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Beilinson [Higher regulators and values of L-functions, Itogi Nauki i Tekhniki Seriya Sovremennye Problemy Matematiki Noveishie Dostizheniya (Current problems in mathematics), vol. 24 (Vserossiisky Institut Nauchnoi i Tekhnicheskoi Informatsii, Moscow, 1984), 181–238] obtained a formula relating the special value of the L-function of H2 of a product of modular curves to the regulator of an element of a motivic cohomology group, thus providing evidence for his general conjectures on special values of L-functions. In this paper we prove a similar formula for the L-function of the product of two Drinfeld modular curves, providing evidence for an analogous conjecture in the case of function fields. [ABSTRACT FROM AUTHOR]

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Abstract: A curve C defined over is modular of level N if there exists a non-constant morphism from onto C defined over for some positive integer N. We provide a sufficient and necessary condition for the existence of a modular non-hyperelliptic curve C of genus 3 and level N such that Jac C is -isogenous to a given three dimensional -quotient of . Using this criterion, we present an algorithm to compute explicitly equations for modular non-hyperelliptic curves of genus 3. Let C be a modular curve of level N, we say that C is new if the corresponding morphism between and Jac C factors through the new part of . We compute equations of 44 non-hyperelliptic new modular curves of genus 3, that we conjecture to be the complete list of this kind of curves. Furthermore, we describe some aspects of non-new modular curves and we present some examples that show the ambiguity of the non-new modular case. [Copyright &y& Elsevier]

The article presents a study which demonstrates that modular functions j1,5 and j1, 6 generate function fields of the modular curves. It mentions that the authors make appropriate changes of variablse to achieve an elliptic curve. It presents several mathematical and transformational formulas to achieve the desired result.

Modular forms for a discrete subgroup Γ of SL.2; R/ can be identified with holomorphic sections of line bundles over the modular curve U corresponding to Γ, and quasimodular forms generalize modular forms. We construct vector bundles over U whose sections can be identified with quasimodular forms for Γ. [ABSTRACT FROM AUTHOR]

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). [ABSTRACT FROM AUTHOR]

We study pullbacks of modular forms of weight 1 from the modular curve X(4) to the modular curve X(4p), where p is an odd prime. We find the extent to which such modular forms separate points on X(4p). Our main result is that these modular forms give rise to a morphism F from the quotient of X(4p) by a certain involution ι to projective space, such that F is a projective embedding of X(4p)/ι away from the cusps. We also report on computer calculations regarding products of such modular forms, going up to weight 4 for p ≤ 13, and up to weight 3 for p ≤ 23, and make a conjecture about these products and the nature of the singularities at the cusps of the image F(X(4p)/ι). [ABSTRACT FROM AUTHOR]

Let E be an elliptic curve of conductor p. Given a cyclic subgroup C of order p in E[p], we construct a modular point PC on E, called self-point, as the image of (E,C) on X0(p) under the modular parametrization X0(p) → E. We prove that the point is of infinite order in the Mordell–Weil group of E over the field of definition of C. One can deduce a lower bound on the growth of the rank of the Mordell–Weil group in its PGL2(ℤp)-tower inside ℚ(E[p∞]). [ABSTRACT FROM AUTHOR]

AbstractIn the predecessor to this article, we used global equidistribution theorems to prove that given a correspondence between a modular curve and an elliptic curve A, the intersection of any finite-rank subgroup of Awith the set of CM-points of Ais finite. In this article we apply local methods, involving the theory of arithmetic differential equations, to prove quantitativeversions of a similar statement. The new methods apply also to certain infinite-rank subgroups, as well as to the situation where the set of CM-points is replaced by certain isogeny classes of points on the modular curve. Finally, we prove Shimura-curve analogues of these results. [ABSTRACT FROM AUTHOR]

Abstract: We address the problem of computing in the group of -torsion rational points of the jacobian variety of algebraic curves over finite fields, with a view toward computing modular representations. [Copyright &y& Elsevier]

The article focuses on the investigation of the non-hyperelliptic modular Jacobians Jac(C[f]) of dimension three. It mentions that the non-hyperelliptic modular Jacobians of dimension of three was describe by computing a model of the curve of genus three which was equipped with canonical polarization. It explores the approach used by Poor to determine if the curve C us hyperelliptic which consists of the values of Riemann's theta function at event two-torsion points.

Abstract: We show that, in geometrically connected modular curves associated with congruence subgroups of , one has equidistribution, towards the hyperbolic probability, of Galois orbits of the modular invariants associated with a level structure on elliptic curves within a given isogeny class. To cite this article: R. Richard, C. R. Acad. Sci. Paris, Ser. I 347 (2009). [Copyright &y& Elsevier]

We compare the asymptotic grows of the number of rational points on modular varieties of 𝒟-elliptic sheaves over finite fields to the grows of their Betti numbers as the degree of the level tends to infinity. This is a generalization to higher dimensions of a well-known result for modular curves. As a consequence of the main result, we produce a new asymptotically optimal sequence of curves. [ABSTRACT FROM AUTHOR]

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+. [ABSTRACT FROM AUTHOR]

Abstract: The purpose of the paper is to determine the cuspidal class number of the modular curve . It is a continuation of the previous work of the determination of the cuspidal class number for the modular curves for the case . [Copyright &y& Elsevier]

Abstract: We find equations for the higher-dimensional analogue of the modular curve using Mumford''s algebraic formalism of algebraic theta functions. As a consequence, we derive a method for the construction of genus 2 hyperelliptic curves over small degree number fields whose Jacobian has complex multiplication and good ordinary reduction at the prime 3. We prove the existence of a quasi-quadratic time algorithm for computing a canonical lift in characteristic 3 based on these equations, with a detailed description of our method in genus 1 and 2. [Copyright &y& Elsevier]

In the studies on the modularity conjecture for rigid Calabi–Yau threefolds several examples with the unique level 8 cusp form were constructed. According to the Tate conjecture, correspondences inducing isomorphisms on the middle cohomologies should exist between these varieties. In the paper, we construct several examples of such correspondences. In the constructions elliptic fibrations play a crucial role. In fact we show that all but three examples are in some sense built upon two modular curves from the Beauville list. [ABSTRACT FROM AUTHOR]

Abstract: We show that Petersson products of any pair of elliptic modular forms of weight and of any pair of Siegel modular forms of degree 2 and weight 1 have an expression as the residue of a Dirichlet series of Rankin–Selberg type. This generalizes a well-known property of cuspidal Petersson products. [Copyright &y& Elsevier]

We develop a theory of abstract arithmetic Chow rings, where the role of the fibres at infinity is played by a complex of abelian groups that computes a suitable cohomology theory. As particular cases of this formalism we recover the original arithmetic intersection theory of Gillet and Soulé for projective varieties. We introduce a theory of arithmetic Chow groups, which are covariant with respect to arbitrary proper morphisms, and we develop a theory of arithmetic Chow rings using a complex of differential forms with log-log singularities along a fixed normal crossing divisor. This last theory is suitable for the study of automorphic line bundles. In particular, we generalize the classical Faltings height with respect to logarithmically singular hermitian line bundles to higher dimensional cycles. As an application we compute the Faltings height of Hecke correspondences on a product of modular curves. [ABSTRACT FROM AUTHOR]

Abstract: We study here the free group generated by isomorphism classes of supersingular elliptic curves in positive characteristic. We compare this -module to the homology of the modular curve . We give an interpretation of Gross formula for special values of L-functions of modular forms in this context. As an application, we obtain a formula for the sum of the squares of the optimal embeddings numbers of quadratic orders in a definite quaternion algebra. [Copyright &y& Elsevier]

Abstract: We state an explicit version of Beilinson''s theorem for the modular curve . We deduce from it, for any elliptic curve E of prime conductor N, a formula giving in terms of the twisted values , where χ is a character modulo N. We illustrate this result and its consequences in the case of the elliptic curve . To cite this article: F. Brunault, C. R. Acad. Sci. Paris, Ser. I 343 (2006). [Copyright &y& Elsevier]

Abstract: We obtain defining equations of modular curves , , and 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 to find explicit modular parameterization of rational elliptic curves of conductor 37. [Copyright &y& Elsevier]

Abstract: In this paper we describe the compactification of the Drinfeld modular curve. This compactification is analogous to the compactification of the classical modular curve given by Katz and Mazur. We show how the Weil pairing on Drinfeld modules that we defined in earlier work gives rise to a map on the Drinfeld modular curve. We introduce the Tate–Drinfeld module and show how this describes the formal neighbourhood of the scheme of cusps of the Drinfeld modular curve. [Copyright &y& Elsevier]

Vatsal has proved recently a result which has consequences for the existence of rational points of odd prime order ℓ on optimal elliptic curves over ℚ. When the conductor N is squarefree, ℓ ∤ N and the local root number wp = -1 for at least one prime p | N, we offer a somewhat different proof, starting from an explicit cuspidal divisor on X0(N). We also prove some results linking the vanishing of L(E,1) with the divisibility by ℓ of the modular parametrization degree, fitting well with the Bloch–Kato conjecture for L(Sym2E,2), and with an earlier construction of elements in Shafarevich–Tate groups. Finally (following Faltings and Jordan) we prove an analogue of the result on ℓ-torsion for cuspidal Hecke eigenforms of level one (and higher weight), thereby strengthening some existing evidence for another case of the Bloch–Kato conjecture. [ABSTRACT FROM AUTHOR]

Abstract: For small odd primes p, we prove that most of the rational points on the modular curve parametrize pairs of elliptic curves having infinitely many supersingular primes. This result extends the class of elliptic curves for which the infinitude of supersingular primes is known. We give concrete examples illustrating how these techniques can be explicitly used to construct supersingular primes for such elliptic curves. Finally, we discuss generalizations to points defined over larger number fields and indicate the types of obstructions that arise for higher level modular curves. [Copyright &y& Elsevier]

Let p > 2 be a prime, and let k be a field of characteristic zero, linearly disjoint from the pth cyclotomic extension of Q. Given a projective Galois representation Gal(k¯/k)→PGL2(Fp) with cyclotomic determinant, two twists Xϱ(p) and X′ϱ(p) of a certain rational model of the modular curve X(p) can be attached to it. The k-rational points of these twists classify the elliptic curves E/k such that ρ¯E,p=ρ, where ρ¯E,p denotes the projective Galois representation associated with the p-torsion module E[p]. The octahedral (p = 3) and icosahedral (p = 5) genus-zero cases are discussed in further detail. 2000 Mathematics Subject Classification 11G05, 14G05, 11R32. [ABSTRACT FROM PUBLISHER]

Let G be a semisimple linear algebraic group and X a nodal curve. We show that if ρ : G → On(ℂ)(Spn(ℂ)) is a faithful representation and if δ⋙ 0, then any δ-semistable singular G-bundle is actually a principal G-bundle away from the node of X. Furthermore, we considerably simplify the notion of δ-(semi)stability. [ABSTRACT FROM AUTHOR]

The maximum modulus test(RE) for simultaneously comparing the means of normal distributions to a known constant is usually presented with the assumption of equal variances. A competing procedure(RI) for use when the unknown variances are not assumed to be equal is presented. Results of simulations are presented to examine various aspects of these procedures, such as comparisons of overall type I error rate and power when the variances are equal and consequences of using RE when the variances are not equal. Both procedures control the overall type I error rate if the variances are equal; however, if the variances are not equal, RI controls the overall type I error rate, but P(type I error) > α using RE for a nominalα-level test. If the variances are equal, the power is higher for RE, but the differences are usually small. If the variances are not equal, the power is usually higher for RE than for RI, but this is associated with the inflated type I error rate. Thus, it is recommended that RI be routinely preferred over RE. [ABSTRACT FROM AUTHOR]

Let E be a modular elliptic curve defined over a rational function field k of odd characteristic. We construct a sequence of Heegner points on E, defined over a Zp∞-tower of finite extensions of k, and show that these Heegner points generate a group of infinite rank. This is a function field analogue of a result of Cornut and Vatsal. [Copyright &y& Elsevier]

We present explicit models for Shimura curves XD of genus two and their Atkin-Lehner quotients XD/〈 ω m 〉. We show that several equations conjectured by Kurihara are correct and compute for them the kernel of Ribet's isogeny J0(D)new → JD between the new part of the Jacobian of the modular curve X0(D) and the Jacobian of XD. [ABSTRACT FROM AUTHOR]

In this work, we determine all Drinfeld modular curves X1(n) that are hyperelliptic or bielliptic. [Copyright &y& Elsevier]

We determine the automorphism group of the modular curve X0*(p) for all prime numbers p. [Copyright &y& Elsevier]

We study the arithmetic properties of higher Weierstrass points on modular curves $X_{0}(p)$ for primes $p$. In particular, for $r\in \{2, 3, 4, 5\}$, we obtain a relationship between the reductions modulo $p$ of the collection of $r$-Weierstrass points on $X_{0}(p)$ and the supersingular locus in characteristic $p$. [ABSTRACT FROM AUTHOR]

We give a criterion to check if, given a prime number N, the only rational points of the modular curve Xsplit(N) are trivial (i.e., cusps or points furnished by complex multiplication). We then prove that this criterion is verified for large enough N satisfying some explicit congruences. To cite this article: P. Parent, C. R. Acad. Sci. Paris, Ser. I 336 (2003). [Copyright &y& Elsevier]

This paper is motivated by a problem that arises in the study of partition functions of Potts models, including as a special case chromatic polynomials. When the underlying graphs have the form of ‘bracelets’, the chromatic polynomials can be expressed in terms of the eigenvalues of a matrix. In this situation a theorem of Beraha, Kahane and Weiss asserts that the zeros of the polynomials approach the curves on which the matrix has two eigenvalues with equal modulus. It is shown here that (in general) these ‘equimodular’ curves comprise a number of segments, the end-points of which are the roots (possibly coincident) of a polynomial equation. The equation represents the vanishing of a discriminant, and the segments are in bijective correspondence with the double roots of another polynomial equation, which is significantly simpler than the discriminant equation. Singularities of the segments can occur, corresponding to the vanishing of a Jacobian. In addition, it is proved by algebraic means that the equimodular curves for a reducible matrix are closed curves. The question of dominance is investigated, and a method of constructing the dominant equimodular curves for a reducible matrix is suggested. These results are illustrated by explicit calculations in a specific case. [Copyright &y& Elsevier]

We prove an analogue of the Andre´–Oort conjecture for the product of two Drinfeld modular curves, following S.J. Edixhoven's approach. To cite this article: F. Breuer, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 867–870. [Copyright &y& Elsevier]

Let N be a prime number, and let J0(N) be the Jacobian of the modular curve X0(N). Let T denote the endomorphism ring of J0(N). In a seminal 1977 article, B. Mazur introduced and studied an important ideal I⊆T, the Eisenstein ideal. In this paper we give an explicit construction of the kernel J0(N)[I] of this ideal (the set of points in J0(N) that are annihilated by all elements of I). We use this construction to determine the action of the group Gal(Q/Q) on J0(N)[I]. Our results were previously known in the special case where N−1 is not divisible by 16. [Copyright &y& Elsevier]

In this note we show that, assuming the generalized Riemann hypothesis for quadratic imaginary fields, an irreducible algebraic curve in ℂŋ is modular if and only if it contains a CM point of sufficiently large height. This is an effective version of a theorem of Edixhoven. [ABSTRACT FROM AUTHOR]

The article discusses the intersection of a curve with the variable algebraic subgroups of multiplicative groups. It indicates the theorem that the sparse set alluded to is actually finite. Specialization estimates reveal that this restriction to rational numbers must be a sparse set if one considers the restriction to the set of algebraic numbers of fixed degree.

Presents a paper based on the parametrizations of elliptic curves through Shimura curves and classical modular curves. Association of a certain Shimura curve whose Jacobian is isogenous to an abelian subvariety; Investigation of the behavior of... as D varies; Reference to tables of John Cremona by anonymous ftp from euclid.ex.ac.uk in pub/cremona/data.

The article discusses the different conjectures for algebraic curves with minimal automorphism group and rational points. It mentions that Jacobian automorphism groups which splits in the two isogenous elliptic curves. It also notes on the higher Jacobian curves which tends to have less rational points to the automorphism.

The article discusses on the linear lower bound on the gonality of modular curves similarly with a quadratic lower bound using the theorem of congruence subgroup and the corresponding modular curve. It mentions on the utilization of the conformal studies of P. Li and S. T. Yau on the nontrivial eigenvalue of the noneuclidean Laplacian. It also notes on the gonality of the modular curves in which algebraic techniques are used to lower the bound of the gonality of equations.

The article describes the projective structures on a Riemann surface. It notes that the are several equivalent notions of a projective structure, which is an equivalence class of coverings by holomorphic coordinate charts such that the transition functions are all Möbius transformations. It also reveals that the space of projective structures on X is canonically identified with the space of all trivialisations of L.