On Congruences of Galois Representations of Number Fields

The purpose of this note is to prove that the Shintani lift of Hilbert modular forms over algebraic number fields commutes with the action of Hecke operators. We show our assertion using the commutativity of the Shimura lift with Hecke operators and some properties of adjoint mappings. This commutativity of the Shintani lift plays an essential role for the proof of the Waldspurger-type theorem concerning the Fourier coefficients of modular forms of half-integral weight over arbitrary algebraic number fields.

The purpose of this paper is to derive that the square of Fourier coefficients a(n) at a square free positive integer n of modular forms f of half integral weight belonging to Kohnen's spaces of arbitrary odd level and of arbitrary primitive character is essentially equal to the critical value of the zeta function attached to the modular form F of integral weight which is the image of f under the Shimura correspondence. Previously, KohnenZagier had obtained an analogous result in the case of Kohnen's spaces of square free level and of trivial character. Our results give some generalizations of them of KohnenZagier. Our method of the proof is similar to that of Shimura's paper concerning Fourier coefficients of Hilbert modular forms of half integral weight over totally real fields.

We extend results of Bringmann and Ono that relate certain generalized traces of Maass–Poincaré series to Fourier coefficients of modular forms of half-integral weight. By specializing to cases in which these traces are usual traces of algebraic numbers, we generalize results of Zagier describing arithmetic traces associated to modular forms. We define correspondences [Formula: see text] and [Formula: see text]. We show that if f is a modular form of non-positive weight 2 - 2 λ and odd level N, holomorphic away from the cusp at infinity, then the traces of values at Heegner points of a certain iterated non-holomorphic derivative of f are equal to Fourier coefficients of the half-integral weight modular forms [Formula: see text].

The quantum degeneracies of Bogomolny‐Prasad‐Sommerfield (BPS) black holes of octonionic magical supergravity in five dimensions are studied. Quantum degeneracy is defined purely number theoretically as the number of distinct states in charge space with a given set of invariant labels. Quantum degeneracies of spherically symmetric stationary BPS black holes are given by the Fourier coefficients of modular forms of exceptional group . Their charges take values in the lattice defined by the exceptional Jordan algebra over integral octonions. The quantum degeneracies of rank 1 and rank 2 BPS black holes are given by the Fourier coefficients of singular modular forms and . The rank 3 (large) BPS black holes will be studied elsewhere. Following the work of N. Elkies and B. Gross on embeddings of cubic rings A into the exceptional Jordan algebra we show that the quantum degeneracies of rank 1 black holes described by such embeddings are given by the Fourier coefficients of the Hilbert modular forms (HMFs) of . If the discriminant of the cubic ring A is with p a prime number then the isotropic lines in the 24 dimensional orthogonal complement of A define a pair of Niemeier lattices which can be taken as charge lattices of some BPS black holes. The current status of the searches for the M/superstring theoretic origins of the octonionic magical supergravity is also reviewed.

This is the third of a series of papers relating intersections of special cycles on the integral model of a Shimura surface to Fourier coefficients of Hilbert modular forms. More precisely, we embed the Shimura curve over ℚ associated to a rational quaternion algebra into the Shimura surface associated to the base change of the quaternion algebra to a real quadratic field. After extending the associated moduli problems over ℤ we obtain an arithmetic threefold with a embedded arithmetic surface, which we view as a cycle of codimension one. We then construct a family, indexed by totally positive algebraic integers in the real quadratic field, of codimension two cycles (complex multiplication points) on the arithmetic threefold. The intersection multiplicities of the codimension two cycles with the fixed codimension one cycle are shown to agree with the Fourier coefficients of a (very particular) Hilbert modular form of weight 3/2. The results are higher dimensional variants of results of Kudla-Rapoport-Yang, which relate intersection multiplicities of special cycles on the integral model of a Shimura curve to Fourier coefficients of a modular form in two variables.

The purpose of this paper is to derive a generalization of Kohnen-Zagier's results concerning Fourier coefficients of modular forms of half integral weight belonging to Kohnen's spaces, and to refine our previous results concerning Fourier coefficients of modular forms of half integral weight belonging to Kohnen's spaces. Employing kernel functions, we construct a correspondence $\varPsi$ from modular forms of half integral weight $k+1/2$ belonging to Kohnen's spaces to modular forms of weight $2k$. We explicitly determine the Fourier coefficients of $\varPsi(f)$ in terms of those of $f$. Moreover, under certain assumptions about $f$ concerning the multiplicity one theorem with respect to Hecke operators, we establish an explicit connection between the square of Fourier coefficients of $f$ and the critical value of the zeta function associated with the image $\varPsi(f)$ of $f$ twisted with quadratic characters, which gives a further refinement of our results concerning Fourier coefficients of modular forms of half integral weight belonging to Kohnen's spaces.

In this paper we consider the hyperbolic Kac–Moody algebra \(\mathcal {F}\) associated with the generalized Cartan matrix Open image in new window. Its connection to Siegel modular forms of genus 2 was first studied by A. Feingold and I. Frenkel. The denominator function of \(\mathcal{F}\) is not an automorphic form. However, Gritsenko and Nikulin extended \(\mathcal{F}\) to a generalized Kac–Moody algebra whose denominator function is a Siegel modular form. Using the Borcherds denominator identity, the denominator function can be written as an infinite product. The exponents that appear in the product are given by Fourier coefficients of a weak Jacobi form. P. Niemann also constructed a generalized Kac–Moody algebra which contains \(\mathcal {F}\) and whose denominator function is related to a product of Dedekind η-functions. In particular, root multiplicities of the generalized Kac–Moody algebra are determined by Fourier coefficients of a modular form. As the main results of this paper, we compute asymptotic formulas for these Fourier coefficients using the method of Hardy–Ramanujan–Rademacher, and obtain an asymptotic bound for root multiplicities of the algebra \(\mathcal{F}\). Our method can be applied to other hyperbolic Kac–Moody algebras and to other modular forms as demonstrated in the later part of the paper.

Serre obtained the p-adic limit of the integral Fourier coefficients of modular forms on SL 2(ℤ) for p = 2, 3, 5, 7. In this paper, we extend the result of Serre to weakly holomorphic modular forms of half integral weight on Γ0(4N) for N = 1, 2, 4. The proof is based on linear relations among Fourier coefficients of modular forms of half integral weight. As applications to our main result, we obtain congruences on various modular objects, such as those for Borcherds exponents, for Fourier coefficients of quotients of Eisentein series and for Fourier coefficients of Siegel modular forms on the Maass Space.

Linear relations among the Fourier coefficients of modular forms on groups [formula omitted] of genus zero and their applications

We prove a formula relating the Fourier coefficients of a modular form of half-integral weight to the special values of L-functions. The form in question is an explicit theta lift from the multiplicative group of an indefinite quaternion algebra over ℚ. This formula has applications to proving the nonvanishing of this lift and to an explicit version of the Rallis inner product formula.

Introduction. The main object of study in this paper with respect to zero-cycles is a special class of Hilbert-Blumenthal surfaces X, which are defined over Q as smooth compactifications of quasi-projective varieties S/Q, more precisely, of coarse moduli schemes S that represent the moduli stack of polarized abelian surfaces with real multiplication by the ring of integers in a real quadratic field F = Q( √ d). We assume that d = q is a prime ≡ 1(4) and that the class number of F is 1. Then S(C), the complex points of S, can be described as H×H/ SL2(OF ), where H is the upper halfplane. In the early seventies, Hirzebruch and Zagier [HZ] defined for each integer N a curve TN on S (called Hirzebruch-Zagier cycles) and showed that their intersection numbers occur as Fourier coefficients of modular forms of level q with Nebentypes eq , the quadratic character of F/ Q. In this connection with modular forms, HirzebruchZagier cycles reveal very similar properties to Hecke correspondences on the selfproduct of the modular curve X0(q). This crucial observation of Hirzebruch and Zagier, together with Tunnell’s proof of the Tate conjecture for a product of modular curves, inspired Harder, Langlands, and Rapoport [HLR] to prove the Tate conjecture for divisors on Hilbert-Blumenthal surfaces over abelian number fields. (The proof of the Tate conjecture was then accomplished by Klingenberg [Kl] and Murty and Ramakrishnan [MR] in the general case.) From the new strategy to study torsion zero-cycles on algebraic surfaces as developed in [LS] and used in [L1], [L2] (compare also [LR]), it is clear that a crucial point is the Tate conjecture in characteristic p at good reduction primes. As one of our main results, we prove the Tate conjecture in characteristic p for primes p that split in F for a certain class of Hilbert-Blumenthal surfaces. For this we recall that to each modular cusp form f of weight 2, level q, and Nebentypes eq , that is, f ∈ S2(� 0(q), eq ) ,w e associate a Hilbert modular cusp form ˆ f ∈ S2(SL2(OF )) under the Doi-Naganuma

In this paper, we give some results on simultaneous non-vanishing and simultaneous sign-changes for the Fourier coefficients of two modular forms. More precisely, given two modular forms [Formula: see text] and [Formula: see text] with Fourier coefficients [Formula: see text] and [Formula: see text] respectively, we consider the following questions: existence of infinitely many primes [Formula: see text] such that [Formula: see text]; simultaneous non-vanishing in the short intervals and in arithmetic progressions; simultaneous sign changes in short intervals.

We characterize Siegel cusp forms in the space of Siegel modular forms of large weight k > 2 ⁢ n {k>2n} on any Siegel congruence subgroup Γ of any degree n and any level N, by a suitable growth of their Fourier coefficients (e.g., by the well-known Hecke bound) at any one of the cusps. For this, we use a ‘local’ approach as compared to our previous results on this topic. We also touch upon the question in the context of vector-valued modular forms.

It is known that if a nonzero cusp form has real Fourier coefficients, then its Fourier coefficients change signs infinitely often. In this paper, we prove that there is a codimension one subspace in the space of holomorphic modular forms of square-free level such that all of its non-zero forms have similar sign change property.

In this paper I will describe some results and open problems connected with noncongruence subgroups of PSL2(z). Most of these have their origins in the fundamental paper of Atkin and Swinnerton-Dyer [1]. Considering how much we know about congruence subgroups and the associated modular forms, it is remarkable how little we can say in the general case (to avoid cumbersome language I shall speak of "congruence modular forms" and "noncongruence modular forms"). The principal difficulty is the absence of a satisfactory theory of Hecke operators. For congruence subgroups, the Hecke operators not only provide a direct interpretation of Fourier coefficients of modular forms in terms of eigenvalues, but also furnish a link with arithmetic, essentially through the representation theory of adèle groups. Moreover, using the action of Hecke operators one can calculate congruence modular forms with relative ease. For a noncongruence subgroup it is possible to define the Hecke algebra using double cosets (as in Ch. 3 of [9]), but it seems difficult to exploit (I take this opportunity to correct the erroneous assertion to the contrary at the beginning of [6]); and there seem to be no good alternative computational devices with which to calculate noncongruence modular forms. This problem is discussed in detail in [1].