Arithmetic on curves

Mazur, Barry

doi:10.1090/s0273-0979-1986-15430-3

Cited by 84 publications

(38 citation statements)

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“…Examples where dim K We have AutC<X) = G; for a more detailed description of L due to Serre, see [Ma,.…”

Section: Examples In Low Dimensionmentioning

confidence: 99%

Group Representations and Lattices

Gross¹

1990

Journal of the American Mathematical Society

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This paper began as an attempt to understand the Euclidean lattices that were recently constructed (using the theory of elliptic curves) by Elkies [E] and Shioda [Sh I,Sh2], from the point of view of group representations. The main idea appears in a note of Thompson [Th2]: if one makes strong irreducibility hypotheses on a rational representation V of a finite group G, then the G-stable Euclidean lattices in V are severely restricted. Unfortunately, these hypotheses are rarely satisfied when V is absolutely irreducible over Q. But there are many examples where the ring End G ( V) is an imaginary quadratic field or a definite quaternion algebra. These representations allow us to construct some of the Mordell-Weillattices considered by Elkies and Shioda, as well as some interesting even unimodular lattices that do not seem to come from the theory of elliptic curves.In § I we discuss lattices and Hermitian forms on T/, and in § §2-4 the strong irreducibility hypotheses we wish to make. In § 5 we show how our hypotheses imply the existence of a finite number (up to isomorphism) of Euclidean I. LATTICES AND HERMITIAN FORMSIn this section, we establish the notation that will be used throughout the paper. Let G be a finite group of order g. Elements of G will be denoted s, t , .... Let V be a finite-dimensional rational vector space that affords a linear representation of Gover Q. We view elements of G as linear operators acting on the right of V, and so have the formula: v S / = (v s / for v E V and s, t E G. Let x(s) = Tracev(s) be the character of V.

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“…Examples where dim K We have AutC<X) = G; for a more detailed description of L due to Serre, see [Ma,.…”

Section: Examples In Low Dimensionmentioning

confidence: 99%

Group Representations and Lattices

Gross¹

1990

Journal of the American Mathematical Society

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“…Deep results in algebraic geometry and diophantine approximation have been employed to address this problem; (see [2,3] for surveys). For instance, in 1929 Siegel [5] used such methods to prove one of the most celebrated results in this area; namely, that if the curve given by ( 1 ) is irreducible and of positive genus then it has at most finitely many points with integral coordinates.…”

Section: Communicated By William Adams)mentioning

confidence: 99%

“…□ Example. It is well known that a polynomial f(x) with integer coefficients takes on the value of an mth power of an integer for all positive integers x if and only if f(x) itself is the mth power of a polynomial with integer coefficients; see [3,Number 114,p. 132].…”

Section: Communicated By William Adams)mentioning

confidence: 99%

The Diophantine equation 𝑓(𝑥)=𝑔(𝑦)

Cochrane¹

1990

Proc. Amer. Math. Soc.

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Abstract.Let f(x),g(y) be polynomials over Z of degrees n and m respectively and with leading coefficients an , b . Suppose that m\n and that an/bm is the mth power of a rational number. We give two elementary proofs that the equation f(x) = g(y) has at most finitely many integral solutions unless f(x) = g(h(x)) for some polynomial h(x) with rational coefficients taking integral values at infinitely many integers.The problem of showing that a given polynomial diophantine equation (1) P(x,y) = 0 has at most finitely many integral solutions has been a central problem of research in number theory. Deep results in algebraic geometry and diophantine approximation have been employed to address this problem; (see [2,3] for surveys). For instance, in 1929 Siegel [5] used such methods to prove one of the most celebrated results in this area; namely, that if the curve given by ( 1 ) is irreducible and of positive genus then it has at most finitely many points with integral coordinates. Moreover, he was able to characterize those curves of genus zero having infinitely many points with integral coordinates. The object of this paper is to obtain the same type of result for a special class of diophantine equations as an easy corollary of two lemmas from elementary complex analysis. A second proof of the result is given at the end of the paper using only elementary methods.We shall restrict our attention to diophantine equations of the type (2) f(x) = anx +an_xx + .-+ a0 = bmy +bm_xy +---+ b0 = g(y)with integer coefficients, an / 0, bm / 0, although the method we use may be applied to a wider variety of equations of type ( 1 ).Theorem. Suppose that m\n and that (an/bm) is the mth power of a rational number. Then either (iCC) f(x) -g(h(x)) for some polynomial h(x) with rational coefficients taking integral values at infinitely many integers ; or (ii) equation (2) has at most finitely many integral solutions, u

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“…Similarly one can construct infinitely many curves C/Q of genus 1 with a fixed locus of bad reduction (and C(Q) = ∅); see [Maz2,p. 241].…”

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confidence: 99%

“…Our sketch follows Deligne's presentation [De1]; see also [Sz], [CS]. More about arithmetic on curves, leading up to Falting's theorem, can be found in Mazur's survey [Maz2].…”

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confidence: 99%

Untitled

2000

Bull. Amer. Math. Soc.

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Arithmetic on curves

Cited by 84 publications

References 40 publications

Group Representations and Lattices

Group Representations and Lattices

The Diophantine equation 𝑓(𝑥)=𝑔(𝑦)

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