Eta-quotients and embeddings of $$X_0(N)$$ in the projective plane

In this paper, we find projective plane models of the modular curves \(X_0(N)\) by constructing maps from \(X_0(N)\) to the projective plane using modular forms. We use eta-quotients of weight 12. We find those eta-quotients in \(M_{12}(\varGamma _0(N))\) which have maximal order of vanishing at infinity.     

A determination of the points which are rational over Q of three modular curves

It is proven that the cusps are the only points which are rational over Q on X₀(N) for N = 53, 113 and 137.     

On the distribution of the zeros of Eisenstein series for $$\Gamma _0^*(5)$$ Γ 0 ∗ ( 5 ) and $$\Gamma _0^*(7)$$ Γ 0 ∗ ( 7 )

The Ramanujan Journal - The goal of the present paper is to show that all the zeros of the Eisenstein series for $$\Gamma _0^*(5)$$ and $$\Gamma _0^*(7)$$ on the standard fundamental domain lie on...     

On primitive elements of algebraic function fields and models of $$X_0(N)$$ X 0 ( N )

The Ramanujan Journal - This paper is a continuation of our previous works where we study maps from $$X_0(N)$$ , $$N\ge 1$$ , into $${\mathbb {P}}^2$$ constructed via modular forms of the same...     

Klingen $${\mathfrak {p}}^2$$ p 2 vectors for $$\mathrm{GSp}(4)$$ GSp ( 4 )

The Ramanujan Journal - We calculate the dimensions of the spaces of invariant vectors under the Klingen congruence subgroup of level $${\mathfrak {p}}^2$$ for all irreducible, admissible...     

On the distribution of integer points of rational curves

Let F(X,Y) be an absolutely irreducible polynomial in such that the algebraic curve C: F(X,Y) = 0 has infinitely many integer points. In this paper we obtain an explicit estimate on the distribution of integer points of C.     

CM points on products of Drinfeld modular curves

Let be a product of Drinfeld modular curves over a general base ring of odd characteristic. We classify those subvarieties of which contain a Zariski-dense subset of CM points. This is an analogue of the André-Oort conjecture. As an application, we construct non-trivial families of higher Heegner points on modular elliptic curves over global function fields.

     

On the denominators of rational points on elliptic curves

Let x(P) = A_P/B²_P denote the x-coordinate of the rational pointP on an elliptic curve in Weierstrass form. We consider whenB_P can be a perfect power or a prime. Using Faltings' theorem,we show that for a fixed f > 1, there are only finitely manyrational points P with B_P equal to an fth power. Where descentvia an isogeny is possible, we show that there are only finitelymany rational points P with B_P equal to a prime, that thesepoints are bounded in number in an explicit fashion, and thatthey are effectively computable. Finally, we prove a strongerversion of this result for curves in homogeneous form.     

Singular rational curves with points of nearly-maximal weight

In this article we study rational curves with a unique unibranch genus-g singularity, which is of κ-hyperelliptic type in the sense of [27]; we focus on the cases $\kappa =0$ and $\kappa =1$, in which the semigroup associated to the singularity is of (sub)maximal weight. We obtain a partial classification of these curves according to the linear series they support, the scrolls on which they lie, and their gonality.     

K3 surfaces, rational curves, and rational points

We prove that for any of a wide class of elliptic surfaces X defined over a number field k, if there is an algebraic point on X that lies on only finitely many rational curves, then there is an algebraic point on X that lies on no rational curves. In particular, our theorem applies to a large class of elliptic K3 surfaces, which relates to a question posed by Bogomolov in 1981.     

On group orders of rational points of elliptic curves

We consider elliptic curves without complex multiplication defined over the rationals or with complex multiplication defined over the Hilbert class field of the endomorphism ring. We examine the distribution of almost prime group orders of these curves when reduced modulo a prime ideal.     

Decidability of the AE-theory of the lattice of $${\varPi }_1^0$$ classes

An AE-sentence is a sentence in prenex normal form with all universal quantifiers preceding all existential quantifiers, and the AE-theory of a structure is the set of all AE-sentences true in the structure. We show that the AE-theory of \((\mathscr {L}({\varPi }_1^0), \cap , \cup , 0, 1)\) is decidable by giving a procedure which, for any AE-sentence in the language, determines the truth or falsity of the sentence in our structure.     

The modular automorphisms of the Drinfeld modular curve $$ X_{1}(\frak n) $$

In this work we determine the group of modular automorphisms of the Drinfeld modular curve for any nonconstant polynomial .Received: 19 September 2002     

p-adicL-functions and rational points on elliptic curves with complex multiplication

Oblatum May 23, 1991     

Estimates for the number of rational points on convex curves and surfaces

Let Γ ⊂ ℝ^d be a bounded strictly convex surface. We prove that the number k_n(Γ) of points of Γ that lie on the lattice satisfies the following estimates: lim inf k_n(Γ)/n^d−2 < ∞ for d ≥ 3 and lim inf k_n(Γ)/log n < ∞ for d = 2. Bibliography: 9 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 344, 2007, pp. 174–189.     

Local geometry of smooth curves passing through rational double points

Partially supported by a fellowship from the University of Nebraska Research Council