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作者：Bennett, Michael A.; Siksek, Samir
摘要：If k is a sufficiently large positive integer, we show that the Diophantine equation n(n + d) center dot center dot center dot (n + (k-1)d) = y(l) has at most finitely many solutions in positive integers n, d, y and, with gcd(n, d) = 1 and l >= 2. Our proof relies upon Frey-Hellegouarch curves and results on supersingular primes for elliptic curves without complex multiplication, derived from upper bounds for short character sums and sieves, analytic and combinatorial.
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作者：Skinner, Christopher
摘要：Let E be a semistable elliptic curve over Q. We prove that if E has non-split multiplicative reduction at at least one odd prime or split multiplicative reduction at at least two odd primes, then rank(z)E(Q) = 1 and #III(E) < infinity double right arrow ords(s=1)L(E, s) = 1. We also prove the corresponding result for the abelian variety associated with a weight 2 newform f of trivial character. These, and other related results, are consequences of our main theorem, which establishes criteria f...
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作者：Masser, David; Zannier, Umberto
摘要：We prove among other things the existence of Hodge generic abelian varieties defined over the algebraic numbers and not isogenous to any Jacobian. Actually, we also show that in various interpretations these abelian varieties make up the majority, and we give certain uniform bounds on the possible degree of the fields of definition. In particular, this yields a new answer (in strong form) to a question of Katz and Oort, compared to previous work of Chai and Oort (2012, conditional on the Andre...
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作者：Piccirillo, Lisa
摘要：A knot is said to be slice if it bounds a smooth properly embedded disk in B-4. We demonstrate that the Conway knot is not slice. This completes the classification of slice knots under 13 crossings and gives the first example of a non-slice knot which is both topologically slice and a positive mutant of a slice knot.
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作者：Couveignes, Jean-Marc
摘要：We construct small models of number fields and deduce a better bound for the number of number fields of given degree and bounded discriminant.
