Modular Elliptic Curves Research Articles

Non-vanishing of $n$-th derivatives of twisted elliptic $L$-functions in the critical point

Let E be a modular elliptic curve over ℚ L (n) (s,E) denote the n-th derivative of its Hasse-Weil L-series. We estimate the number of twisted elliptic curves E d ,d≤D such that L (n) (1,E d )≠0.

Torsion zero-cycles on the self-product of a modular elliptic curve

Torsion zero-cycles on the self-product of a modular elliptic curve

Deninger's conjecture onL-functions of elliptic curves ats=3

I compute explicitly the regulator map on $K_4(X)$ for an arbitrary curve $X$ over a number field. Using this and Beilinson's theorem about regulators for modular curves ([B2]) I prove a formula expressing the value of the $L$-function $L(E,s)$ of a modular elliptic curve $E$ over $\Bbb Q$ at $s=3$ by the double Eisenstein-Kronecker series.

Heegner points on modular elliptic curves

Heegner points on modular elliptic curves

The sign of the functional equation of the L-function of an orthogonal motive

We prove that the sign of the functional equation of the L-function of an orthogonal motive of even weight is always positive, if we admit the conjecture of the product formula for the epsilon factor of Deligne [D3] (5.2). More precisely, we define the sign independently of any conjecture by using the product formula as the definition and by working with one chosen "good" prime f and we prove its positivity. The positivity has been established for the Artin motives [Fr-Q][D2], for the symmetric square of a modular elliptic curve [C-Sc] and in positive characteristic by Serre [Fr-Q]. As in the proof in [D2], we consider the second Stiefel-Whitney class in zBr(K) of an orthogonal representation of the absolute Galois group G~: of an algebraic number field K. The positivity of the sign is then a consequence of the reciprocity law of the square residue symbol. Fixing a "good" prime f, we work with an f-adic representation and compare the local epsilon factor and the local Stiefel-Whitney class. For a prime vXr it is a theorem of Deligne [D2]. For a prime vie , we compute the Stiefel-Whitney class assuming the representation is cristalline (Theorem 2) using a generalization of a theorem of Fr6hlich [Fr] to f-adic representation of Hodge-Tate type (Theorem 1). By combining them with an elementary computation for the Hodge structures at r iot , we conclude that the sign is positive (Theorem 3). The crucial step in the proof of Theorem 2, given in section 1, is the determination by Fontaine and Lafaille [Fo-L] of the action of the tame inertia on the semisimplification of reduction modulo p of a cristalline representation. The theory of orthogonal representations of Galois group including a proof of Fr6hlich's theorem is reviewed in a preliminary section 0.

Crystalline cohomology and GL(2, ℚ)

This paper applies recent advances in crystalline cohomology to the classical case of open elliptic modular curves. In so doing control is gained over the action of inertia in the Galois representations attached to modular forms. Our aim is to study the modular Galois representations attached to automorphic forms modp of weightk≥2. We generalize to higher weightk several results which were previously accessible only in the case of weight 2 where jacobian varieties can be invoked. Additionally we reconsider Gross’s theorem on companion forms in a crystalline context.

Computing the Degree of the Modular Parametrization of a Modular Elliptic Curve

The Weil-Taniyama conjecture states that every elliptic curve $E/\mathbb {Q}$ of conductor N can be parametrized by modular functions for the congruence subgroup ${\Gamma _0}(N)$ of the modular group $\Gamma = PSL(2,\mathbb {Z})$. Equivalently, there is a nonconstant map $\varphi$ from the modular curve ${X_0}(N)$ to E. We present here a method of computing the degree of such a map $\varphi$ for arbitrary N. Our method, which works for all subgroups of finite index in $\Gamma$ and not just ${\Gamma _0}(N)$, is derived from a method of Zagier published in 1985; by using those ideas, together with techniques which have recently been used by the author to compute large tables of modular elliptic curves, we are able to derive an explicit and general formula which is simpler to implement than Zagier’s. We discuss the results obtained, including a table of degrees for all the modular elliptic curves of conductors up to 200.

Algorithms for modular elliptic curves, by J. E. Cremona. Pp. 343. 1992. SBN 0-521-41813-5 (Cambridge University Press)

Algorithms for modular elliptic curves, by J. E. Cremona. Pp. 343. 1992. SBN 0-521-41813-5 (Cambridge University Press)

Modular Elliptic Curves and Fermat's Last Theorem

When Andrew John Wiles was 10 years old, he read Eric Temple Bell’s The Last Problem and was so impressed by it that he decided that he would be the first person to prove Fermat’s Last Theorem. This theorem states that there are no nonzero integers a, b, c, n with n > 2 such that an + bn = cn. The object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular. Fermat’s Last Theorem follows as a corollary by virtue of previous work by Frey, Serre and Ribet.

On thep-adic height of Heegner cycles

(0.1) In [Gr Za], Gross and Zagier proved a remarkable formula, which relates the first derivative of the L-function of a modular form f of weight 2 on Γ0(N) (over a suitable imaginary quadratic field) and the Neron-Tate height of a “Heegner point” (over the same quadratic field) on the f -part of the Jacobian of the modular curve X0(N). Later, Perrin-Riou [PR 2] proved a p-adic version of this formula. Kolyvagin’s method of Euler systems [Ko], combined with the formula of Gross-Zagier, proves the conjecture of Birch and Swinnerton-Dyer (up to a controlled rational factor) for all modular elliptic curves over Q with analytic rank ≤ 1.

Computing the degree of the modular parametrization of a modular elliptic curve

The Weil-Taniyama conjecture states that every elliptic curve E / Q E/\mathbb {Q} of conductor N can be parametrized by modular functions for the congruence subgroup Γ 0 ( N ) {\Gamma _0}(N) of the modular group Γ = P S L ( 2 , Z ) \Gamma = PSL(2,\mathbb {Z}) . Equivalently, there is a nonconstant map φ \varphi from the modular curve X 0 ( N ) {X_0}(N) to E. We present here a method of computing the degree of such a map φ \varphi for arbitrary N. Our method, which works for all subgroups of finite index in Γ \Gamma and not just Γ 0 ( N ) {\Gamma _0}(N) , is derived from a method of Zagier published in 1985; by using those ideas, together with techniques which have recently been used by the author to compute large tables of modular elliptic curves, we are able to derive an explicit and general formula which is simpler to implement than Zagier’s. We discuss the results obtained, including a table of degrees for all the modular elliptic curves of conductors up to 200.

Iwasawa Theory of Modular Elliptic Curves of Analytic Rank at most 1

Iwasawa Theory of Modular Elliptic Curves of Analytic Rank at most 1

On Modular Elliptic Curves

The number of isogeny classes of elliptic curves E/ Q p , having potentially good reduction at p, and which are quotients of the jacobian J 0( p 2), is bounded by ( p/4 + 4).

The analytic order of III for modular elliptic curves

The analytic order of III for modular elliptic curves

Elliptic Curves and Modular Forms

AbstractThis is a survey of some recent developments in the theory of elliptic curves. After an informal discussion of the main theorems of the arithmetic side of the theory and the open problems confronting the subject, we describe the recent work of K. Rubin, V. Koly vagin, K. Murty and the author which establishes the finiteness of the Shafarevic-Tate group for modular elliptic curves of rank zero and one.

On the structure of Selmer groups

The paper contains some applications of explicit cohomology classes (which the author has constructed earlier using Heegner points) to the theory of Selmer groups of a modular elliptic curve. Moreover, some generalizations of Selmer groups are considered. The case when the Heegner point over the imaginary quadratic field has infinite order was studied in the work [1]. In fact, the theory of [1] is valid under a more general assumption which is, hypothetically, always true and discussed below. For the convenience of the reader, we recall in part 1 the definitions of the Selmer groups and of our explicit cohomology classes, and formulate some of our results. The second part is essentially based on the work [1] and requires some familiarity with it. The second part contains proofs of results for ` ∈ B(E) (see below for notations), formulations of corresponding results for ` 6∈ B(E), and some global consequences of these results.

Elliptic Tate curves over local ?-extensions

The group A(K)/N is computed, where A(K) is the group of points of a Tate curve over a local field while N is the group of universal norms from the group of points over a Γ-extension. As an application, the Mazurl-modulus of modular elliptic curves is computed for values ofl dividing the denominator of the absolute invariant.