Elliptic Curves

Many problems in number theory have simple statements, but their solutions require a deep understanding of algebra, algebraic geometry, complex analysis, group representations, or a combination of all four. The original simply stated problem can be obscured in the depth of the theory developed to understand it. This book is an introduction to some of these problems, and an overview of the theories used nowadays to attack them, presented so that the number theory is always at the forefront of the discussion. Lozano-Robledo gives an introductory survey of elliptic curves, modular forms, and $L$-functions. His main goal is to provide the reader with the big picture of the surprising connections among these three families of mathematical objects and their meaning for number theory. As a case in point, Lozano-Robledo explains the modularity theorem and its famous consequence, Fermat's Last Theorem. He also discusses the Birch and Swinnerton-Dyer Conjecture and other modern conjectures. The book begins with some motivating problems and includes numerous concrete examples throughout the text, often involving actual numbers, such as 3, 4, 5, $\frac{3344161}{747348}$, and $\frac{2244035177043369699245575130906674863160948472041} {8912332268928859588025535178967163570016480830}$. The theories of elliptic curves, modular forms, and $L$-functions are too vast to be covered in a single volume, and their proofs are outside the scope of the undergraduate curriculum. However, the primary objects of study, the statements of the main theorems, and their corollaries are within the grasp of advanced undergraduates. This book concentrates on motivating the definitions, explaining the statements of the theorems and conjectures, making connections, and providing lots of examples, rather than dwelling on the hard proofs. The book succeeds if, after reading the text, students feel compelled to study elliptic curves and modular forms in all their glory.

A subexponential algorithm for discrete logarithms over hyperelliptic curves of large genus over GF( q)

Let G be a unitary group of rank one over a non-archimedean local field K (whose residue field has a characteristic ≠ 2). We consider the action of G on the projective plane. A G(K) equivariant map from the set of points in the projective plane that are semistable for every maximal K split torus in G to the set of convex subsets of the building of G(K) is constructed. This map gives rise to an equivariant map from the set of points that are stable for every maximal K split torus to the building. Using these maps one describes a G(K) invariant pure affinoid covering of the set of stable points. The reduction of the affinoid covering is given.

Acting within the framework of geometric quantum mechanics, an interpretation of quantum uncertainty is discussed in terms of Jacobi fields, and a connection with the theory of elliptic curves is outlined, via classical integrability of Schrödinger's dynamics and the cross-ratio interpretation of quantum transition probabilities. Furthermore, a thoroughly geometrical construction of all special unitary representations of the 3-strand braid group on the quantum 1-qubit space is given, and the connection of one of them with elliptic curves admitting complex multiplication automorphisms — the physically relevant one corresponding to the anharmonic ratio — is shown. Also, contact is made with the Temperley–Lieb algebra theoretic constructions of Kauffman and Lomonaco, and it is shown that the standard trace relative to one of the above representations computes the Jones polynomial for particular values of the parameter, for knots arising as closures of 3-strand braids. Subsequently, a geometric entanglement criterion (in terms of Segre embeddings) is discussed, together with a projective geometrical portrait for quantum 2-gates. Finally, Aravind's idea of describing quantum states via knot theory is critically analyzed, and a geometrical picture — involving a blend of SU(2)-representation theory, classical projective geometry, binary trees and Brunnian and Hopf links — is set up in order to describe successive measurements made upon generalized GHZ states, close in spirit to the quantum knot picture again devised by Kauffman and Lomonaco.

This book represents the proceedings of a workshop on elliptic curves held in St. Adele, Quebec, in February 1992. Containing both expository and research articles on the theory of elliptic curves, this collection covers a range of topics, from Langlands's theory to the algebraic geometry of elliptic curves, from Iwasawa theory to computational aspects of elliptic curves. This book is especially significant in that it covers topics comprising the main ingredients in Andrew Wiles's recent result on Fermat's Last Theorem.

The complex AGM, periods of elliptic curves over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mi mathvariant="double-struck">C</mml:mi></mml:math> and complex elliptic logarithms

This 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.

In this paper, we construct an authentication scheme for multi-receivers and multiple messages based on a linear code C. This construction can be regarded as a generalization of the authentication scheme given by Safavi-Naini and Wang [1]. Actually, we notice that the scheme of Safavi-Naini and Wang is constructed with Reed-Solomon codes. The generalization to linear codes has the similar advantages as generalizing Shamir's secret sharing scheme to linear secret sharing sceme based on linear codes [2—6]. For a fixed message base field ${\mathbb F}_q$ , our scheme allows arbitrarily many receivers to check the integrity of their own messages, while the scheme of Safavi-Naini and Wang has a constraint on the number of verifying receivers $V\leqslant q$ . We further introduce access structure in our scheme. Masseya[4] characterized the access structure of linear secret sharing scheme by minimal codewords in the dual code whose first component is 1. We slightly modify the definition of minimal codewords in [4]. Let C be a [V,k] linear code. For any coordinate i ∈ {1,2, i¾? ,V}, a codeword c in C is called minimal respect to i if the codeword c has component 1 at the i-th coordinate and there is no other codeword whose i-th component is 1 with support strictly contained in that of c. Then the security of receiver R i in our authentication scheme is characterized by the minimal codewords respect to i in the dual code $C^\bot$ . Finally, we illustrate our authentication scheme based on the elliptic curve codes, a special class of algebraic geometry codes. We use the group of rational points on the elliptic curve to determine all the malicious groups that can successfully make a substitution attack to any fixed receiver.

The growth of the Mordell-Weil rank of an elliptic curve in a tower of number fields can be discussed at many levels.On the one hand, the issues raised can be embedded in a broader Iwasawa theory of elliptic curves (Mazur [10]); on the other hand, they can be crystallized in a single easily stated question, namely whether the rank of the elliptic curve over subextensions of finite degree is bounded in the tower.But whatever one's point of view, the case of abelian towers has seen important advances in recent years, including notably the results of Kato on cyclotomic towers, of Cornut [2] and Vatsal [18] on anticyclotomic towers, and of Skinner and Urban on Mazur's conjecture.In the case of nonabelian towers, by contrast, virtually nothing is known, and even a conjectural framework has begun to emerge only recently [1].However Greenberg [7] observed more than twenty years ago that some insight could already be gained from the classical conjectures about L-functions: They imply that for certain pairs (E 1 , E 2 ) of elliptic curves of relatively prime conductor there exist primes p such that the rank of E 1 is unbounded in the p-division tower of E 2 (which can be chosen not to have complex multiplication, so that the division tower is nonabelian).Greenberg's idea was elaborated further by L. Howe [8], who gave a conjectural lower bound for the growth of the rank of E 1 in the division tower of E 2 .The aim of the present note is to show that the standard conjectures also have some bearing on the growth of the rank of an elliptic curve in its own division tower.For example, we shall see that if E is a semistable elliptic curve over Q and p is a sufficiently large prime congruent to 3 mod 4 then the rank of E should be unbounded in its p-division tower.As in the papers of Greenberg and Howe, the mechanism for arriving at such conclusions is a root number calculation.Fix a number field F , an elliptic curve E over F , and a prime p, and put, where E[p ∞ ] denotes the group of points on E of p-power order.The root numbers at issue here are associated to the L-functions L(s, E, τ ), where τ runs over irreducible self-dual representations of Gal(F ∞ /F ).For a precise definition of L(s, E, τ ) we refer the reader to [12], pp.151 and 156, or to [1], §5, but we should at least spell out our group-theoretic conventions: First of all, a representation of a topological group is understood to be continuous, finite-dimensional, and defined over the complex numbers.In particular, in the case of a profinite group like Gal(F ∞ /F ) a representation is trivial on an open subgroup and hence factors through a finite quotient.It also follows that when the term character is used in the sense of "one-dimensional representation" we are

The arithmetic theory of elliptic curves enters the new century with some of its major secrets intact. Most notably, the Birch and Swinnerton-Dyer conjecture, which relates the arithmetic of an elliptic curve to the analytic behaviour of its associated L-series, is still unproved in spite of important advances in the last decades, some of which are recalled in Chapter 1.

The Birch and Swinnerton-Dyer (BSD) conjecture establishes a deep connection between the arithmetic structure of elliptic curves and the analytic behavior of their associated L-functions, yet its conceptual interpretation remains elusive despite extensive partial results. In this work, we propose an informational–geometric reinterpretation of the conjecture within the framework of Viscous Time Theory (VTT), in which arithmetic invariants are mapped to measurable quantities governing informational coherence. Within this framework, the canonical height is interpreted as a coherence potential, the Mordell–Weil rank as the dimensionality of stable coherence directions, and the BSD regulator as an informational volume. The analytic behavior of the L-function near the critical point s = 1 is reinterpreted as a global coherence response of the underlying informational manifold. This leads naturally to a regime-dependent conservation principle for informational coherence, under which the classical BSD identity emerges as a stable balance condition. To test this formulation, we perform independent numerical validation on benchmark elliptic curves with established BSD data. An informational L-function is fitted to empirical analytic profiles, from which informational curvature and volume are derived. The results show strong quantitative agreement between informational volumes and arithmetic regulators, with correlation coefficients exceeding 0.99, and demonstrate robust, rank-dependent stability behavior across curves of varying complexity. The framework is further explored under increasing geometric and topological complexity through higher-genus and synthetic informational models, which exhibit systematic coherence suppression consistent with known arithmetic phenomena. While this work does not claim a proof of the Birch and Swinnerton-Dyer conjecture, it offers a coherent explanatory framework that clarifies its internal structure, geometrically aligns analytic and arithmetic invariants, and opens new avenues for numerical and conceptual investigation.

Let 𝐸 be an elliptic curve over a quartic field 𝐾. By the Mordell–Weil theorem, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>E</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>K</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> E(K) is a finitely generated group. We determine all the possibilities for the torsion group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>E</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>K</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>tors</m:mi> </m:msub> </m:mrow> </m:math> E(K)_{\mathrm{tors}} , where 𝐾 ranges over all quartic fields 𝐾 and 𝐸 ranges over all elliptic curves over 𝐾. We show that there are no sporadic torsion groups, or in other words, that all torsion groups either do not appear or they appear for infinitely many non-isomorphic elliptic curves 𝐸. Proving this requires showing that numerous modular curves <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>X</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>m</m:mi> <m:mo>,</m:mo> <m:mi>n</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> X_{1}(m,n) have no non-cuspidal degree 4 points. We deal with almost all the curves using one of 3 methods: a method for the rank 0 cases requiring no computation; the Hecke sieve, a local method requiring computer-assisted computations; and the global method, an argument for the positive rank cases also requiring no computation. We deal with the handful of remaining cases using ad hoc methods.

In this chapter we will prove Mordell’s theorem that the group of rational points on a non-singular cubic is finitely generated. There is a tool used in the proof called the height. In brief, the height of a rational point measures how complicated the point is from the viewpoint of number theory.

We prove that the integral points are potentially Zariski dense in the complement of a reduced effective singular anticanonical divisor in a smooth del Pezzo surface, with the exception of ${{\mathbb {P}}}^{2}$ minus three concurrent lines (for which potential density does not hold). This answers positively a question raised by Hassett and Tschinkel and, combined with previous results, completes the proof of the potential density of integral points for complements of anticanonical divisors in smooth del Pezzo surfaces. We then classify the complements that are simply connected and for these we prove that the set of integral points is potentially not thin, as predicted by a conjecture of Corvaja and Zannier.

The constructive theory of hyperelliptic curves has been advanced significantly during the last year. It is intended to give here an indication of the current level of progress, and an outline of the main methods employed. The emphasis in Sections 1 to 4 will be on the group of rational points on the Jacobian of a hyperelliptic curve. Section 5 will concern itself with the use of the Jacobian to help to determine the rational points on the curve itself.KeywordsAbelian GroupElliptic CurveRational PointElliptic CurfNumber FieldThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.