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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：Lie, Victor
摘要：We prove affirmatively the one-dimensional case of a conjecture of Stein regarding the L-p-boundedness of the Polynomial Carleson operator for 1 < p < infinity. Our proof relies on two new ideas: (i) we develop a framework for higher-order wave-packet analysis that is consistent with the time-frequency analysis of the (generalized) Carleson operator, and (ii) we introduce a local analysis adapted to the concepts of mass and counting function, which yields a new tile discretization of the time-...
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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：Xu, Chenyang; Zhuang, Zhiquan
摘要：In this paper, we consider the CM line bundle on the K-moduli space, i.e., the moduli space parametrizing K-polystable Fano varieties. We prove it is ample on any proper subspace parametrizing reduced uniformly K-stable Fano varieties that conjecturally should be the entire moduli space. As a corollary, we prove that the moduli space parametrizing smoothable K-polystable Fano varieties is projective. During the course of proof, we develop a new invariant for filtrations that can be used to tes...
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作者：Xu, Chenyang
摘要：We prove a version of Jonsson-Mustata's Conjecture, which says for any graded sequence of ideals, there exists a quasi-monomial valuation computing its log canonical threshold. As a corollary, we confirm Chi Li's conjecture that a minimizer of the normalized volume function is always quasi-monomial. Applying our techniques to a family of klt singularities, we show that the volume of klt singularities is a constructible function. As a corollary, we prove that in a family of klt log Fano pairs, ...
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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.
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作者：Mellit, Anton
摘要：We prove an explicit formula for the Poincare polynomials of parabolic character varieties of Riemann surfaces with semisimple local monodromies, which was conjectured by Hausel, Letellier and Rodriguez-Villegas. Using an approach of Mozgovoy and Schiffmann the problem is reduced to counting pairs of a parabolic vector bundle and a nilpotent endomorphism of prescribed generic type. The generating function counting these pairs is shown to be a product of Macdonald polynomials and the function c...
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作者：Bridson, M. R.; McReynolds, D. B.; Reid, A. W.; Spitler, R.
摘要：We construct arithmetic Kleinian groups that are profinitely rigid in the absolute sense: each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. The Bianchi group PSL(2, Z[omega]) with omega(2) + omega + 1 = 0 is rigid in this sense. Other examples include the non-uniform lattice of minimal co-volume in PSL(2, C) and the fundamental group of the Weeks manifold (the closed hyperbolic 3-manifold of minimal volume).
