On some entire modular forms of weights 5 and 6 for the congruence group Г(in0)(4N)

Two entire modular forms of weight 5 and two of weight 6 for the congruence subgroup Γ₀ (4N)are constructed, which will be useful for revealing the arithmetical sense of additional terms in formulas for the number of representations of positive integers by quadratic forms in 10 and 12 variables.     

Shintani lifts and fractional derivatives for harmonic weak Maass forms

In this paper, we construct Shintani lifts from integral weight weakly holomorphic modular forms to half-integral weight weakly holomorphic modular forms. Although defined by different methods, these coincide with the classical Shintani lifts when restricted to the space of cusp forms. As a side effect, this gives the coefficients of the classical Shintani lifts as new cycle integrals. This yields new formulas for the L-values of Hecke eigenforms. When restricted to the space of weakly holomorphic modular forms orthogonal to cusp forms, the Shintani lifts introduce a definition of weakly holomorphic Hecke eigenforms. Along the way, auxiliary lifts are constructed from the space of harmonic weak Maass forms which yield a “fractional derivative” from the space of half-integral weight harmonic weak Maass forms to half-integral weight weakly holomorphic modular forms. This fractional derivative complements the usual ξ-operator introduced by Bruinier and Funke.     

On Atkin and Swinnerton-Dyer congruence relations (3)

It is now well known that Hecke operators defined classically act trivially on genuine cuspforms for noncongruence subgroups of SL₂(Z). Atkin and Swinnerton-Dyer speculated the existence of p-adic Hecke operators so that the Fourier coefficients of their eigenfunctions satisfy three-term congruence recursions. In the previous two papers with the same title ([W.C. Li, L. Long, Z. Yang, On Atkin and Swinnerton-Dyer congruence relations, J. Number Theory 113 (1) (2005) 117-148] by W.C. Li, L. Long, Z. Yang and [A.O.L. Atkin, W.C. Li, L. Long, On Atkin and Swinnerton-Dyer congruence relations (2), Math. Ann. 340 (2) (2008) 335-358] by A.O.L. Atkin, W.C. Li, L. Long), the authors have studied two exceptional spaces of noncongruence cuspforms where almost all p-adic Hecke operators can be diagonalized simultaneously or semi-simultaneously. Moreover, it is shown that the l-adic Scholl representations attached to these spaces are modular in the sense that they are isomorphic, up to semisimplification, to the l-adic representations arising from classical automorphic forms.In this paper, we study an infinite family of spaces of noncongruence cuspforms (which includes the cases in [W.C. Li, L. Long, Z. Yang, On Atkin and Swinnerton-Dyer congruence relations, J. Number Theory 113 (1) (2005) 117-148; A.O.L. Atkin, W.C. Li, L. Long, On Atkin and Swinnerton-Dyer congruence relations (2), Math. Ann. 340 (2) (2008) 335-358]) under a general setting. It is shown that for each space in this family there exists a fixed basis so that the Fourier coefficients of each basis element satisfy certain weaker three-term congruence recursions. For a new case in this family, we will exhibit that the attached l-adic Scholl representations are modular and the p-adic Hecke operators can be diagonalized semi-simultaneously.     

Construction of entire modular forms of weights 5 and 6 for the congruence group Γ0(4N)

Two classes of entire modular forms of weight 5 and two of weight 6 are constructed for the congruence subgroup ₀(4N). The constructed modular forms as well as the modular forms from [1] will be helpful in the theory of representation of numbers by the quadratic forms in 10 and 12 variables.     

On type numbers of split orders of definite quaternion algebras

In this paper, we shall give a new relation between the arithmetic of quaternion algebras and modular forms; we shall express the type numberT _{q, N} of a split order of type (q, N) as the sums of dimensions of some subspaces of the space of cusp forms of weight 2 with respect to Γ₀(qN) which are common eigenspaces of Atkin-Lehner's involutions.     

Weight-dependent congruence properties of modular forms

In this paper, we study congruence properties of modular forms in various ways. By proving a weight-dependent congruence property of modular forms, we give some sufficient conditions, in terms of the weights of modular forms, for a modular form to be non-p-ordinary. As applications of our main theorem we derive a linear relation among coefficients of new forms. Furthermore, congruence relations among special values of Dedekind zeta functions of real quadratic fields are derived.     

Moduli for pairs of elliptic curves¶with isomorphic N-torsion

We study the moduli surface for pairs of elliptic curves together with an isomorphism between their N-torsion groups. The Weil pairing gives a “determinant” map from this moduli surface to (Z/N Z)^*; its fibers are the components of the surface. We define spaces of modular forms on these components and Hecke correspondences between them, and study how those spaces of modular forms behave as modules for the Hecke algebra. We discover that the component with determinant −1 is somehow the “dominant” one; we characterize the difference between its spaces of modular forms and the spaces of modular forms on the other components using forms with complex multiplication. In addition, we prove Atkin–Lehner-style results about these spaces of modular forms. Finally, we show some simplifications that arise when N is prime, including a complete determination of such CM-forms, and give numerical examples. Received: 20 September 2000 / Revised version: 7 February 2001     

Some Siegel modular standard L-values, and Shafarevich-Tate groups

We explain how the Bloch-Kato conjecture leads us to the following conclusion: a large prime dividing a critical value of the L-function of a classical Hecke eigenform f of level 1, should often also divide certain ratios of critical values for the standard L-function of a related genus two (and in general vector-valued) Hecke eigenform F. The relation between f and F (Harder?s conjecture in the vector-valued case) is a congruence involving Hecke eigenvalues, modulo the large prime. In the scalar-valued case we prove the divisibility, subject to weak conditions. In two instances in the vector-valued case, we confirm the divisibility using elaborate computations involving special differential operators. These computations do not depend for their validity on any unproved conjecture.     

On Atkin and Swinnerton-Dyer congruence relations (2)

In this paper we give an example of a noncongruence subgroup whose three-dimensional space of cusp forms of weight 3 has the following properties. For each of the four residue classes of odd primes modulo 8 there is a basis whose Fourier coefficients at infinity satisfy a three-term Atkin and Swinnerton-Dyer congruence relation, which is the p-adic analogue of the three-term recursion satisfied by the coefficients of classical Hecke eigenforms. We also show that there is an automorphic L-function over whose local factors agree with those of the l-adic Scholl representations attached to the space of noncongruence cusp forms. The research of the second author was supported in part by an NSA grant #MDA904-03-1-0069 and an NSF grant #DMS-0457574. Part of the research was done when she was visiting the National Center for Theoretical Sciences in Hsinchu, Taiwan. She would like to thank the Center for its support and hospitality. The third author was supported in part by an NSF-AWM mentoring travel grant for women. She would further thank the Pennsylvania State University and the Institut des Hautes études Scientifiques for their hospitality.     

On simultaneous non-vanishing of modular <Emphasis Type="Italic">L</Emphasis>-functions

For i = 1, , r, let f _i be newforms of weight 2k _i for Γ₀(N _i ) with trivial character. We consider the simultaneous non-vanishing problem for the central values of twisted L-functions of f _i . By using the Shimura correspondence, we give a certain relation between this problem and the kernel fields of 2-adic Galois representations associated to modular forms. Received: 28 January 2006     

On the representation of integers as sums of triangular numbers

Summary In this survey article we discuss the problem of determining the number of representations of an integer as sums of triangular numbers. This study yields several interesting results. Ifn 0 is a non-negative integer, then thenth triangular number isT _n =n(n + 1)/2. Letk be a positive integer. We denote by _k (n) the number of representations ofn as a sum ofk triangular numbers. Here we use the theory of modular forms to calculate _k (n). The case wherek = 24 is particularly interesting. It turns out that, ifn 3 is odd, then the number of points on the 24 dimensional Leech lattice of norm 2n is 2¹²(2¹² – 1) ₂₄(n – 3). Furthermore the formula for ₂₄(n) involves the Ramanujan(n)-function. As a consequence, we get elementary congruences for(n). In a similar vein, whenp is a prime, we demonstrate ₂₄(p ^k – 3) as a Dirichlet convolution of ₁₁(n) and(n). It is also of interest to know that this study produces formulas for the number of lattice points insidek-dimensional spheres.     

Arithmetic expressions of Selberg's zeta functions for congruence subgroups

In [P. Sarnak, Class numbers of indefinite binary quadratic forms, J. Number Theory 15 (1982) 229-247], it was proved that the Selberg zeta function for SL₂(Z) is expressed in terms of the fundamental units and the class numbers of the primitive indefinite binary quadratic forms. The aim of this paper is to obtain similar arithmetic expressions of the logarithmic derivatives of the Selberg zeta functions for congruence subgroups of SL₂(Z). As applications, we study the Brun-Titchmarsh type prime geodesic theorem and the asymptotic formula of the sum of the class number.     

On mod <Emphasis Type="Italic">p</Emphasis> properties of Siegel modular forms

We prove two results on mod p properties of Siegel modular forms. First, we use theta series in order to construct of a Siegel modular form of weight p−1 which is congruent to 1 mod p. Second, we define a theta operator on q-expansions and show that the algebra of Siegel modular forms mod p is stable under , by exploiting the relation between and generalized Rankin-Cohen brackets.     

Defining equations of modular curves

We obtain defining equations of modular curves X₀(N), X₁(N), and X(N) by explicitly constructing modular functions using generalized Dedekind eta functions. As applications, we describe a method of obtaining a basis for the space of cusp forms of weight 2 on a congruence subgroup. We also use our model of X₀(37) to find explicit modular parameterization of rational elliptic curves of conductor 37.     

Exceptional Congruences for Powers of the Partition Function

In Journal of London Math. Soc. 31 (1956), 350–359, Morris Newman studied vector spaces of functions arising from lifts to ₀(p) of certain eta-products on the group ₀(pQ), Q = p ⁿ. In this paper, the author considers vector spaces of modular functions obtained as lifts of more general eta-products from ₀(pQ) to ₀(p), (Q, p) = 1. Specifically considered are functions arising as lifts of functions of the form

2-affine complete algebras need not be affine complete

For each we exhibit a finite algebra R _k such that R _k is k-affine complete, but not (k+1)-affine complete; this means that every k-ary congruence preserving function on R _k lies in , but there is a (k +1)-ary congruence preserving function of R _k that does not lie in . Received September 27, 2001; accepted in final form February 9, 2002.     

The 2-primary torsion on elliptic curves in the Z p -extensions of Q

Let E be an elliptic curve over Q and p a prime number. Denote by Q_p,∞ the Z_p-extension of Q. In this paper, we show that if p≠3, then where E(Q_p,∞)₍₂₎ is the 2-primary part of the group E(Q_p,∞) of Q_p,∞-rational points on E. More precisely, in case p=2, we completely classify E(Q_2,∞)₍₂₎ in terms of E(Q)₍₂₎; in case p≥5 (or in case p=3 and E(Q)₍₂₎≠{O}), we show that E(Q_p,∞)₍₂₎=E(Q)₍₂₎.     

Q-curves of degree 5 and jacobian surfaces of GL2-type

We construct a parametric family {E ^(±)(s,t,u)} of minimal Q-curves of degree 5 over the quadratic fields Q , and the family {C(s,t,u)} of genus two curves over Q covering E ^{(+)(s,t,u) whose jacobians are abelian surfaces of GL₂-type. We also discuss the modularity for them and the sign change between E ^{(+)(s,t,u) and its twist E ⁽⁻⁾(s,t,u), which correspond by modularity to cusp forms of trivial and non-trivial Neben type characters, respectively. We find in {C(s,t,u)} concrete equations of curves over Q whose jacobians are isogenous over cyclic quartic fields to Shimura's abelian surfaces A _f attached to cusp forms of Neben type character of level N= 29, 229, 349, 461, and 509. Received: 23 September 1997 / Revised version: 26 May 1998     

Mixed Siegel Modular Forms and Special Valuesof Certain Dirichlet Series

 Let be a Siegel modular form of weight ?, and let be an Eichler embedding, where denotes the Siegel upper half space of degree n. We use the notion of mixed Siegel modular forms to construct the linear map of the spaces of Siegel cusp forms for the congruence subgroup and express the Fourier coefficients of the image of an element under in terms of special values of a certain Dirichlet series. We also discuss connections between mixed Siegel cusp forms and holomorphic forms on a family of abelian varieties. (Received 28 February 2000; in revised form 11 July 2000)     

Nonvanishing of Koecher–Maass series attached to Siegel cusp forms

We prove a nonvanishing result for Koecher–Maass series attached to Siegel cusp forms of weight k and degree n   in certain strips on the complex plane. When n=2$n=2$, we prove such a result for forms orthogonal to the space of the Saito–Kurokawa lifts ‘up to finitely many exceptions’, in bounded regions.