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Manins conjecture predicts the asymptotic behavior of the number of rational points of bounded height on algebraic varieties. For toric varieties, it was proved by Batyrev and Tschinkel via height zeta functions and an application of the Poisson formula. An alternative approach to Manins conjecture via universal torsors was used so far mainly over the field Q of rational numbers. In this note, we give a proof of Manins conjecture over the Gaussian rational numbers Q(i) and over other imaginary quadratic number fields with class number 1 for the singular toric cubic surface defined by t^3=xyz.
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We investigate the average number of solutions of certain quadratic congruences. As an application, we establish Manins conjecture for a cubic surface whose singularity type is A_5+A_1.
We prove Manins conjecture over imaginary quadratic number fields for a cubic surface with a singularity of type E_6.
In 1922, Mordell conjectured the striking statement that for a polynomial equation $f(x,y)=0$, if the topology of the set of complex number solutions is complicated enough, then the set of rational number solutions is finite. This was proved by Faltings in 1983, and again by a different method by Vojta in 1991, but neither proof provided a way to provably find all the rational solutions, so the search for other proofs has continued. Recently, Lawrence and Venkatesh found a third proof, relying on variation in families of $p$-adic Galois representations; this is the subject of the present exposition.
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