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61
Elliptic Curves with Complex Multiplication and the Conjecture of Birch and SwinnertonDyer
 Birch and SwinnertonDyer, Invent. Math
, 1981
"... A stronger version of (ii) (with no assumption that E have good reduction above p) was proved in [Ru2]. The program to prove (ii) was also begun by Coates and Wiles; it can ? Partially supported by the National Science Foundation. The author also gratefully acknowledges the CIME for its hospitality ..."
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A stronger version of (ii) (with no assumption that E have good reduction above p) was proved in [Ru2]. The program to prove (ii) was also begun by Coates and Wiles; it can ? Partially supported by the National Science Foundation. The author also gratefully acknowledges the CIME for its hospitality. ?? current address: Department of Mathematics, Stanford University, Stanford, CA 94305 USA, rubin@math.stanford.edu now be completed thanks to the recent Euler system machinery of Kolyvagin [Ko]. This proof will be given in x12, Corollary 12.13 and Theorem 12.19. The material through x4 is background which was not in the Cetraro lectures but is included here for completeness. In those sections we summarize, generally with references to [Si] instead of proofs, the basic properties of elliptic curves that will be needed later. For more details, including proofs, see Silverman's b
Efficient Solution of Rational Conics
 Math. Comp
, 1998
"... this paper (section 2), and to Denis Simon for the reference [10]. ..."
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this paper (section 2), and to Denis Simon for the reference [10].
Solving the Pell Equation
, 2008
"... We illustrate recent developments in computational number theory by studying their implications for solving the Pell equation. We shall see that, if the solutions to the Pell equation are properly represented, the traditional continued fraction method for solving the equation can be significantly a ..."
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We illustrate recent developments in computational number theory by studying their implications for solving the Pell equation. We shall see that, if the solutions to the Pell equation are properly represented, the traditional continued fraction method for solving the equation can be significantly accelerated. The most promising method depends on the use of smooth numbers. As with many algorithms depending on smooth numbers, its run time can presently only conjecturally be established; giving a rigorous analysis is one of the many open problems surrounding the Pell equation.
Verifying nonlinear real formulas via sums of squares
 Theorem Proving in Higher Order Logics, TPHOLs 2007, volume 4732 of Lect. Notes in Comp. Sci
, 2007
"... Abstract. Techniques based on sums of squares appear promising as a general approach to the universal theory of reals with addition and multiplication, i.e. verifying Boolean combinations of equations and inequalities. A particularly attractive feature is that suitable ‘sum of squares ’ certificates ..."
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Abstract. Techniques based on sums of squares appear promising as a general approach to the universal theory of reals with addition and multiplication, i.e. verifying Boolean combinations of equations and inequalities. A particularly attractive feature is that suitable ‘sum of squares ’ certificates can be found by sophisticated numerical methods such as semidefinite programming, yet the actual verification of the resulting proof is straightforward even in a highly foundational theorem prover. We will describe our experience with an implementation in HOL Light, noting some successes as well as difficulties. We also describe a new approach to the univariate case that can handle some otherwise difficult examples. 1 Verifying nonlinear formulas over the reals Over the real numbers, there are algorithms that can in principle perform quantifier elimination from arbitrary firstorder formulas built up using addition, multiplication and the usual equality and inequality predicates. A classic example of such a quantifier elimination equivalence is the criterion for a quadratic equation to have a real root: ∀a b c. (∃x. ax 2 + bx + c = 0) ⇔ a = 0 ∧ (b = 0 ⇒ c = 0) ∨ a � = 0 ∧ b 2 ≥ 4ac
Some old problems and new results about quadratic forms
 Notices Amer. Math. Soc
, 1997
"... It may be a challenging problem to describe the integer solutions to a polynomial equation in several variables. Which integers, for example, are represented by a quadratic polynomial? This ..."
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It may be a challenging problem to describe the integer solutions to a polynomial equation in several variables. Which integers, for example, are represented by a quadratic polynomial? This
Archimedes' Cattle Problem
 American Mathematical Monthly
, 1998
"... this paper is to take the Cattle Problem out of the realm of the \astronomical" and put it into manageable form. This is achieved in formulas (12) and (13) which give explicit forms for the solution. For example, the smallest possible value for the total number of cattle satisfying the conditions of ..."
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this paper is to take the Cattle Problem out of the realm of the \astronomical" and put it into manageable form. This is achieved in formulas (12) and (13) which give explicit forms for the solution. For example, the smallest possible value for the total number of cattle satisfying the conditions of the problem is
Superelliptic curves
, 1993
"... A detailed study is made of super elliptic curves, namely super Riemann surfaces of genus one considered as algebraic varieties, particularly their relation with their Picard groups. This is the simplest setting in which to study the geometric consequences of the fact that certain cohomology groups ..."
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A detailed study is made of super elliptic curves, namely super Riemann surfaces of genus one considered as algebraic varieties, particularly their relation with their Picard groups. This is the simplest setting in which to study the geometric consequences of the fact that certain cohomology groups of super Riemann surfaces are not freely generated modules. The divisor theory of Rosly, Schwarz, and Voronov gives a map from a supertorus to its Picard group, but this map is a projection, not an isomorphism as it is for ordinary tori. The geometric realization of the addition law on Pic via intersections of the supertorus with superlines in projective space is described. The isomorphisms of Pic with the Jacobian and the divisor class group are verified. All possible isogenies, or surjective holomorphic maps between supertori, are determined and shown to induce homomorphisms of the Picard groups. Finally, the solutions to the new super Kadomtsev–Petviashvili (super KP) hierarchy of Mulase–Rabin which arise from super elliptic curves via the Krichever construction are exhibited. 1
EUCLIDEAN QUADRATIC FORMS AND ADCFORMS: I
"... Abstract. A classical result of Aubry, Davenport and Cassels gives conditions for an integral quadratic form to integrally represent every integer that it rationally represents. We give a generalization which allows one to pass from rational to integral representations for suitable quadratic forms o ..."
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Abstract. A classical result of Aubry, Davenport and Cassels gives conditions for an integral quadratic form to integrally represent every integer that it rationally represents. We give a generalization which allows one to pass from rational to integral representations for suitable quadratic forms over a normed ring. The CasselsPfister theorem follows as another special case. This motivates a closer study of the classes of forms which satisfy the hypothesis of our theorem (“Euclidean forms”) and its conclusion (“ADC forms”). 1.1. Normed rings. Let R be a commutative, unital ring. We write R • for R \ {0}. A norm on R is a function   : R → N such that (N0) x  = 0 ⇐ ⇒ x = 0, (N1) ∀x ∈ R, x  = 1 ⇐ ⇒ x ∈ R × , and (N2) ∀x, y ∈ R, xy  = xy. A norm   is nonArchimedean if for all x, y ∈ R, x + y  ≤ max(x, y). A normed ring is a pair (R,  ) where   is a norm on R. A ring admitting a norm is necessarily an integral domain. We denote the fraction field by K. The norm extends uniquely to a homomorphism of groups (K×, ·) → (Q>0, ·). Example 1: The usual absolute value   ∞ (inherited from R) is a norm on Z. Example 2: Let k be a field, R = k[t], and let a ≥ 2 be an integer. Then the map f ∈ k[t] • deg f ↦ → a is a nonArchimedean norm  a on R. When k is finite, it is most natural to take a = #k (see below). Otherwise, we may as well take a = 2. Example 3: An abstract number ring is an infinite integral domain R such that for every nonzero ideal I of R, R/I is finite. (In particular, R may be an order in a number field, the ring of regular functions on an irreducible affine algebraic curve over a finite field, or any localization or completion thereof.) The map x ∈ R • ↦ → #R/(x) gives a norm on R [Cl10, Prop. 5], which we will call the canonical norm. The standard norm   ∞ on Z is canonical, as is the norm  q on the polynomial ring Fq[t]. Example 4: Let R be a discrete valuation ring (DVR) with valuation v: K × → Z
Mathematical method and proof
"... Abstract. On a traditional view, the primary role of a mathematical proof is to warrant the truth of the resulting theorem. This view fails to explain why it is very often the case that a new proof of a theorem is deemed important. Three case studies from elementary arithmetic show, informally, that ..."
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Abstract. On a traditional view, the primary role of a mathematical proof is to warrant the truth of the resulting theorem. This view fails to explain why it is very often the case that a new proof of a theorem is deemed important. Three case studies from elementary arithmetic show, informally, that there are many criteria by which ordinary proofs are valued. I argue that at least some of these criteria depend on the methods of inference the proofs employ, and that standard models of formal deduction are not wellequipped to support such evaluations. I discuss a model of proof that is used in the automated deduction community, and show that this model does better in that respect.
Group Representations and Harmonic Analysis from Euler to Langlands, Part II
, 1996
"... T he essence of harmonic analysis is to decompose complicated expressions into pieces that reflect the structure of a group action when there is one. The goal is to make some difficult analysis manageable. In the seventeenth and eighteenth centuries, the groups that arose in this connection were t ..."
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T he essence of harmonic analysis is to decompose complicated expressions into pieces that reflect the structure of a group action when there is one. The goal is to make some difficult analysis manageable. In the seventeenth and eighteenth centuries, the groups that arose in this connection were the circle R/2#Z , the line R , and finite abelian groups. Embedded in applications were decompositions of functions in terms of multiplicative characters, continuous homomorphisms of the group into the nonzero complex numbers. In the case of the circle, the decomposition is just the expansion of a function on (#,#) into its Fourier series Anthony W. Knapp is professor of mathematics at the State University of New York, Stony Brook. His email address is aknapp@ccmail.sunysb.edu. The author expresses his appreciation to Sigurdur Helgason,