Universal Elliptic Curve Research Articles

Eisenstein cocycles in motivic cohomology

Several authors have studied homomorphisms from first homology groups of modular curves to $K_2(X)$ , with $X$ either a cyclotomic ring or a modular curve. These maps send Manin symbols in the homology groups to Steinberg symbols of cyclotomic or Siegel units. We give a new construction of these maps and a direct proof of their Hecke equivariance, analogous to the construction of Siegel units using the universal elliptic curve. Our main tool is a $1$ -cocycle from $\mathrm {GL}_2(\mathbb {Z})$ to the second $K$ -group of the function field of a suitable group scheme over $X$ , from which the maps of interest arise by specialization.

On equivariant topological modular forms

Following ideas of Lurie, we give a general construction of equivariant elliptic cohomology without restriction to characteristic zero. Specializing to the universal elliptic curve we obtain, in particular, equivariant spectra of topological modular forms. We compute the fixed points of these spectra for the circle group and more generally for tori.

Hodge theory for elliptic curves and the Hopf element ν$\nu$

AbstractWe show that the vector bundle on the moduli stack of elliptic curves associated to the 2‐cell complex is isomorphic to the de Rham cohomology sheaf of the universal elliptic curve . We use this to calculate the homotopy groups of the ‐quotient of by , called the spectrum of ‘topological quasimodular forms’, by relating its Adams–Novikov spectral sequence to the cohomology of the moduli stack of cubic curves with a chosen splitting of the Hodge–de Rham filtration.

Chern–Weil Theory for Line Bundles with the Family Arakelov Metric

We prove a result of Chern–Weil type for canonically metrized line bundles on one-parameter families of smooth complex curves. Our result generalizes a result due to J. I. Burgos Gil, J. Kramer, and U. Kühn that deals with a line bundle of Jacobi forms on the universal elliptic curve over the modular curve with full level structure, equipped with the Petersson metric. Our main tool, as in the work by Burgos Gil, Kramer, and Kühn, is the notion of a b-divisor.

Poisson structures on loop spaces of [formula omitted] and an [formula omitted]-matrix associated with the universal elliptic curve

We construct a family of Poisson structures of hydrodynamic type on the loop space of ℂPn−1. This family is parametrized by the moduli space of elliptic curves or, in other words, by the modular parameter τ. This family can be lifted to a homogeneous Poisson structure on the loop space of ℂn but in order to do that we need to upgrade the modular parameter τ to an additional field τ(x) with Poisson brackets {τ(x),τ(y)}=0,{τ(x),za(y)}=2πiza(y)δ′(x−y) where z1,…,zn are coordinates on ℂn. These homogeneous Poisson structures can be written in terms of an elliptic r-matrix of hydrodynamic type.

The Syntomic Realization of the Elliptic Polylogarithm via the Poincaré Bundle

We give an explicit description of the syntomic elliptic polylogarithm on the universal elliptic curve over the ordinary locus of the modular curve in terms of certain p -adic analytic moment functions associated to Katz' two-variable p -adic Eisenstein measure. The present work generalizes previous results of Bannai-Kobayashi-Tsuji and Bannai-Kings on the syntomic Eisenstein classes.

EISENSTEIN–KRONECKER SERIES VIA THE POINCARÉ BUNDLE

A classical construction of Katz gives a purely algebraic construction of Eisenstein–Kronecker series using the Gauß–Manin connection on the universal elliptic curve. This approach gives a systematic way to study algebraic and$p$-adic properties of real-analytic Eisenstein series. In the first part of this paper we provide an alternative algebraic construction of Eisenstein–Kronecker series via the Poincaré bundle. Building on this, we give in the second part a new conceptional construction of Katz’ two-variable$p$-adic Eisenstein measure through$p$-adic theta functions of the Poincaré bundle.

Descent for the punctured universal elliptic curve, and the average number of integral points on elliptic curves

We show that the average number of integral points on elliptic curves, counted modulo the natural involution on a punctured elliptic curve, is bounded from above by $2.1 \times 10^8$. To prove it, we design a descent map, whose prototype goes at least back to Mordell, which associates a pair of binary forms to an integral point on an elliptic curve. Other ingredients of the proof include the upper bounds for the number of solutions of a Thue equation by Evertse and Akhtari-Okazaki, and the estimation of the number of binary quartic forms by Bhargava-Shankar. Our method applies to $S$-integral points to some extent, although our present knowledge is insufficient to deduce an upper bound for the average number of them. We work out the numerical example with $S=\{2\}$.

Elliptic curves with complex multiplication and {u039B}-structures

This thesis examines the relationship between elliptic curves with complex multiplication and Lambda structures. Our main result is to show that the moduli stack of elliptic curves with complex multiplication, and the universal elliptic curve with complex multiplication over it, both admit Lambda structures and that the structure morphism is a Lambda morphism. This implies that elliptic curves with complex multiplication can be canonically lifted to the Witt vectors of the base (these are big and global Witt vectors). We also show that elliptic curves with complex multiplication of Shimura type are precisely those admitting Lambda structures and that a large class of these elliptic curves admit global minimal models. Along the way, we present a detailed study of families of elliptic curves with complex multiplication over arbitrary bases, give new derivations of the reciprocity maps associated to local fields and imaginary quadratic fields, construct a new flat, affine and pro-smooth rigidification of the moduli stack of elliptic curves with complex multiplication and exhibit a relationship between perfect Lambda schemes and periods, both $p$-adic and analytic.

Painlevé VI equations in p-adic time

Using the description of Painleve VI family of differential equations in terms of a universal elliptic curve, going back to R. Fuchs, we translate it into the realm of Buium’s p-adic Arithmetic Differential Equations, where the role of derivative is played by a version of Fermat quotient.

Some explicit expressions concerning BP

Abstract This paper provides some explicit expressions concerning the formal group laws of the following cohomology theories: BP, the Brown–Peterson cohomology, G(s), the cohomology theory obtained from BP by putting v i = 0 ${v_i=0}$ for all i ≥ 1 with i ≠ s ${i \ne s}$ , the Morava K-theory and the Abel cohomology. The coefficient rings of Weierstrass universal elliptic curve formal group law are computed in low dimensions.

A 0.5 (half) overconvergent Eichler-Shimura isomorphism

In this article we construct a Galois and Hecke equivariant morphism connecting the first cohomology group on Faltings’ site of a formal strict neighborhood of the ordinary locus in a formal modular curve of level prime to p, with coefficients in the analytic distributions of a certain analytic weight k on the p-adic Tate module of the universal elliptic curve to the overconvergent modular forms of weight \(k+2\). We prove that this morphism is an isomorphism on the finite slope parts.

ARITHMETIC PROGRESSIONS ON CONIC SECTIONS

The set {1, 25, 49} is a 3-term collection of integers which forms an arithmetic progression of perfect squares. We view the set {(1, 1), (5, 25), (7, 49)} as a 3-term collection of rational points on the parabola y = x2whose y-coordinates form an arithmetic progression. In this exposition, we provide a generalization to 3-term arithmetic progressions on arbitrary conic sections [Formula: see text] with respect to a linear rational map [Formula: see text]. We explain how this construction is related to rational points on the universal elliptic curve Y2+ 4XY + 4kY = X3+ kX2classifying those curves possessing a rational 4-torsion point.

Unlikely Intersections in Poincaré Biextensions over Elliptic Schemes

This paper concerns the relations between the relative Manin–Mumford conjecture and Pink’s conjecture on unlikely intersections in mixed Shimura varieties. The variety under study is the 4-dimensional Poincare biextension attached to a universal elliptic curve. A detailed list of its special subvarieties is drawn up, providing partial verifications of Pink’s conjecture in this case, and two open problems are stated in order to complete its proof.

Commutativity conditions for truncated Brown-Peterson spectra of height 2

An algebraic criterion, in terms of closure under power operations, is determined for the existence and uniqueness of generalized truncated Brown–Peterson spectra of height 2 as E∞-ring spectra. The criterion is checked for an example at the prime 2 derived from the universal elliptic curve equipped with a level Γ1(3) structure.

A differential equation satisfied by the arithmetic Kodaira–Spencer class

The arithmetic Kodaira–Spencer class of the universal elliptic curve was introduced in [A. Buium, Differential modular forms, J. Reine Angew. Math. 520 (2000) 95–167]; its reduction mod p was explicitly computed by Hurlburt [C. Hurlburt, Isogeny covariant differential modular forms modulo p, Compos. Math. 128 (1) (2001) 17–34]. In this paper the complicated expression of Hurlburt is shown to be the unique solution of a simple partial differential equation subject to a certain initial condition and weight condition.

Visualizing elements of order two in the Weil–Châtelet group

Let E be an elliptic curve over an infinite field K with characteristic ≠ 2 , and σ ∈ H 1 ( G K , E ) [ 2 ] a two-torsion element of its Weil–Châtelet group. We prove that σ is always visible in infinitely many abelian surfaces up to isomorphism, in the sense put forward by Cremona and Mazur in their article (J. Exp. Math. 9(1) (2000) 13). Our argument is a variant of Mazur's proof, given in (Asian J. Math. 3(1) (1999) 221), for the analogous statement about three-torsion elements of the Shafarevich–Tate group in the setting where K is a number field. In particular, instead of the universal elliptic curve with full level-three-structure, our proof makes use of the universal elliptic curve with full level-two-structure and an invariant differential.

Gaps in √n mod 1 and ergodic theory

Cut the unit circle $S^1=\mathbb{R}/\mathbb{Z}$ at the points $\{\sqrt{1}\}, \{\sqrt{2}\},\ldots,\{\sqrt{N}\}$, where $\{x\} = x \bmod 1$, and let $J_1, \ldots, J_N$ denote the complementary intervals, or \emph{gaps}, that remain. We show that, in contrast to the case of random points (whose gaps are exponentially distributed), the lengths $|J_i|/N$ are governed by an explicit piecewise real-analytic distribution $F(t) \,dt$ with phase transitions at $t=1/2$ and $t=2$. The gap distribution is related to the probability $p(t)$ that a random unimodular lattice translate $\Lambda \subset \mathbb{R}^2$ meets a fixed triangle $S_t$ of area $t$; in fact, $p''(t) = -F(t)$. The proof uses ergodic theory on the universal elliptic curve \[ E = \big(\mathrm{SL}_2(\mathbb{R}) \ltimes \mathbb{R}^2\big)/ \big(\mathrm{SL}_2(\mathbb{Z}) \ltimes \mathbb{Z}^2\big) \] and Ratner's theorem on unipotent invariant measures.

Modular units and the surjectivity of a Galois representation

For a prime p⩾7 the pth roots of certain modular units are shown to generate the second layer of the extension of function fields cut out by the universal Galois deformation of the representation on p-division points of a universal elliptic curve. It follows that certain Galois representations obtained by specializing the modular invariant to a rational number have large image.

Superpotentials forM-theory on aG2holonomy manifold and triality symmetry

For $M$-theory on the $G_2$ holonomy manifold given by the cone on ${\bf S^3}\x {\bf S^3}$ we consider the superpotential generated by membrane instantons and study its transformations properties, especially under monodromy transformations and triality symmetry. We find that the latter symmetry is, essentially, even a symmetry of the superpotential. As in Seiberg/Witten theory, where a flat bundle given by the periods of an universal elliptic curve over the $u$-plane occurs, here a flat bundle related to the Heisenberg group appears and the relevant universal object over the moduli space is related to hyperbolic geometry.