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作者：Burungale, Ashay A.; Kobayashi, Shinichi; Ota, Kazuto
摘要：We prove a fundamental conjecture of Rubin on the structure of local units in the anticyclotomic Z(p)-extension of the unramified quadratic extension of Q(p) for p >= 5 a prime. Rubin's conjecture underlies Iwasawa theory of the anticyclotomic deformation of a CM elliptic curve over the CM field at primes p of good supersingular reduction, notably the Iwasawa main conjecture in terms of the p-adic L -function. As a consequence, we prove an inequality in the p-adic Birch and Swinnerton-Dyer con...
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作者：Jiang, Zilin; Tidor, Jonathan; Yao, Yuan; Zhang, Shengtong; Zhao, Yufei
摘要：Solving a longstanding problem on equiangular lines, we determine, for each given fixed angle and in all sufficiently large dimensions, the maximum number of lines pairwise separated by the given angle. Fix 0 < alpha < 1. Let N-alpha(d) denote the maximum number of lines through the origin in R-d with pairwise common angle arccos alpha. Let k denote the minimum number (if it exists) of vertices in a graph whose adjacency matrix has spectral radius exactly (1 - alpha)/(2 alpha). If k < infinity...
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作者：Gardam, Giles
摘要：The unit conjecture, commonly attributed to Kaplansky, predicts that if K is a field and G is a torsion-free group, then the only units of the group ring K[G] are the trivial units, that is, the non-zero scalar multiples of group elements. We give a concrete counterexample to this conjecture; the group is virtually abelian and the field is order two.
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作者：Alweiss, Ryan; Lovett, Shachar; Wu, Kewen; Zhang, Jiapeng
摘要：A sunflower with r petals is a collection of r sets so that the intersection of each pair is equal to the intersection of all of them. Erdos and Rado proved the sunflower lemma: for any fixed r, any family of sets of size w, with at least about w(w) sets, must contain a sunflower with r petals. The famous sunflower conjecture states that the bound on the number of sets can be improved to c(w) for some constant c. In this paper, we improve the bound to about (log w)(w). In fact, we prove the re...
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作者：Li, Chao; Liu, Yifeng
摘要：In this article, we study the Chow group of the motive associated to a tempered global L-packet pi of unitary groups of even rank with respect to a CM extension, whose global root number is -1. We show that, under some restrictions on the ramification of pi, if the central derivative L'(1/2, pi) is nonvanishing, then the pi-nearly isotypic localization of the Chow group of a certain unitary Shimura variety over its reflex field does not vanish. This proves part of the Beilinson-Bloch conjectur...
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作者：Aspero, David; Schindler, Ralf
摘要：We show that Martin's Maximum(++) implies Woodin's P-max axiom (*). This answers a question from the 1990s and amalgamates two prominent axioms of set theory which were both known to imply that there are aleph(2) many real numbers.
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作者：Leuzinger, Enrico; Young, Robert
摘要：The Dehn function and its higher-dimensional generalizations measure the difficulty of filling a sphere in a space by a ball. In nonpositively curved spaces, one can construct fillings using geodesics, but fillings become more complicated in subsets of nonpositively curved spaces, such as lattices in symmetric spaces. In this paper, we prove sharp filling inequalities for (arithmetic) lattices in higher rank semisimple Lie groups. When n is less than the rank of the associated symmetric space,...
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作者：Elgindi, Tarek M.
摘要：It has been known since work of Lichtenstein and Gunther in the 1920s that the 3D incompressible Euler equation is locally well-posed in the class of velocity fields with Ho center dot lder continuous gradient and suitable decay at infinity. It is shown here that these local solutions can develop singularities in finite time, even for some of the simplest three-dimensional flows.
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作者：Fresan, Javier; Jossen, Peter
摘要：We answer in the negative Siegel's question whether all E-functions are polynomial expressions in hypergeometric E-functions. Namely, we show that if an irreducible differential operator of order three annihilates an E-function in the hypergeometric class, then the singularities of its Fourier transform are constrained to satisfy a symmetry property that generically does not hold. The proof relies on Andre ''s theory of E-operators and Katz's computation of the Galois group of hypergeometric d...
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作者：Bader, Uri; Fisher, David; Miller, Nicholas; Stover, Matthew
摘要：Let Gamma be a lattice in SO0(n, 1). We prove that if the associated locally symmetric space contains infinitely many maximal totally geodesic subspaces of dimension at least 2, then Gamma is arithmetic. This answers a question of Reid for hyperbolic n-manifolds and, independently, McMullen for hyperbolic 3-manifolds. We prove these results by proving a superrigidity theorem for certain representations of such lattices. The proof of our superrigidity theorem uses results on equidistribution fr...
