Modular Forms. The Hypergeometric Function

We give examples of infinite order rational transformations that leave linear differential equations covariant. These examples are non-trivial yet simple enough illustrations of exact representations of the renormalization group. We first illustrate covariance properties on order-two linear differential operators associated with identities relating the same hypergeometric function with different rational pullbacks. These rational transformations are solutions of a differentially algebraic equation that already emerged in a paper by Casale on the Galoisian envelopes. We provide two new and more general results of the previous covariance by rational functions: a new Heun function example and a higher genus hypergeometric function example. We then focus on identities relating the same hypergeometric function with two different algebraic pullback transformations: such remarkable identities correspond to modular forms, the algebraic transformations being solution of another differentially algebraic Schwarzian equation that also emerged in Casale’s paper. Further, we show that the first differentially algebraic equation can be seen as a subcase of the last Schwarzian differential condition, the restriction corresponding to a factorization condition of some associated order-two linear differential operator. Finally, we also explore generalizations of these results, for instance, to , hypergeometric functions, and show that one just reduces to the previous cases through a Clausen identity. The question of the reduction of these Schwarzian conditions to modular correspondences remains an open question. In a hypergeometric framework the Schwarzian condition encapsulates all the modular forms and modular equations of the theory of elliptic curves, but these two conditions are actually richer than elliptic curves or hypergeometric functions, as can be seen on the Heun and higher genus example. This work is a strong incentive to develop more differentially algebraic symmetry analysis in physics.

There are close relations between Bernoulli numbers and quadratic fields or quadratic forms. In order to explain those, we give a survey of the ideal theory of quadratic fields and quadratic forms. Since Gauss , it is well known that there is a deep relation between the ideal theory of quadratic fields (i.e. quadratic extensions of the rational number field) and integral quadratic forms. This is obvious for specialists, but textbooks which explain this in detail are rare. In most textbooks, they treat the correspondence of ideals only in the case of maximal orders (the ring of all integers) for which the description is simple and easy. This is very disappointing. We cannot see the whole picture of the theory of quadratic forms by such a restricted treatment and sometimes it causes misunderstanding. We need the full description of this kind of relation when we explain later the relation between L-functions of prehomogeneous vector spaces and the Bernoulli numbers. So it would be a good chance to try to explain the full relation here.

Mirror maps, modular relations and hypergeometric series II

For primes p ≡ 1 (mod 12), we present an explicit relation between the traces of Frobenius on a family of elliptic curves with j-invariant 1728 / t and values of a particular 2 F 1 -hypergeometric function over F p . We also give a formula for traces of Hecke operators on spaces of cusp forms of weight k and level 1 in terms of the same traces of Frobenius. This leads to formulas for Ramanujan's T -function in terms of hypergeometric functions.

Sharp Landen transformation inequalities for hypergeometric functions, with applications

If R(ω,q) denotes Dyson’s partition rank generating function, due to work of Bringmann and Ono, it is known that for roots of unity ω≠1, R(ω,q) is the “holomorphic part” of a harmonic weak Maass form. Dating back to Ramanujan, it is also known that \(\widehat{R}(\omega,q):=R(\omega,q^{-1})\) is given by Eichler integrals and modular forms. In analogy to these results, more recently Monks and Ono have shown that modular forms arise in a natural way from G(ω,q), the generating function for ranks of partitions into distinct parts. Moreover, Monks and Ono pose the following problem: determine whether the function \(\widehat{G}(\omega,q):=G(\omega,q^{-1})\) appears naturally in the theory of modular forms. Here we answer this question of Monks and Ono, and show that \(\widehat{G}(\omega,q)\) , when combined with \(\widehat{G}(\omega^{-1},q)\) and a twisted third-order mock theta of Ramanujan, form a weight 1 modular form. We provide a more general result on the modularity of certain expressions involving basic hypergeometric series and then show that our result on \(\widehat {G}(\omega,q)\) may be deduced from this as a special case.

Our purpose in this chapter is to introduce three algebras of importance in the theory of quadratic forms, the Clifford algebra, the quaternion algebra, and the Hasse algebra. The Clifford algebra will be developed from first principles and its main use for us will be in the definition of an invariant called the spinor norm. The quaternion algebra and the Hasse algebra play an important role in the arithmetic theory of quadratic forms. The definition of the Hasse algebra depends on some of the structure theory of central simple algebras, in particular it needs Wedderburn’s theorem and the theory of similarity of algebras that is normally used in defining the Brauer group. We have therefore included a proof of Wedderburn’s theorem and some of its consequences. Also included as a convenience to the reader is a brief discussion of the tensor product of finite dimensional vector spaces1.

We consider a class of singular, zero-range perturbations of the quantum Hamiltonian of a system consisting of a test particle and N harmonic oscillators (Rayleigh gas). Using the theory of quadratic forms we construct the self-adjoint and bounded from below perturbed Hamiltonian and we give representation formulas for the resolvent and the unitary group. The one-dimensional, two-dimensional and three-dimensional cases are discussed.

Classifications and representations are two main topics in the theory of quadratic forms. In this paper, we consider these topics of ternary quadratic forms. For a given squarefree integer N N , first we give the classification of positive definite ternary quadratic forms of level 4 N 4N explicitly. Second, we give explicit formulas of the weighted sum of representations over each class in every genus of ternary quadratic forms of level 4 N 4N , which are involved with modified Hurwitz class number. In the proof of the main results, we use the relations among ternary quadratic forms, quaternion algebras, and Jacobi forms. As a corollary, we get the formula for the class number of positive ternary quadratic forms of level 4 N 4N . As applications, we derive an explicit base of Eisenstein series space of modular forms of weight 3 / 2 3/2 and level 4 N 4N , and give new proofs of some interesting identities involving representation number of ternary quadratic forms.

In this note (cf. sections 3, 4) I shall give an axiomatization of those fields (of characteristic ≠ 2) which have a theory of quadratic forms like the -adic numbers or like the real numbers. This leads then, for instance, to a generalization of the well-known theorems on -adic forms to a wider class of fields, including non-local ones. The main purpose of the exercise is, however, to separate out the roles of the arithmetic in the underlying field, on the one hand, which solely enters into the verification of the axioms, and of the ordinary algebra of quadratic forms on the other hand. The resulting clarification of the structure of the theory is of interest even in the known -adic case.

A q-integral is a definite integral of a function of q having an expansion in non-negative powers of q for {|q|<1} (q-series). In his book on hypergeometric series, N. J. Fine [N. J. Fine, Basic Hypergeometric Series and Applications, Math. Surveys Monogr. 27, American Mathematical Society, Providence, 1988] explicitly evaluated three q-integrals. For example, he showed that \int_{0}^{e^{-\pi}}\prod_{n=1}^{\infty}\frac{(1-q^{2n})^{20}}{(1-q^{n})^{16}}% dq=\frac{1}{16}. In this paper, we prove a general theorem which allows us to determine a wide class of integrals of this type. This class includes the three q-integrals evaluated by Fine as well as some of those evaluated by L.-C. Zhang [L.-C. Zhang, Some q-integrals associated with modular forms, J. Math Anal. Appl. 150 1990, 264–273]. It also includes many new evaluations of q-integrals.

Introduction Classical theory of symmetric bilinear forms and quadratic forms: Bilinear forms Quadratic forms Forms over rational function fields Function fields of quadrics Bilinear and quadratic forms and algebraic extensions $u$-invariants Applications of the Milnor conjecture On the norm residue homomorphism of degree two Algebraic cycles: Homology and cohomology Chow groups Steenrod operations Category of Chow motives Quadratic forms and algebraic cycles: Cycles on powers of quadrics The Izhboldin dimension Application of Steenrod operations The variety of maximal totally isotropic subspaces Motives of quadrics Appendices Bibliography Notation Terminology.

We show that the theorem of Ellenberg and Venkatesh on representation of integral quadratic forms by integral positive definite quadratic forms is valid under weaker conditions on the represented form.

Field invariants under totally positive quadratic extensions

We give a new construction of overconvergent modular forms of arbitrary weights, defining them in terms of functions on certain affinoid subsets of Scholze's infinite-level modular curve. These affinoid subsets, and a certain canonical coordinate on them, play a role in our construction which is strongly analogous with the role of the upper half-plane and its coordinate `$z$' in the classical analytic theory of modular forms. As one application of these ideas, we define and study an overconvergent Eichler-Shimura map in the context of compact Shimura curves over $\mb{Q}$, proving stronger analogues of results of Andreatta-Iovita-Stevens.