Abstract We give a conditional bound for the average analytic rank of elliptic curves over an arbitrary number field. In particular, under the assumptions that all elliptic curves over a number field K are modular and have L-functions which satisfy the Generalized Riemann Hypothesis, we show that the average analytic rank of isomorphism classes of elliptic curves over K is bounded above by $(9\deg (K)+1)/2$ , when ordered by naive height. A key ingredient in the proof is giving asymptotics for the number of elliptic curves over an arbitrary number field with a prescribed local condition; these results are obtained by proving general results for counting points of bounded height on weighted projective stacks with a prescribed local condition, which may be of independent interest.

For each t∈Q∖{−1,0,1}, define an elliptic curve over Q by Et:y2=x(x+1)(x+t2).Using a formula for the root number W(Et) as a function of t and assuming some standard conjectures about ranks of elliptic curves, we determine (up to a set of density zero) the set of isomorphism classes of elliptic curves E/Q whose Mordell–Weil group contains Z×Z/2Z×Z/4Z, and the set of rational numbers that can be written as a product of the slopes of two rational right triangles.

Abstract We discuss the$\ell $-adic case of Mazur’s ‘Program B’ over$\mathbb {Q}$: the problem of classifying the possible images of$\ell $-adic Galois representations attached to elliptic curvesEover$\mathbb {Q}$, equivalently, classifying the rational points on the corresponding modular curves. The primes$\ell =2$and$\ell \ge 13$are addressed by prior work, so we focus on the remaining primes$\ell = 3, 5, 7, 11$. For each of these$\ell $, we compute the directed graph of arithmetically maximal$\ell $-power level modular curves$X_H$, compute explicit equations for all but three of them and classify the rational points on all of them except$X_{\mathrm {ns}}^{+}(N)$, for$N = 27, 25, 49, 121$and two-level$49$curves of genus$9$whose Jacobians have analytic rank$9$.Aside from the$\ell $-adic images that are known to arise for infinitely many${\overline {\mathbb {Q}}}$-isomorphism classes of elliptic curves$E/\mathbb {Q}$, we find only 22 exceptional images that arise for any prime$\ell $and any$E/\mathbb {Q}$without complex multiplication; these exceptional images are realised by 20 non-CM rationalj-invariants. We conjecture that this list of 22 exceptional images is complete and show that any counterexamples must arise from unexpected rational points on$X_{\mathrm {ns}}^+(\ell )$with$\ell \ge 19$, or one of the six modular curves noted above. This yields a very efficient algorithm to compute the$\ell $-adic images of Galois for any elliptic curve over$\mathbb {Q}$.In an appendix with John Voight, we generalise Ribet’s observation that simple abelian varieties attached to newforms on$\Gamma _1(N)$are of$\operatorname {GL}_2$-type; this extends Kolyvagin’s theorem that analytic rank zero implies algebraic rank zero to isogeny factors of the Jacobian of$X_H$.

We consider the moduli of elliptic curves with G-structures, where G is a finite 2-generated group. When G is abelian, a G-structure is the same as a classical congruence level structure. There is a natural action of $$\text {SL}_2(\mathbb {Z})$$ on these level structures. If $$\Gamma $$ is a stabilizer of this action, then the quotient of the upper half plane by $$\Gamma $$ parametrizes isomorphism classes of elliptic curves equipped with G-structures. When G is sufficiently nonabelian, the stabilizers $$\Gamma $$ are noncongruence. Using this, we obtain arithmetic models of noncongruence modular curves as moduli spaces of elliptic curves equipped with nonabelian G-structures. As applications we describe a link to the Inverse Galois Problem, and show how our moduli interpretations explains the bad primes for the Unbounded Denominators Conjecture, and allows us to translate the conjecture into the language of geometry and Galois theory.

Let E / Q E/\mathbb {Q} be an elliptic curve and let Q ( 3 ∞ ) \mathbb {Q}(3^\infty ) be the compositum of all cubic extensions of Q \mathbb {Q} . In this article we show that the torsion subgroup of E ( Q ( 3 ∞ ) ) E(\mathbb {Q}(3^\infty )) is finite and we determine 20 possibilities for its structure, along with a complete description of the Q ¯ \overline {\mathbb {Q}} -isomorphism classes of elliptic curves that fall into each case. We provide rational parameterizations for each of the 16 torsion structures that occur for infinitely many Q ¯ \overline {\mathbb {Q}} -isomorphism classes of elliptic curves, and a complete list of j j -invariants for each of the 4 that do not.

On the number of isomorphism classes of CM elliptic curves defined over a number field

Let $\varphi$ be an endomorphism of $\mathbb{P}^1_{\overline{\Q}}$ defined over a number field $K$. Given a discrete valuation $v$ of $K$, we consider here two notions of good reduction of $\varphi$ at $v$, called Standard Good Reduction (S.G.R., for short) and Critically Good Reduction (C.G.R.). If we consider the reduced map $\varphi_v$, in general its degree is smaller or equal to the degree of $\varphi$. We say that the map $\varphi$ has S.G.R. at $v$ if the degree of the reduced map $\varphi_v$ is equal to the degree of $\varphi$. This notion is frequently used in the study of arithmetical dynamical systems, allowing to reduce a global problem to a local problem. Another notion of good reduction has been recently introduced by Szpiro and Tucker to prove a finitess result about equivalence classes of endomorphisms of the projective line. We say that $\varphi$ has C.G.R. at $v$ if every pair of ramification points of $\varphi$ do not coincide modulo $v$ and the same holds for every pair of branch points. As an application of their result, Szpiro and Tucker showed that their theorem implies the well-known Shafarevich-Faltings theorem about the finiteness of the isomorphism classes of elliptic curves defined over a number field $K$ having good reduction outside a prescribed finite set of discrete valuations of $K$. Szpiro and Tucker already in their paper showed with same examples that these two notions are not equivalent. We prove here that if $\varphi$ has C.G.R. at $v$ and the reduced map $\varphi_v$ is separable, then $\varphi$ has S.G.R. at $v$.

We propose a public-key encryption scheme and key agreement protocols based on a group action on a set. We construct an implementation of these schemes for the action of the class group $\mathcal{CL}(\mathcal{O}_K)$ of an imaginary quadratic field $K$ on the set $\mathcal{ELL}$p,n$(\mathcal{O}_K)$ of isomorphism classes of elliptic curves over $\mathbb{F}_p$ with $n$ points and the endomorphism ring $\mathcal{O}_K$.This introduces a novel way of using elliptic curves for constructing asymmetric cryptography.

* Author writes in a clear and engaging style * Contains never before published elementary proofs * Author provides new results and detailed exposition * Self-contained, and suitable for use in a classroom setting or for self-study * A highly creative contribution to the theory of modular forms and dirichlet series The main topics of the book are the critical values of Dirichlet L-functions and Hecke L-functions of an imaginary quadratic field, and various problems on elliptic modular forms. As to the values of Dirichlet L-functions, all previous papers and books reiterate a single old result with a single old method. After a review of elementary Fourier analysis, the author presents completely new results with new methods, though old results will also be proved. No advanced knowledge of number theory is required up to this point. As applications, new formulas for the second factor of the class number of a cyclotomic field will be given. The second half of the book assumes familiarity with basic knowledge of modular forms. However, all definitions and facts are clearly stated, and precise references are given. The notion of nearly holomorphic modular forms is introduced and applied to the determination of the critical values of Hecke L-functions of an imaginary quadratic field. Other notable features of the book are: (1) some new results on classical Eisenstein series; (2) the discussion of isomorphism classes of elliptic curves with complex multiplication in connection with their zeta function and periods; (3) a new class of holomorphic differential operators that send modular forms to those of a different weight. The book will be of interest to graduate students and researchers who are interested in special values of L-functions, class number formulae, arithmetic properties of modular forms (especially their values), and the arithmetic properties of Dirichlet series. It treats in detail, from an elementary viewpoint, the simplest cases of a fundamental area of ongoing research, the only prerequisite being a basic course in algebraic number theory.

Elliptic curves, modular forms, and sums of Hurwitz class numbers

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.

where q = e. The values of j(z) at imaginary quadratic arguments in the upper half of the complex plane are known as singular moduli. Singular moduli are algebraic integers which play prominent roles in classical and modern number theory (see [C, BCSH]). For example, Hilbert class fields of imaginary quadratic fields are generated by singular moduli. Furthermore, isomorphism classes of elliptic curves with complex multiplication are distinguished by singular moduli. Throughout, let d ≡ 0, 3 (mod 4) be a positive integer (so that −d is the discriminant of an order in an imaginary quadratic field), and let H(d) be the Hurwitz-Kronecker class number for the discriminant −d. Let Qd be the set of positive definite integral binary quadratic forms (note. including imprimitive forms, if there are any)

We prove the following.Theorem. Let be a number field, and the Jacobian of the curve parametrizing the elliptic curves with distinguished cyclic subgroups of order . If the number is written as , where contains a -simple abelian subvariety such that {\operatorname{rk}} A_k,$ SRC=http://ej.iop.org/images/0025-5734/11/2/A12/tex_sm_2058_img8.gif/> then the set of -isomorphism classes of elliptic curves over the field possessing -points of order is finite.Bibliography: 4 items.

For a prime p and a given square box, B, we consider all elliptic curves Er,s : Y 2 = X 3 + rX + s defined over a field Fp of p elements with coefficients (r, s) ∈ B. We obtain a nontrivial upper bound for the number of such curves which are isomorphic to ag iven one overFp, in terms of the size of B. We also give an optimal lower bound on the number of distinct isomorphic classes represented.