scholarly journals On the representation of a number as a sum of squares

1978 ◽  
Vol 19 (2) ◽  
pp. 173-197 ◽  
Author(s):  
Karl-Bernhard Gundlach
Keyword(s):  
Positive Integer ◽  
Modular Forms ◽  
Remainder Term ◽  
Modular Group ◽  
Congruence Subgroup ◽  
Hilbert Modular Group ◽  
Totally Positive ◽  
Hilbert Modular ◽  
Image Position ◽  
Totally Real

It is well known that the number Ak(m) of representations of a positive integer m as the sum of k squares of integers can be expressed in the formwhere Pk(m) is a divisor function, and Rk(m) is a remainder term of smaller order. (1) is a consequence of the fact thatis a modular form for a certain congruence subgroup of the modular group, andwithwhere Ek(z) is an Eisenstein series and is a cusp form (as was first pointed out by Mordell [9]). The result (1) remains true if m is taken to be a totally positive integer from a totally real number field K and Ak(m) is the number of representations of m as the sum of k squares of integers from K (at least for 2|k, k>2, and for those cases with 2+k which have been investigated). then are replaced by modular forms for a subgroup of the Hilbert modular group with Fourier expansions of the form (10) (see section 2).

scholarly journals Automorphy factors for a Hilbert modular group

1988 ◽  
Vol 30 (2) ◽  
pp. 231-236
Author(s):  
Shigeaki Tsuyumine
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Rational Number ◽  
Modular Group ◽  
Algebraic Number Field ◽  
Maximal Order ◽  
Hilbert Modular Group ◽  
Hilbert Modular ◽  
Totally Real

Let K be a totally real algebraic number field of degree n > 1, and let OK be the maximal order. We denote by гk, the Hilbert modular group SL2(OK) associated with K. On the extent of the weight of an automorphy factor for гK, some restrictions are imposed, not as in the elliptic modular case. Maass [5] showed that the weight is integral for K = ℚ(√5). It was shown by Christian [1] that for any Hilbert modular group it is a rational number with the bounded denominator depending on the group.

scholarly journals Multiplier systems for Hilbert's and Siegel's modular groups

1985 ◽  
Vol 27 ◽  
pp. 57-80 ◽  
Author(s):  
Karl-Bernhard Gundlach
Keyword(s):  
Modular Forms ◽  
Functional Equations ◽  
Modular Group ◽  
Meromorphic Functions ◽  
Invariance Property ◽  
Modular Functions ◽  
Multiplier System ◽  
Modular Groups ◽  
Image Position ◽  
Totally Real

The classical generalizations (already investigated in the second half of last century) of the modular group SL(2, ℤ) are the groups ГK = SL(2, o)(o the principal order of a totally real number field K, [K:ℚ]=n), operating, originally, on a product of n upper half-planes or, for n=2, on the product 1×− of an upper and a lower half-plane by(where v(i), for v∈K, denotes the jth conjugate of v), and Гn = Sp(n, ℤ), operating on n={Z∣Z=X+iY∈ℂ(n,n),tZ=Z, Y>0} byNowadays ГK is called Hilbert's modular group of K and Гn Siegel's modular group of degree (or genus) n. For n=1 we have Гℚ=Г1= SL(2, ℤ). The functions corresponding to modular forms and modular functions for SL(2, ℤ) and its subgroups are holomorphic (or meromorphic) functions with an invariance property of the formJ(L, t) for fixed L (or J(M, Z) for fixed M) denoting a holomorphic function without zeros on ) (or on n). A function J;, defined on ℤK×or ℤn×n to be able to appear in (1.3) with f≢0, has to satisfy certain functional equations (see below, (2.3)–(2.5) for ГK, (5.7)–(5.9) for Гn) and is called an automorphic factor (AF) then. In close analogy to the case n=1, mainly AFs of the following kind have been used:with a complex number r, the weight of J, and complex numbers v(L), v(M). AFs of this kind are called classical automorphic factors (CAP) in the sequel. If r∉ℤ, the values of the function v on ГK (or Гn) depend on the branch of (…)r. For a fixed choice of the branch (for each L∈ГK or M∈Гn) the functional equations for J, by (1.4), (1.5), correspond to functional equations for v. A function v satisfying those equations is called a multiplier system (MS) of weight r for ГK (or Гn).

On the elliptic points of the Hilbert modular group of the totally real cyclotomic cubic field ℚ(ζ9)+

2013 ◽  
Vol 143 (5) ◽  
pp. 893-903
Author(s):  
A. Arenas
Keyword(s):  
Modular Group ◽  
Root Of Unity ◽  
Cubic Field ◽  
Hilbert Modular Group ◽  
Hilbert Modular ◽  
Totally Real

We determine explicitly the elliptic points with respect to the Hilbert modular group associated with the totally real cyclotomic cubic field ℚ(ζ + ζ−1), where ζ stands for a primitive 9th root of unity.

The coefficients of certain integral modular forms

1954 ◽  
Vol 50 (2) ◽  
pp. 305-308 ◽  
Author(s):  
R. A. Rankin ◽  
J. M. Rushforth
Keyword(s):  
Positive Integer ◽  
Vector Space ◽  
Modular Forms ◽  
Modular Group ◽  
Finite Basis ◽  
Cusp Forms ◽  
Negative Dimension ◽  
Image Position

The notation which we use is that of a recent paper by one of us, and we quote results from that paper as they are required. It is known (see R, Theorem 1, for example) that the vector space k of all cusp-forms f(z) of even negative dimension – k (k ≥ 12), belonging to the full modular group Γ(1), possesses a finite basis of formswhere k is defined by (2·10) of R and the coefficients possess the following properties:for a prime p, where p is a positive integer.

RESTRICTIONS OF HILBERT MODULAR FORMS

2009 ◽  
Vol 05 (01) ◽  
pp. 67-80
Author(s):  
NAJIB OULED AZAIEZ
Keyword(s):  
Modular Forms ◽  
Modular Group ◽  
Quadratic Field ◽  
Hilbert Modular Forms ◽  
Real Quadratic Field ◽  
Modular Surface ◽  
Modular Curve ◽  
Hilbert Modular Group ◽  
Hilbert Modular ◽  
Real Quadratic

Let Γ ⊂ PSL (2, ℝ) be a discrete and finite covolume subgroup. We suppose that the modular curve [Formula: see text] is "embedded" in a Hilbert modular surface [Formula: see text], where ΓK is the Hilbert modular group associated to a real quadratic field K. We define a sequence of restrictions (ρn)n∈ℕ of Hilbert modular forms for ΓK to modular forms for Γ. We denote by Mk1, k2(ΓK) the space of Hilbert modular forms of weight (k1, k2) for ΓK. We prove that ∑n∈ℕ ∑k1, k2∈ℕ ρn(Mk1, k2(ΓK)) is a subalgebra closed under Rankin–Cohen brackets of the algebra ⊕k∈ℕ Mk(Γ) of modular forms for Γ.

scholarly journals Projective varieties and rings of Thetanullwerte

1990 ◽  
Vol 118 ◽  
pp. 165-176
Author(s):  
Riccardo Salvati Manni
Keyword(s):  
Positive Integer ◽  
Holomorphic Function ◽  
Half Space ◽  
Modular Forms ◽  
Congruence Subgroup ◽  
Integral Closure ◽  
Graded Ring ◽  
Projective Varieties ◽  
Entry Vector

Let r denote an even positive integer, m an element of Q2g such that r·m ≡ 0 mod 1 and ϑm the holomorphic function on the Siegel upper-half space Hg defined by(1) ,in which e(t) stands for exp and m′ and m″ are the first and the second entry vector of m. Let Θg(r) denote the graded ring generated over C by such Thetanullwerte; then it is a well known fact that the integral closure of Θg(r) is the ring of all modular forms relative to Igusa’s congruence subgroup Γg(r2, 2r2) cf. [6]. We shall denote this ring by A(Γg(r2, 2r2)).

scholarly journals Minimal weights of Hilbert modular forms in characteristic

2017 ◽  
Vol 153 (9) ◽  
pp. 1769-1778 ◽  
Author(s):  
Fred Diamond ◽  
Payman L Kassaei
Keyword(s):  
Modular Forms ◽  
Real Field ◽  
Hilbert Modular Forms ◽  
Hilbert Modular ◽  
Hilbert Modular Varieties ◽  
Totally Real ◽  
Modular Varieties ◽  
Totally Real Field

We consider mod $p$ Hilbert modular forms associated to a totally real field of degree $d$ in which $p$ is unramified. We prove that every such form arises by multiplication by partial Hasse invariants from one whose weight (a $d$-tuple of integers) lies in a certain cone contained in the set of non-negative weights, answering a question of Andreatta and Goren. The proof is based on properties of the Goren–Oort stratification on mod $p$ Hilbert modular varieties established by Goren and Oort, and Tian and Xiao.

scholarly journals Eichler Maps and Hyperbolic Fourier Expansion

1970 ◽  
Vol 40 ◽  
pp. 173-192 ◽  
Author(s):  
Toyokazu Hiramatsu
Keyword(s):  
Positive Integer ◽  
Modular Forms ◽  
Quadratic Forms ◽  
Fourier Expansion ◽  
Automorphic Forms ◽  
Hilbert Modular Forms ◽  
Ternary Quadratic Forms ◽  
Hilbert Modular ◽  
Lecture Notes

In his lecture notes ([1, pp. 33-35], [2, pp. 145-152]), M. Eichler reduced ‘quadratic’ Hilbert modular forms of dimension —k {k is a positive integer) to holomorphic automorphic forms of dimension — 2k for the reproduced groups of indefinite ternary quadratic forms, by means of so-called Eichler maps.

scholarly journals The normalizer of Γ0(N) in PSL(2, ℝ)

1990 ◽  
Vol 32 (3) ◽  
pp. 317-327 ◽  
Author(s):  
M. Akbas ◽  
D. Singerman
Keyword(s):  
Positive Integer ◽  
Modular Group ◽  
Prime Divisors ◽  
Möbius Transformations ◽  
The Matrix ◽  
Image Position ◽  
Mobius Transformations

Let Γ denote the modular group, consisting of the Möbius transformationsAs usual we denote the above transformation by the matrix remembering that V and – V represent the same transformation. If N is a positive integer we let Γ0(N) denote the transformations for which c ≡ 0 mod N. Then Γ0(N) is a subgroup of indexthe product being taken over all prime divisors of N.

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