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作者：Razborov, Alexander A.
摘要：If A is a finite subset of a free group with at least two noncommuting elements, then vertical bar A.A.A vertical bar >= vertical bar A vertical bar(2)/(log vertical bar A vertical bar)(O(1)). More generally, the same conclusion holds in an arbitrary virtually free group, unless A generates a virtually cyclic subgroup. The central part of the proof of this result is carried on by estimating the number of collisions in multiple products A(1) . . . . A(k). We include a few simple observations sh...
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作者：Medvedev, Alice; Scanlon, Thomas
摘要：We study algebraic dynamical systems (and, more generally, a-varieties) (Phi : A(c)(n) given by coordinatewise univariate polynomials by refining an old theorem of Ritt on compositional identities amongst polynomials. More precisely, we find a nearly canonical way to write a polynomial as a composition of clusters from which one may easily read off possible compositional identities. Our main result is an explicit description of the (weakly) skew-invariant varieties, that is, for a fixed field ...
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作者：Gaussent, Stephane; Rousseau, Guy
摘要：We define the spherical Hecke algebra H for an almost split Kac-Moody group G over a local non-archimedean field. We use the hovel I associated to this situation, which is the analogue of the Bruhat-Tits building for a reductive group. The stabilizer K of a special point on the standard apartment plays the role of a maximal open compact subgroup. We can define H as the algebra of K-bi-invariant functions on G with almost finite support. As two points in the hovel are not always in a same apart...
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作者：Poonen, Bjorn; Stoll, Michael
摘要：Consider the smooth projective models C of curves y(2) = f(x) with f (x) is an element of Z[x] monic and separable of degree 2g + 1. We prove that for g >= 3, a positive fraction of these have only one rational point, the point at infinity. We prove a lower bound on this fraction that tends to 1 as g -> infinity. Finally, we show that C(Q) can be algorithmically computed for such a fraction of the curves. The method can be summarized as follows: using p-adic analysis and an idea of McCallum, w...
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作者：Lei, Antonio; Loeffler, David; Zerbes, Sarah Livia
摘要：We construct a Euler system in the cohomology of the tensor product of the Galois representations attached to two modular forms, using elements in the higher Chow groups of products of modular curves. We use these elements to prove a finiteness theorem for the strict Selmer group of the Galois representation when the associated p-adic Rankin-Selberg L-function is nonvanishing at s=1.
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作者：Green, Ben; Tao, Terence
摘要：This is an erratum to the paper The quantitative behaviour of polynomial orbits on nilmanifolds by the authors, published as Ann. of Math. (2) 175 (2012), no. 2, 465-540. The proof of Theorem 8.6 of that paper, which claims a distribution result for multiparameter polynomial sequences on nilmanifolds, was incorrect. We provide two fixes for this issue here. First, we deduce the equal sides case N-1 = . . . = N-t = N of this result from the 1-parameter results in the paper. This is the same bas...
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作者：Hacon, Christopher D.; McKernan, James; Xu, Chenyang
摘要：We show that log canonical thresholds satisfy the ACC.
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作者：Ekedahl, Torsten; Skjelnes, Roy
摘要：We give an explicit construction, for a flat map X -> S of algebraic spaces, of an ideal in the n'th symmetric product of X over S. Blowing up this ideal is then shown to be isomorphic to the schematic closure in the Hilbert scheme of length n subschemesof the locusof n distinct points.This generalizes Haiman's corresponding result for the affine complex plane.However, our construction of the ideal is very different from that of Haiman, using the formalism of divided powers rather than represe...
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作者：Ullmo, Emmanuel; Yafaev, Andrei
摘要：In this paper we develop a strategy and some technical tools for proving the Andre-Oort conjecture. We give lower bounds for the degrees of Galois orbits of geometric components of special subvarieties of Shimura varieties, assuming the Generalised Riemann Hypothesis. We proceed to show that sequences of special subvarieties whose Galois orbits have bounded degrees are equidistributed in a suitable sense.
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作者：Barnet-Lamb, Thomas; Gee, Toby; Geraghty, David; Taylor, Richard
摘要：We prove an automorphy lifting theorem for l-adic representations where we impose a new condition at l, which we call potentential diagonalizability. This result allows for change of weight and seems to be substantially more flexible than previous theorems along the same lines. We derive several applications. For instance we show that any irreducible, totally odd, essentially self-dual, regular, weakly compatible system of l-adic representations of the absolute Galois group of a totally real f...
