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作者：Petridis, Yiannis N.; Risager, Morten S.
作者单位：University of London; University College London; University of Copenhagen
摘要：Mazur, Rubin, and Stein have recently formulated a series of conjectures about statistical properties of modular symbols in order to understand central values of twists of elliptic curve L-functions. Two of these conjectures relate to the asymptotic growth of the first and second moments of the modular symbols. We prove these on average by using analytic properties of Eisenstein series twisted by modular symbols. Another of their conjectures predicts the Gaussian distribution of normalized mod...
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作者：Boehm, Christoph; Lafuente, Ramiro A.
作者单位：University of Munster
摘要：We show that for an immortal homogeneous Ricci flow solution any sequence of parabolic blow-downs subconverges to a homogeneous expanding Ricci soliton. This is established by constructing a new Lyapunov function based on curvature estimates which come from real geometric invariant theory.
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作者：Ma, Linquan; Schwede, Karl
作者单位：Purdue University System; Purdue University; Utah System of Higher Education; University of Utah
摘要：Using perfectoid algebras we introduce a mixed characteristic analog of the multiplier ideal, respectively test ideal, from characteristic zero, respectively in the case of a regular ambient ring. We prove several properties about this ideal such as subadditivity. We then use these techniques to derive a uniform bound on the growth of symbolic powers of radical ideals in all excellent regular rings. The analogous result was shown in equal characteristic by Ein-Lazarsfeld-Smith and Hochster-Hun...
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作者：Achar, Pramod N.; Riche, Simon
作者单位：Louisiana State University System; Louisiana State University; Universite Clermont Auvergne (UCA); Centre National de la Recherche Scientifique (CNRS)
摘要：In this paper we prove equivalences of categories relating the derived category of a block of the category of representations of a connected reductive algebraic group over an algebraically closed field of characteristic p bigger than the Coxeter number and a derived category of equivariant coherent sheaves on the Springer resolution (or a parabolic counterpart). In the case of the principal block, combined with previous results, this provides a modular version of celebrated constructions due t...
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作者：Kurylev, Yaroslav; Lassas, Matti; Uhlmann, Gunther
作者单位：University of London; University College London; University of Helsinki; University of Washington; University of Washington Seattle
摘要：We study two inverse problems on a globally hyperbolic Lorentzian manifold (M, g). The problems are: Passive observations in spacetime: consider observations in an open set . The light observation set corresponding to a point source at is the intersection of V and the light-cone emanating from the point q. Let be an unknown open, relatively compact set. We show that under natural causality conditions, the family of light observation sets corresponding to point sources at points determine uniqu...
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作者：Ma, Shouhei
作者单位：Institute of Science Tokyo; Tokyo Institute of Technology
摘要：We prove that up to scaling there are only finitely many integral lattices L of signature (2, n) with or such that the modular variety defined by the orthogonal group of L is not of general type. In particular, when , every modular variety defined by an arithmetic group for a rational quadratic form of signature (2, n) is of general type. We also obtain similar finiteness in for the stable orthogonal groups. As a byproduct we derive finiteness of lattices admitting reflective modular form of b...
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作者：Bhatt, Bhargav
作者单位：University of Michigan System; University of Michigan
摘要：Andre recently gave a beautiful proof of Hochster's direct summand conjecture in commutative algebra using perfectoid spaces; his two main results are a generalization of the almost purity theorem (the perfectoid Abhyankar lemma) and a construction of certain faithfully flat extensions of perfectoid algebras where discriminants acquire all p-power roots. In this paper, we explain a quicker proof of Hochster's conjecture that circumvents the perfectoid Abhyankar lemma; instead, we prove and use...
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作者：Viet-Anh Nguyen
作者单位：Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite de Lille
摘要：We study the holonomy cocycle H of a holomorphic foliation F by Riemann surfaces defined on a compact complex projective surface X satisfying the following two conditions: its singularities E are all hyperbolic; there is no holomorphic non-constant map C -> X such that out of E the image of C is locally contained in a leaf. Let T be a harmonic current tangent to F which does not give mass to any invariant analytic curve. Using the leafwise Poincare metric, we show that H is integrable with res...
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作者：Calegari, Frank; Geraghty, David
作者单位：University of Chicago; Boston College
摘要：We prove new modularity lifting theorems for p-adic Galois representations in situations where the methods of Wiles and Taylor-Wiles do not apply. Previous generalizations of these methods have been restricted to situations where the automorphic forms in question contribute to a single degree of cohomology. In practice, this imposes several restrictions-one must be in a Shimura variety setting and the automorphic forms must be of regular weight at infinity. In this paper, we essentially show h...
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作者：Bourque, Maxime Fortier
作者单位：University of Toronto
摘要：We prove that if two conformal embeddings between Riemann surfaces with finite topology are homotopic, then they are isotopic through conformal embeddings. Furthermore, we show that the space of all conformal embeddings in a given homotopy class is homotopy equivalent to a point, a circle, a torus, or the unit tangent bundle of the codomain, depending on the induced homomorphism on fundamental groups. Quadratic differentials play a central role in the proof.
