Search

We generalize Birch's construction of the Heegner points on X 0 ( N ) to construct new points X 1 ( N ) (and therefore construct points on the abelian varieties associated to J 1 ( N ) ). Then, we show that these points form an Euler system, and we improve Kolyvagin's Euler system techniques to show that for our point P τ K / c and any ring class character χ of the extended ring class field of conductor c satisfying χ = χ ‾ , if P τ K / c χ is non-torsion and G K → Aut A f [ π ] is surjective, then the corank of Sel ( A χ / K ) is 1, which implies the rank of A f ( K ) χ is 1. more...

We construct bases of the space of harmonic weak Maass forms of weight $$\kappa \in \frac{1}{2}{\mathbb {Z}}$$ . Using these bases, we obtain a Shintani lift from a positive integral weight harmonic weak Maass form to a half-integral weight harmonic weak Maass form, which reduces to the classical Shintani lift on the space of cusp forms. more...

Recently, a weak converse theorem for Borcherds' lifting operator of type $O(2,1)$ for $\G_0(N)$ is proved and the logarithmic derivative of a modular form for $\G_0(N)$ is explicitly described in terms of the values of Niebur-Poincar\'e series at its divisors in the complex upper half-plane. In this paper, we prove that the generalized Borcherds' lifting operator of type $O(2,1)$ is Hecke equivariant under the extension of Guerzhoy's multiplicative Hecke operator on the integral weight meromorphic modular forms and the Hecke operator on half-integral weight vector-valued harmonic weak Maass forms. Additionally, we show that the logarithmic differential operator is also Hecke equivariant under the multiplicative Hecke operator and the Hecke operator on integral weight meromorphic modular forms. As applications of Hecke equivariance of the two operators, we obtain relations for twisted traces of singular moduli modulo prime powers and congruences for twisted class numbers modulo primes, including those associated to genus $1$ modular curves., Comment: 18 pages more...

We show that the values of a certain family of weakly holomorphic modular functions at points in the divisors of any meromorphic modular form with algebraic Fourier coefficients are algebraic. We use this to extend the classical result of Schneider by proving that zeros or poles of any non-zero meromorphic modular form with algebraic Fourier coefficients are either transcendental or imaginary quadratic irrational. more...

In this work, we determine all modular curves X 0 + ( N ) that are bielliptic, and then find all numbers N for which the curves X 0 + ( N ) admit infinitely many quadratic points over Q .

For a square-free positive integer N, we study the normalizer of ΓΔ(N) in PSL2(ℝ) and investigate the group structure of its quotient by ΓΔ(N) under certain conditions.

In this paper, we find all Weierstrass points on the hyperelliptic modular curves $X_{0}(N)$ whose hyperelliptic involutions are non-exceptional, i.e., induced by matrices in $\mathrm{GL}_{2}(\mathbf{R})$. more...

Zagier defined lifts of weakly holomorphic modular functions to weakly holomorphic modular forms of weight 3/2. Duke and Jenkins extended Zagier-lifts for weakly holomorphic modular forms of negative-integral weights. In this paper, we show that half-integral weight harmonic weak Maass forms can be interpreted as Zagier-lifts of negative-integral weight harmonic weak Maass forms. more...

Let $E$ be an elliptic curve defined over $\Q$, and let $G$ be the torsion group $E(K)_{tors}$ for some cubic field $K$ which does not occur over $\Q$. In this paper, we determine over which types of cubic number fields (cyclic cubic, non-Galois totally real cubic, complex cubic or pure cubic) $G$ can occur, and if so, whether it can occur infinitely often or not. Moreover, if it occurs, we provide elliptic curves $E/\Q$ together with cubic fields $K$ so that $G= E(K)_{tors}$., Comment: 15 pages more...

We investigate Weierstrass points of the modular curve X Δ ( N ) of genus ≥2 when Δ is a proper subgroup of ( Z / N Z ) ⁎ . Let N = p 2 M where p is a prime number and M is a positive integer. Modifying Atkin's method in the case ± ( 1 + p M ) ∈ Δ , we find conditions for the cusp 0 to be a Weierstrass point on the modular curve X Δ ( p 2 M ) . Moreover, applying Schoneberg's theorem we show that except for finitely many N , the fixed points of the Fricke involutions W N are Weierstrass points on X Δ ( N ) . more...

In this paper we propose a method of computing thenumber of points on the reduction of non-hyperelliptic modularcurves of genus greater than or equal to 3 over flnite flelds. 1. IntroductionLet N be a positive integer, and leti 0 ( N ) :=‰µ a bc d ¶ 2 SL 2 (Z) jc · 0 mod N ¾ : Let X 0 ( N ) denote the modular curve corresponding to i 0 ( N ) and g 0 ( N )denote its genus. The modular curve X 0 ( N )(with cusps removed) parametrizes isomorphism classes of pairs ( E;C ), where E is an elliptic curveand C is a cyclic subgroup of E of order N .A curve X deflned over an algebraically closed fleld k is called d-gonal if it admits a map ` : X ! P 1 over k of degree d: The smallest possible d is called the gonality of X denoted by Gon( X ). If a curve X is 2-gonaland its genus g ( X ) ‚ 2, then X is said to be hyperelliptic . If a curve X is 3-gonal, then we call X trigonal .Ogg [4] determined all values of N for which X 0 ( N ) is hyperelliptic,and Hasegawa and Shimura [2] determined all the trigonal curves more...

We construct infinite families of elliptic curves with cyclic torsion groups over quartic number fields K such that the Galois closure of K is dihedral of degree 8; such a quartic number field K is called a dihedral quartic number field. In fact, all the cyclic torsion groups of elliptic curves which occur over quartic number fields (but not over quadratic number fields) are Z/NZ with N=17,20,21,22,24. The cases of N=20 and 24 are treated in the previous work of the authors, and the current work completes the construction of infinite families of elliptic curves over dihedral quartic number fields with cyclic torsion groups (which do not occur over quadratic number fields). more...

We determine which of the modular curves $X_\Delta(N)$, that is, curves lying between $X_0(N)$ and $X_1(N)$, are bielliptic. Somewhat surprisingly, we find that one of these curves has exceptional automorphisms. Finally we find all $X_\Delta(N)$ that have infinitely many quadratic points over $\mathbb{Q}$., Comment: 35 pages, published version, improved exposition, added many details more...

The author with C. H. Kim and Y. Lee constructedinflnite families of elliptic curves over cubic number flelds K withprescribed torsion groups which occur inflnitely often. In this paper,we examine the Galois structures of such cubic number flelds K forthe families of elliptic curves with cyclic torsion. 1. IntroductionOver cubic number flelds K , it is proved in [4] that all the groupstructures occurring inflnitely often as torsion groups E ( K ) tors are ex-actly the following 38 types:(1.1)Z =N 1 Z ; N 1 = 1 i 16 ; 18 ; 20Z = 2Z ' Z = 2 N 2 Z ; N 2 = 1 i 7where K varies over all cubic number flelds and E varies over all ellipticcurves over K . In [3] the author with C. H. Kim and Y. Lee constructinflnite families of elliptic curves with torsion structures in Eq. (1.2) overcubic number flelds K .In this paper we examine the Galois structures of deflning flelds ofthe constructed inflnite families of elliptic curves with cyclic torsion in[3]. For our purpose, it is su–cient to consider the families of ellipticcurves with cyclic torsion groups which do not occur over Q. That is,we consider the following groups(1.2) Z more...

In this paper, we give a new method to get defining equations of modular curves $X_1(2N)$ which show the moduli problems.

In this paper, we explain how to get defining equations of the modular curves which show the moduli problems and present defining equations of for $N

The primary goal of this paper is to construct the basis of the space of weight 3/2 mock modular forms, which is an extension of the Borcherds–Zagier basis of weight 3/2 weakly holomorphic modular forms. The shadows of the members of this basis form the Borcherds–Zagier basis of the space of weight 1/2 weakly holomorphic modular forms. In the course of the construction, we provide a full computation of the Fourier coefficients for the weak Maass–Poincare series in most general form for the purpose of future reference. more...

In this study, we determine all modular curves X 0 ( N ) that admit infinitely many cubic points.

A class invariant is the value of a modular function that generates a ring class field of an imaginary quadratic number field such as the singular moduli of level 1. In this paper, using Shimura Reciprocity Law, we compute the Galois actions of a class invariant from a generalized Weber function over quadratic number fields with discriminant . more...

We find infinite families of elliptic curves over quartic number fields with torsion group Z/NZ with N=20,24. We prove that for each elliptic curve Et in the constructed families, the Galois group Gal(L/Q) is isomorphic to the Dihedral group D4 of order 8 for the Galois closure L of K over Q, where K is the defining field of (Et,Qt) and Qt is a point of Et of order N. We also notice that the plane model for the modular curve X1(24) found in Jeon et al. (2011) [1] is in the optimal form, which was the missing case in Sutherlandʼs work (Sutherland, 2012 [12]). more...

In this paper, we present a explicit procedure to compute a basis for the spaces of cuspforms of weight 2 on which consists of eigenforms for the Atkin-Lehner involutions.

Zagier showed that the Galois traces of the values of j-invariant at CM points are Fourier coefficients of a weakly holomorphic modular form of weight 3/2 and Bruinier–Funke expanded his result to the sums of the values of arbitrary modular functions at Heegner points. In this paper, we identify the Galois traces of real-valued class invariants with modular traces of the values of certain modular functions at Heegner points so that they are Fourier coefficients of weight 3/2 weakly holomorphic modular forms. more...

In this paper, we determine the automorphism groups of the hyperelliptic modular curves $X_{1}(N)$, and determine explicit forms for the actions of all automorphisms on certain defining equations of $X_{1}(N)$. more...

We determine which groups ${\mathbb{Z}}/M{\mathbb{Z}}\oplus{\mathbb{Z}}/N{\mathbb{Z}}$ occur infinitely often as torsion groups $E(K)_{\operatorname{tors}}$ when $K$ varies over all quartic number fields and $E$ varies over all elliptic curves over $K$ . more...

In this work, we find plane models for certain Drinfeld modular curves X 0 ( n ) which have better properties than the plane models derived from the usual Drinfeld modular equations. As an application, we construct ring class fields over imaginary quadratic fields by using singular values of generators of the function field of X 0 ( n ) . more...

In this work, we determine all modular curves X 1(M,N) which are bielliptic, and then we also discuss the problem of finding the curves X 1(M,N) which admit infinitely many quartic points over .

In this work, we determine all modular curves X1(M,N) which are bielliptic, and then we also discuss the problem of finding the curves X1(M,N) which admit infinitely many quartic points over Open image in new window. more...

In this work, we determine all Drinfeld modular curves X 1 ( n ) that are hyperelliptic or bielliptic.

An error in Proposition 5.2 of “Weak Maass–Poincare series and weight 3/2 mock modular forms” [J. Number Theory 133 (8) (2013) 2567–2587] is corrected.

Borcherds-Zagier bases of the spaces of weakly holomorphic modular forms of weights 1/2 and 3/2 share the Fourier coefficients which are traces of singular moduli. Recently, Duke, Imamo\={g}lu, and T\'{o}th have constructed a basis of the space of weight 1/2 mock modular forms, each member in which has Zagier's generating series of traces of singular moduli as its shadow. They also showed that Fourier coefficients of their mock modular forms are sums of cycle integrals of the $j$-function which are real quadratic analogues of singular moduli. In this paper, we prove the Fourier coefficients of a basis of the space of weight 3/2 mock modular forms are sums of cycle integrals of a sesqui-harmonic Maass form of weight zero whose image under hyperbolic Laplacian is the $j$-function. Furthermore, we express these sums as regularized inner products of weakly holomorphic modular forms of weight 1/2., Comment: 14 pages more...

A conjecture of Goldfeld implies that a positive proportion of quadratic twists of an elliptic curve E/Q has (analytic) rank 1. This assertion has been confirmed by Vatsal (V1) and the first author (By) for only two elliptic curves. Here we confirm this assertion for infinitely many elliptic curves E/Q using the Heegner divisors, the 3-part of the class groups of quadratic fields, and a variant of the binary Goldbach problem for polynomials. more...

In this work, we estimate the genus of the intermediate modular curves between $X_1(N)$ and $X_0(N),$ and determine all the trigonal ones., 18 pages