questions related to the finer structure of the Principal Ideal Theorem. The optimistic atmosphere, and the prospect of contributing at the frontiers of class field theory, sparked in Olga a deep interest in this problem which lasted her entire career. She returned to the questions of “capitulation”, a term coined by one of her co-authors Arnold Scholz, several times in her life, always with the sense that these were questions of deep arithmetic significance. I believe that it was the experience of her early contact with class field theory which dominated her research interests in number theory and which provided her a fertile source of inspiration for most of her career. I will describe some of her work in this area below. I would like to take this opportunity to thank the referee for many valuable suggestions. For a number field F , (i.e., an extension of finite degree over the rational field Q) its Hilbert class field H = H(F ) is the maximal Galois extension of F which is everywhere unramified and whose Galois group Gal(H/F ) is abelian. It is a consequence of Artin’s reciprocity law that Gal(H/F ) is isomorphic to the ideal class group C(F ) of F. The Principal Ideal Theorem is the statement that every ideal of F becomes a principal ideal when considered

Dedekind symbols generalize the classical Dedekind sums (symbols). These symbols are determined uniquely, up to additive constants, by their reciprocity laws. For k ≧ 2, there is a natural isomorphism between the space of Dedekind symbols with Laurent polynomial reciprocity laws of degree 2k − 2 and the space of modular forms of weight 2k for the full modular group \(PSL_2 (\mathbb{Z}).\) However, this is not the case when k = 1 as there is no modular form of weight two; nevertheless, there exists a unique (up to a scalar multiple) quasi-modular form (Eisenstein series) of weight two. The purpose of this note is to define the Dedekind symbol associated with this quasi-modular form, and to prove its reciprocity law. Furthermore we show that the odd part of this Dedekind symbol is nothing but a scalar multiple of the classical Dedekind sum. This gives yet another proof of the reciprocity law for the classical Dedekind sum in terms of the quasi-modular form.

The workshop brought together leading experts in Algebraic Number Theory. The talks presented new methods and results that intertwine a multitude of topics ranging from classical diophantine themes to modern arithmetic geometry, modular forms and p-adic aspects in number theory.

Cohomology of Drinfeld Modular Varieties provides an introduction, in two volumes, both to this subject and to the Langlands correspondence for function fields. These varieties are the analogues for function fields of the Shimura varieties over number fields. The Langlands correspondence is a conjectured link between automorphic forms and Galois representations over a global field. By analogy with the number-theoretic case, one expects to establish the conjecture for function fields by studying the cohomology of Drinfeld modular varieties, which has been done by Drinfeld himself for the rank two case. This second volume is concerned with the Arthur-Selberg trace formula, and with the proof in some cases of the Rmamanujan-Petersson conjecture and the global Langlands conjecture for function fields. It is based on graduate courses taught by the author, who uses techniques which are extensions of those used to study Shimura varieties. Though the author considers only the simpler case of function rather than number fields, many important features of the number field case can be illustrated. Several appendices on background material keep the work reasonably self-contained. It is the first book on this subject and will be of much interest to all researchers in algebraic number theory and representation theory.

Brauer groups of curves and reciprocity law in Brauer groups of their function fields

We prove a certain Riemann–Roch-type formula for symmetric powers of Galois modules on Dedekind schemes which, in the number field or function field case, specializes to a formula of Burns and Chinburg for Cassou–Noguès–Taylor operations.

Reflection principles for class groups

Let $X$ be a Berkovich space over a valued field. We prove that every finite group is a Galois group over $\Ms(B)(T)$, where $\Ms(B)$ is the field of meromorphic functions over a part $B$ of $X$ satisfying some conditions. This gives a new geometric proof that every finite group is a Galois group over $K(T)$, where $K$ is a complete valued field with non-trivial valuation. Then we switch to Berkovich spaces over ${\bf Z}$ and use a similar strategy to give a new proof of the following theorem by D. Harbater: every finite group is a Galois group over a field of convergent arithmetic power series. We believe our proof to be more geometric and elementary that the original one. We have included the necessary background on Berkovich spaces over ${\bf Z}$.

Let K = k(C) be the function field of an algebraic curve C over an algebraically closed ground field k. Let Γ/K be a smooth projective curve of genus g > 0 with a k-rational point O ∈ Γ(K), and let J/K denote the Jacobian variety of Γ/K. Further let (τ, B) be the K/k:-trace of J (see §2 below and).