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作者：Dasgupta, Samit; Kakde, Mahesh; Ventullo, Kevin
摘要：In 1980, Gross conjectured a formula for the expected leading term at s = 0 of the Deligne-Ribet p-adic L-function associated to a totally even character psi of a totally real field F. The conjecture states that after scaling by L(psi omega(-1), 0), this value is equal to a p-adic regulator of units in the abelian extension of F cut out by psi omega(-1). In this paper, we prove Gross's conjecture.
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作者：Jitomirskaya, Svetlana; Liu, Wencai
摘要：We determine exact exponential asymptotics of eigenfunctions and of corresponding transfer matrices of the almost Mathieu operators for all frequencies in the localization regime. This uncovers a universal structure in their behavior, governed by the continued fraction expansion of the frequency, explaining some predictions in physics literature. In addition it proves the arithmetic version of the frequency transition conjecture. Finally, it leads to an explicit description of several non-regu...
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作者：Frantzikinakis, Nikos; Host, Bernard
摘要：The Mobius disjointness conjecture of Sarnak states that the Mobius function does not correlate with any bounded sequence of complex numbers arising from a topological dynamical system with zero topological entropy. We verify the logarithmically averaged variant of this conjecture for a large class of systems, which includes all uniquely ergodic systems with zero entropy. One consequence of our results is that the Liouville function has super-linear block growth. Our proof uses a disjointness ...
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作者：Irie, Kei; Marques, Fernando C.; Neves, Andre
摘要：For almost all Riemannian metrics (in the C-infinity Baire sense) on a closed manifold Mn+1, 3 <= (n + 1) <= 7, we prove that the union of all closed, smooth, embedded minimal hypersurfaces is dense. This implies there are infinitely many minimal hypersurfaces, thus proving a conjecture of Yau (1982) for generic metrics.
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作者：Bourgain, Jean; Dyatlov, Semyon
摘要：For all convex co-compact hyperbolic surfaces, we prove the existence of an essential spectral gap, that is, a strip beyond the unitarity axis in which the Selberg zeta function has only finitely many zeroes. We make no assumption on the dimension delta of the limit set; in particular, we do not require the pressure condition delta <= 1/2. This is the first result of this kind for quantum Hamiltonians. Our proof follows the strategy developed by Dyatlov and Zahl. The main new ingredient is the...
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作者：Fujino, Osamu
摘要：We prove some semipositivity theorems for singular varieties coming from graded polarizable admissible variations of mixed Hodge structure. As an application, we obtain that the moduli functor of stable varieties is semipositive in the sense of Kollar. This completes Kollar's projectivity criterion for the moduli spaces of higher-dimensional stable varieties.
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作者：Barlow, Martin T.; Murugan, Mathav
摘要：We prove that the elliptic Harnack inequality (on a manifold, graph, or suitably regular metric measure space) is stable under bounded perturbations, as well as rough isometrics.
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作者：Gan, Wee Teck; Ichino, Atsushi
摘要：We generalize the Shimura-Waldspurger correspondence, which describes the generic part of the automorphic discrete spectrum of the metaplectic group Mp(2), to the metaplectic group Mp(2n) of higher rank. To establish this, we transport Arthur's endoscopic classification of representations of the odd special orthogonal group SO2r+1 with r >> 2n by using a result of J.-S. Li on global theta lifts in the stable range.
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作者：Georgieva, Penka; Zinger, Aleksey
摘要：We construct positive-genus analogues of Welschinger's invariants for many real symplectic manifolds, including the odd-dimensional projective spaces and the renowned quintic threefold. In some cases, our invariants provide lower bounds for counts of real positive-genus curves in real algebraic varieties. Our approach to the orientability problem is based entirely on the topology of real bundle pairs over symmetric surfaces; the previous attempts involved direct computations for the determinan...
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作者：Mirkovic, I.; Vilonen, K.
