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151
Zeroes of Zeta Functions and Symmetry
, 1999
"... Hilbert and Polya suggested that there might be a natural spectral interpretation of the zeroes of the Riemann Zeta function. While at the time there was little evidence for this, today the evidence is quite convincing. Firstly, there are the “function field” analogues, that is zeta functions of cur ..."
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Cited by 180 (2 self)
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Hilbert and Polya suggested that there might be a natural spectral interpretation of the zeroes of the Riemann Zeta function. While at the time there was little evidence for this, today the evidence is quite convincing. Firstly, there are the “function field” analogues, that is zeta functions of curves over finite fields and their generalizations. For these a spectral interpretation for their zeroes exists in terms of eigenvalues of Frobenius on cohomology. Secondly, the developments, both theoretical and numerical, on the local spacing distributions between the high zeroes of the zeta function and its generalizations give striking evidence for such a spectral connection. Moreover, the lowlying zeroes of various families of zeta functions follow laws for the eigenvalue distributions of members of the classical groups. In this paper we review these developments. In order to present the material fluently, we do not proceed in chronological order of discovery. Also, in concentrating entirely on the subject matter of the title, we are ignoring the standard body of important work that has been done on the zeta function and Lfunctions.
Cycles of quadratic polynomials and rational points on a genus 2 curve
, 1996
"... It has been conjectured that for N sufficiently large, there are no quadratic polynomials in Q[z] with rational periodic points of period N. Morton proved there were none with N = 4, by showing that the genus 2 algebraic curve that classifies periodic points of period 4 is birational to X1(16), who ..."
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Cited by 45 (13 self)
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It has been conjectured that for N sufficiently large, there are no quadratic polynomials in Q[z] with rational periodic points of period N. Morton proved there were none with N = 4, by showing that the genus 2 algebraic curve that classifies periodic points of period 4 is birational to X1(16), whose rational points had been previously computed. We prove there are none with N = 5. Here the relevant curve has genus 14, but it has a genus 2 quotient, whose rational points we compute by performing a 2descent on its Jacobian and applying a refinement of the method of Chabauty and Coleman. We hope that our computation will serve as a model for others who need to compute rational points on hyperelliptic curves. We also describe the three possible Gal(Q/Q)stable 5cycles, and show that there exist Gal(Q/Q)stable Ncycles for infinitely many N. Furthermore, we answer a question of Morton by showing that the genus 14 curve and its quotient are not modular. Finally, we mention some partial results for N = 6.
Density of rational points on elliptic K3 surfaces
 Asian J. Math
"... Let X be a smooth projective algebraic variety defined over a number field K. We will say that rational points on X are potentially dense if there exists a finite extension K ′ /K such that the set X(K ′ ) of K ′rational points is Zariski dense. What are possible strategies to propagate rational po ..."
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Cited by 41 (3 self)
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Let X be a smooth projective algebraic variety defined over a number field K. We will say that rational points on X are potentially dense if there exists a finite extension K ′ /K such that the set X(K ′ ) of K ′rational points is Zariski dense. What are possible strategies to propagate rational points on
RATIONAL POINTS ON QUARTICS
 VOL. 104, NO. 3 DUKE MATHEMATICAL JOURNAL
, 2000
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Ranks of twists of elliptic curves and Hilbert’s tenth problem, arxiv:0904.3709v2 [math.NT
"... Abstract. In this paper we investigate the 2Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2Selmer rank, and we give lower bounds for the number of twists (with bound ..."
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Cited by 35 (4 self)
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Abstract. In this paper we investigate the 2Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) that have a given 2Selmer rank. As a consequence, under appropriate hypotheses we can find many twists with trivial MordellWeil group, and (assuming the ShafarevichTate conjecture) many others with infinite cyclic MordellWeil group. Using work of Poonen and Shlapentokh, it follows from our results that if the ShafarevichTate conjecture holds, then Hilbert’s Tenth Problem has a negative answer over the ring of integers of every number field. 1. Introduction and
A linear lower bound on the gonality of modular curves
 Internat. Math. Res. Notices
, 1996
"... 0.1. Statement of result. In this note we prove the following: Theorem 0.1. Let Γ ⊂ PSL2(Z) be a congruence subgroup, and XΓ the corresponding modular curve. Let DΓ = [PSL2(Z) : Γ] and let dC(XΓ) be the Cgonality of XΓ. Then 7 ..."
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Cited by 31 (0 self)
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0.1. Statement of result. In this note we prove the following: Theorem 0.1. Let Γ ⊂ PSL2(Z) be a congruence subgroup, and XΓ the corresponding modular curve. Let DΓ = [PSL2(Z) : Γ] and let dC(XΓ) be the Cgonality of XΓ. Then 7
Integral points on elliptic curves and 3torsion in class groups
"... We give new bounds for the number of integral points on elliptic curves. The method may be said to interpolate between approaches via diophantine techniques ([BP], [HBR]) and methods based on quasiorthogonality in the MordellWeil lattice ([Sil6], [GS], [He]). We apply our results to break previous ..."
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Cited by 26 (6 self)
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We give new bounds for the number of integral points on elliptic curves. The method may be said to interpolate between approaches via diophantine techniques ([BP], [HBR]) and methods based on quasiorthogonality in the MordellWeil lattice ([Sil6], [GS], [He]). We apply our results to break previous