Abstract
These notes fall into two parts. The first part, which goes up to the end of Sect. 5, is a general survey of some of the topics in the theory of Diophantine equations which interest me and on which I hope to see progress within the next 10 years.
Keywords
- Elliptic Curve
- Elliptic Curf
- Algebraic Number
- Diophantine Equation
- Weak Approximation
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References
Silverberg, A.: Open questions in arithmetic algebraic geometry. In: Arithmetic algebraic geometry, pp. 85–142. AMS, RI (2002)
Vaughan, R.C.: The Hardy-Littlewood method, 2nd edn. Cambridge (1997)
Davis, M., Matijasevič, Y., Robinson, J.: Hilbert’s tenth problem: Diophantine equations: positive aspects of a negative solution. In: Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math. Vol. XXVIII, Northern Illinois Univ., De Kalb, Ill., 1974), pp. 323–378. (loose erratum) Amer. Math. Soc., Providence, RI (1976)
Matiyasevich, Y.: Hilbert’s tenth problem: what was done and what is to be done. In: Hilbert’s tenth problem: relations with arithmetic and algebraic geometry (Ghent, 1999), Contemp. Math., vol. 270, pp. 1–47, Amer. Math. Soc., Providence, RI (2000)
Sir Swinnerton-Dyer P.: Weak approximation and R-equivalence on Cubic Surfaces. In: Peyre, E., Tschinkel, Y. (eds.) Rational points on algebraic varieties, Progress in Mathematics, vol. 199, pp. 357–404. Birkhäuser (2001)
Skorobogatov, A.N.: Beyond the Manin obstruction. Invent. Math. 135, 399–424 (1999)
Harari, D.: Obstructions de Manin transcendantes. In: Number theory (Paris, 1993–1994). London Mathematical Society Lecture Note Series, vol. 235, pp. 75–87. Cambridge University Press, Cambridge (1996)
Bright, M., Sir Peter Swinnerton-Dyer: Computing the Brauer-Manin obstructions. Math. Proc. Camb. Phil. Soc. 137, 1–16 (2004)
Manin, Yu.I.: Cubic forms. North-Holland, Amsterdam (1974)
Colliot-Thélène, J.-L., Sansuc, J.-J., Sir Peter Swinnerton-Dyer: Intersections of two quadrics and Châtelet surfaces. J. reine angew. Math. 373, 37–107 (1987); 374,
Colliot-Thélène, J.-L., Sir Peter Swinnerton-Dyer: Hasse principle and weak approximation for pencils of Severi-Brauer and similar varieties. J. reine angew. Math. 453, 49–112 (1994)
Sir Peter Swinnerton-Dyer: Rational points on pencils of conics and on pencils of quadrics. J. Lond. Math. Soc. 50(2), 231–242 (1994)
Skorobogatov, A.: Torsors and rational points. Cambridge Tracts in Mathematics, vol. 144. Cambridge University Press, Cambridge (2001)
Wittenberg, O.: Intersections de deux quadriques et pinceaux de courbes de genre 1. Springer, Heidelberg (2007)
Bender, A.O., Sir Peter Swinnerton-Dyer: Solubility of certain pencils of curves of genus 1, and of the intersection of two quadrics in P 4. Proc. Lond. Math. Soc. 83(3), 299–329 (2001)
Colliot-Thélène, J.-L., Skorobogatov, A.N., Sir Peter Swinnerton-Dyer: Hasse principle for pencils of curves of genus one whose Jacobians have rational 2-division points. Invent. Math. 134, 579–650 (1998)
Sir Peter Swinnerton-Dyer: Arithmetic of diagonal quartic surfaces II. Proc. Lond. Math. Soc. 80(3), 513–544 (2000)
Sir Peter Swinnerton-Dyer: The solubility of diagonal cubic surfaces. Ann. Scient. Éc. Norm. Sup. 34(4), 891–912 (2001)
Kleiman, S.L.: Algebraic cycles and the Weil conjectures. In: Grothendieck, A., Kuiper, N.H. (eds.) Dix exposés sur la cohomologie des schémas, pp. 359–386. North-Holland, Amsterdam (1968)
Serre, J.-P.: Facteurs locaux des fonctions zêta des variétés algébriques (définitions et conjectures). Séminaire Delange-Pisot-Poitou 1969/70, exp. 19
Borel, A.: Cohomologie de SL n et valeurs de fonctions zeta aux points entiers. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 4(4), 613–636 (1977)
Tate, J.T.: On the conjectures of Birch and Swinnerton-Dyer and a geometric analog. Sém. Bourbaki 306 (1966)
Swinnerton-Dyer, P.: The conjectures of Birch and Swinnerton-Dyer, and of Tate. In: Springer, T.A. (ed.) Proceedings of a Conference on Local Fields, Driebergen, 1966, pp. 132–157. Springer, NY (1967)
Rapaport, M., Schappacher, N., Schneider, P. (eds.): Beilinson’s conjectures on special values of L-functions. Academic, NY (1988)
Hulsbergen, W.W.J.: Conjectures in arithmetic algebraic geometry. Vieweg, Braunschweig (1992)
Bloch, S.J.: Higher regulators, algebraic K-theory and zeta-functions of elliptic curves. CRM Monograph Series 11. AMS, RI (2000)
Bloch, S.J., Kato, K.: L-functions and Tamagawa numbers of motives. In: The Grothendieck Festschrift, vol. I, pp. 333–400. Birkhauser, Boston (1990)
Siegel, C.L.: Normen algebraischer Zahlen, Werke, Band IV, pp. 250–268
Raghavan, S.: Bounds for minimal solutions of Diophantine equations. Nachr. Akad. Wiss. Göttingen Math. -Phys. Kl. II, (9):109–114 (1975)
Faltings, G., Wüstholz, G.(eds.): Rational points, 3rd edn. Vieweg, Braunschweig (1992)
Mazur, B.: Rational isogenies of prime degree. Invent. Math. 44, 129–162 (1978)
Merel, L.: Bornes pour la torsion des courbes elliptiques sur les corps de nombres. Invent. Math. 124, 437–449 (1996)
Rubin, K., Silverberg, A.: Ranks of elliptic curves. Bull. Amer. Math. Soc. 39, 455–474 (2002)
Gebel, J., Zimmer, H.G.: Computing the Mordell-Weil group of an elliptic curve over Q. In: Kisilevsky, H., Ram Murthy, M. (eds.) Elliptic curves and related topics. CRM Proceedings and Lecture Notes, vol. 4, pp. 61–83. AMS, RI (1994)
Gross, B., Kohnen, W., Zagier, D.: Heegner points and derivatives of L-series II. Math. Ann. 278, 497–562 (1987)
Gross, B., Zagier, D.: Heegner points and derivatives of L-series. Invent. Math. 84, 225–320 (1986)
Gross, B.: Kolyvagin’s work for modular elliptic curves. In: Coates, J., Taylor, M.J. (eds.) L-functions and arithmetic, pp. 235–256. Cambridge (1991)
Kolyvagin, V.A.: Finiteness of E(Q) and III (E∕Q) for a class of Weil curves. Izv. Akad. Nauk SSSR 52, 522–540 (1988); translation in Math. USSR-Izv. 33, 523–541 (1989)
Rubin, K.: Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer. In: Viola, C. (ed.) Arithmetic theory of elliptic curves, pp. 167–234. Springer Lecture Notes 1716 (1999)
Beauville, A.: Complex algebraic surfaces, 2nd edn. London Mathematical Society Student Texts, 34, Cambridge University Press, Cambridge (1996)
Birch, B.J.: Homogeneous forms of odd degree in a large number of variables. Mathematika 4, 102–105 (1957)
Hooley, C.: On nonary cubic forms. J. Reine Angew. Math. 386, 32–98 (1988); 415, 95–165 (1991); 456, 53–63 (1994)
Browning, T.D.: An overview of Manin’s conjecture for Del Pezzo surfaces. In: Duke, W., Tschinkel, Y. (eds.) Analytic number theory: A tribute to Gauss and Dirichlet. AMS, RI (2007)
Peyre, E.: Hauteurs et mesures de Tamagawa sur les variétés de Fano. Duke J. Math. 79, 101–218 (1995)
Slater, J.B., Sir Peter Swinnerton-Dyer: Counting points on cubic surfaces I. Astérisque 251, 1–11 (1998)
de la Brèteche, R., Browning, T.D.: On Manin’s conjecture for singular Del Pezzo surfaces of degree four, I. Mich. Math. J (to appear)
dela Brèteche, R., Browning, T.D.: On Manin’s conjecture for singular del Pezzo surfaces of degree four. II. Math. Proc. Cambridge Philos. Soc., 143(3), 579–605 (2007)
dela Brèteche, R., Browning, T.D., Derenthal, U.: On Manin’s conjecture for a certain singular cubic surface. Ann. Sci. École Norm. Sup. (4), 40(1), 1–50 (2007)
de la Brèteche, R.: Sur le nombre de points de hauteur bornée d’une certaine surface cubique singulière. Astérisque 251, 51–77 (1998)
dela Brèteche, R., Swinnerton-Dyer, P.: Fonction zêta des hauteurs associée à une certaine surface cubique. Bull. Soc. Math. France, 135(1), (2007)
Swinnerton-Dyer, P.: A canonical height on X 0 3=X 1 X 2 X 3. In: Diophantine geometry. CRM Series, vol.4, pp.309–322. Ed. Norm., Pisa (2007)
Colliot-Thélène, J.-L., Skorobogatov, A.N. Sir Peter Swinnerton-Dyer: Rational points and zero-cycles on fibred varieties: Schinzel’s Hypothesis and Salberger’s device. J. reine angew. Math. 495, 1–28 (1998)
Sir Peter Swinnerton-Dyer: Some applications of Schinzel’s hypothesis to diophantine equations. In: Györy, K., Iwaniec, H., Urbanowicz, J. (eds.) Number theory in progress, vol.1, pp.503–530. de Gruyter, Berlin (1999)
Milne, J.S.: Arithmetic duality theorems. Perspectives in Mathematics 1, Academic Press Inc., Boston (1986)
Skorobogatov, A.N., Swinnerton-Dyer, P.: 2-descent on elliptic curves and rational points on certain Kummer surfaces. Adv. Math. 198, 448–483 (2005)
Colliot-Thélène, J.-L., Skorobogatov, A.N., Sir Peter Swinnerton-Dyer: Double fibres and double covers: paucity of rational points. Acta Arith. 79, 113–135 (1997)
Bright, M.: Ph.D. dissertation. Cambridge (2002)
Cassels, J.W.S.: Second descents for elliptic curves. J. reine angew. Math. 494, 101–127 (1998)
Colliot-Thélène, J.-L.: Hasse principle for pencils of curves of genus one whose Jacobians have a rational 2-division point (close variation on a paper of Bender and Swinnerton-Dyer). In: Rational points on algebraic varieties. Progr. Math. 199, 117–161 (2001)
Salberger, P., Skorobogatov, A.N.: Weak approximation for surfaces defined by two quadratic forms. Duke J. Math. 63, 517–536 (1991)
Swinnerton-Dyer, P.: Weak approximation on Del Pezzo surfaces of degree 4. In: Arithmetic of higher-dimensional algebraic varieties; Progr. Math. 226, 235–257 (2004)
Coray, D.: Points algébriques sur les surfaces de Del Pezzo. C. R. Acad. Sci. Paris 284, 1531–1534 (1977)
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Swinnerton-Dyer, P. (2011). Topics in Diophantine Equations. In: Corvaja, P., Gasbarri, C. (eds) Arithmetic Geometry. Lecture Notes in Mathematics(), vol 2009. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15945-9_2
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