Modular Curve X0 Research Articles

Singular moduli and the distribution of partition ranks modulo 2

AbstractIn this paper, we prove an asymptotic formula with a power saving error term for traces of weight zero weakly holomorphic modular forms of level N along Galois orbits of Heegner points on the modular curve X0(N). We use this result to study the distribution of partition ranks modulo 2. In particular, we give an asymptotic formula with a power saving error term for the number of partitions of a positive integer n with even (respectively, odd) rank. We use these results to deduce a strong quantitative form of equidistribution of partition ranks modulo 2.

Local to global trace questions and twists of genus one curves

— Let E be an elliptic curve defined over a number field F and K/F a quadratic extension. For a point P ∈ E(F) that is a local trace for every completion of K/F, we find necessary and sufficient conditions for P to lie in the image of the global trace map. These conditions can then be used to determine whether a quadratic twist of E, as a genus one curve, has rational points. In the case of quadratic twists of genus one modular curves X0(N) with squarefree N , the existence of rational points corresponds to the existence of Q-curves of degree N defined over K.

Computing homology using generalized Gröbner bases

A well-known theorem due to Manin gives a relationship between modular symbols for a congruence subgroup Γ0(N) of SL2(Z) and the homology of the modular curve X0(N), making the homology easier to compute. A corresponding theorem of Ash (1992) allows for explicit computation of the homology of congruence subgroups of SL3(Z) with coefficients in a given representation V. Applying Ashʼs theorem requires finding the invariants of an ideal in the group algebra Z[SL3(Z)] on V. We employ a generalized notion of Gröbner bases for a non-commutative group algebra in order to determine a minimal generating set for the desired ideal.

Quadratic modular symbols

We study a special kind of homology cycles of the modular curve X0(N). For a newform of weight 2 for Γ0(N), we construct a p-adic L-function by using these cycles. If the newform is defined over \({\mathbb{Q}}\), this p-adic L-function gives rise to algebraic points of the attached elliptic curve.

The Galois closure of Drinfeld modular towers

In this article we study Drinfeld modular curves X0(pn) associated to congruence subgroups Γ0(pn) of GL(2,Fq[T]) where p is a prime of Fq[T]. For n>r>0 we compute the extension degrees and investigate the structure of the Galois closures of the covers X0(pn)→X0(pr) and some of their variations. The results have some immediate implications for the Galois closures of two well-known optimal wild towers of function fields over finite fields introduced by Garcia and Stichtenoth, for which the modular interpretation was given by Elkies.

ON ASYMPTOTICALLY OPTIMAL TOWERS OVER QUADRATIC FIELDS RELATED TO GAUSS HYPERGEOMETRIC FUNCTIONS

We define two asymptotically optimal towers over quadratic fields, and give the explicit descriptions of the ramification loci and the sets of places splitting completely, which relate to the elliptic modular curves X0(4n) and X0(3n), respectively. Moreover, in the last section, we determine completely the modularity of a tower given by Maharaj and Wulftange in [18].

Automorphisms of hyperelliptic modular curves X0(N) in positive characteristic

AbstractWe study the automorphism groups of the reduction$X_0(N) \times \bar {\mathbb {F}}_p$of a modular curveX0(N) over primespN.

Equidistribution of Heegner points and the partition function

Let p(n) denote the number of partitions of a positive integer n. In this paper we study the asymptotic growth of p(n) using the equidistribution of Galois orbits of Heegner points on the modular curve X0(6). We obtain a new asymptotic formula for p(n) with an effective error term which is \({O(n^{-(\frac{1}{2}+\delta)})}\) for some δ > 0. We then use this asymptotic formula to sharpen the classical bounds of Hardy and Ramanujan, Rademacher, and Lehmer on the error term in Rademacher’s exact formula for p(n).

FOURIER EXPANSIONS WITH MODULAR FORM COEFFICIENTS

In this paper, we study the Fourier expansion where the coefficients are given as the evaluation of a sequence of modular forms at a fixed point in the upper half-plane. We show that for prime levels l for which the modular curve X0(l) is hyperelliptic (with hyperelliptic involution of the Atkin–Lehner type) then one can choose a sequence of weight k (any even integer) forms so that the resulting Fourier expansion is itself a meromorphic modular form of weight 2-k. These sequences have many interesting properties, for instance, the sequence of their first nonzero next-to-leading coefficient is equal to the terms in the Fourier expansion of a certain weight 2-k form. The results in the paper generalizes earlier work by Asai, Kaneko, and Ninomiya (for level one), and Ahlgren (for the cases where X0(l) has genus zero).

Twisted Gross–Zagier Theorems

Abstract.The theorems of Gross–Zagier and Zhang relate the Néron–Tate heights of complex multiplication points on the modular curve X0(N) (and on Shimura curve analogues) with the central derivatives of automorphic L-function. We extend these results to include certain CM points on modular curves of the form X (Ⲅ0(M ) ∩ Ⲅ1(S)) (and on Shimura curve analogues). These results are motivated by applications to Hida theory that can be found in the companion article “Central derivatives of L -functions in Hida families”,Math. Ann. 399(2007), 803–818.

Bounds on Faltings’s delta function through covers

Let X be a compact Riemann surface of genus gX 1. In 1984, G. Faltings introduced a new invariant Fal(X) associated to X. In this paper we give explicit bounds for Fal(X) in terms of fundamental dierential geometric invariants arising from X, when gX > 1. As an application, we are able to give bounds for Faltings’s delta function for the family of modular curves X0(N) in terms of the genus only. In combination with work of A. Abbes, P. Michel and E. Ullmo, this leads to an asymptotic formula for the Faltings height of the Jacobian J0(N) associated to X0(N).

AN "ANTI-HASSE PRINCIPLE" FOR PRIME TWISTS

Given an algebraic curve C/ℚ having points everywhere locally and endowed with a suitable involution, we show that there exists a positive density family of prime quadratic twists of C violating the Hasse principle. The result applies in particular to wN-Atkin–Lehner twists of most modular curves X0(N) and to wp-Atkin–Lehner twists of certain Shimura curves XD+.

Intersection Numbers of Heegner Divisors on Shimura Curves

In foundational papers, Gross, Zagier, and Kohnen established two formulas for arithmetic intersection numbers of certain Heegner divisors on integral models of modular curves. In [GZ1], only one imaginary quadratic discriminant plays a role. In [GZ2] and [GKZ], two quadratic discriminants play a role. In this paper we generalize the two-discriminant formula from the modular curves X0(N) to certain Shimura curves defined over Q. Our intersection formula was stated in [Ro], but the proof was only outlined there. Independently, the general formula was given, in a weaker and less explicit form, in [Ke2]; there it was proved completely. This paper is thus a synthesis of parts of [Ro] and [Ke2]. The intersection multiplicities computed here were used in [Ku] to derive a relation between height pairings and special values of the derivatives of certain Eisenstein series. We note also that Zhang [Zh] has generalized all of [GZ1] from ground field Q to general totally real ground fields F , working with general Shimura curves. So we certainly expect that all of [GKZ] should generalize similarly. Our work here can be viewed as a step in this direction. We are happy to thank B. Gross and S. Kudla for their encouragement during the long period in which this work was done.

Beilinson–Kato elements in K2of modular curves

This article investigates explicit linear dependence relations in the K2-group of modular curves. In particular, it is shown that the Beilinson-Kato elements in K2 of the modular curve Y (N) satisfy the Manin relations whenN is not divisible by 3. Similar results are obtained for the modular curves X1(N) and X0(N) when N is prime. Finally we exhibit explicit generators of K2, assuming the Beilinson conjecture.

ON THE EISENSTEIN IDEAL OF DRINFELD MODULAR CURVES

Let 𝔈(𝔭) denote the Eisenstein ideal in the Hecke algebra 𝕋(𝔭) of the Drinfeld modular curve X0(𝔭) parameterizing Drinfeld modules of rank two over 𝔽q[T] of general characteristic with Hecke level 𝔭-structure, where 𝔭 ◃ 𝔽q[T] is a non-zero prime ideal. We prove that the characteristic p of the field 𝔽qdoes not divide the order of the quotient 𝕋(𝔭)/𝔈(𝔭) and the Eisenstein ideal 𝔈(𝔭) is locally principal.

Weierstrass points on X 0(p ℓ) and arithmetic properties of Fourier coefficients of cusp forms

Generally it is unknown, whether or not ∞ is a Weierstrass point on the modular curve X0(N) if N is squarefree. A classical result of Atkin and Ogg states that ∞ is not a Weierstrass point on X0(N), if N=pM with p prime, p∤M and the genus of X0(M) zero. We use results of Kohnen and Weissauer to show that there is a connection between this question and the p-adic valuation of cusp forms under the Atkin–Lehner involution. This gives, in a sense, a generalization of Ogg’s Theorem in some cases.

Rational expression for J-invariant function in terms of generators of modular function fields

We give a method for expressing the modular j-invariant function J in a rational function of generators of the modular function field with respect to the modular group Γ0(N ). In the case the genus of the modular function field is positive, using this expression, we can determine isomorphism classes of elliptic curves corresponding to solutions of the defining equation, deduced from these generators, of the modular curve X0(N ). For every N from 6 to 50 and further for N = 52, we give computational results for the expression of J, the generators and the defining equation.

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.

Towards the triviality of $X_0^+ (p^r) (\mathbb{Q})$ for r > 1

We give a criterion to check if, given a prime power pr with r > 1, the only rational points of the modular curve X0+ (pr) are trivial (i.e. cusps or points furnished by complex multiplication). We then prove that this criterion is verified if p satisfies explicit congruences. This applies in particular to the modular curves Xsplit (p), which intervene in the problem of Serre concerning uniform surjectivity of Galois representations associated to division points of elliptic curves.

A Diophantine Equation Associated to X0(5)

AbstractSeveral classes of Fermat-type diophantine equations have been successfully resolved using the method of galois representations and modularity. In each case, it is possible to view the proper solutions to the diophantine equation in question as corresponding to suitably defined integral points on a modular curve of level divisible by 2 or 3. Motivated by this point of view, an example of a diophantine equation associated to the modular curve X0(5) is discussed in this paper. The diophantine equation has four terms rather than the usual three terms characteristic of generalized Fermat equations.