3,130,335 research outputs found

    On Semi-Periods

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    The periods of the three-form on a Calabi-Yau manifold are found as solutions of the Picard-Fuchs equations; however, the toric varietal method leads to a generalized hypergeometric system of equations which has more solutions than just the periods. This same extended set of equations can be derived from symmetry considerations. Semi-periods are solutions of this extended system. They are obtained by integration of the three-form over chains; these chains can be used to construct cycles which, when integrated over, give periods. In simple examples we are able to obtain the complete set of solutions for the extended system. We also conjecture that a certain modification of the method will generate the full space of solutions in general.Comment: 18 pages, plain TeX. Revised derivation of Δ\Delta^* system of equations; version to appear in Nuclear Physics

    Geometric Waldspurger periods II

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    In this paper we extend the calculation of the geometric Waldspurger periods from our paper math/0510110 to the case of ramified coverings. We give some applications to the study of Whittaker coefficients of the theta-lifting of automorphic sheaves from PGL_2 to the metaplectic group Mp_2, they agree with our conjectures from arXiv:1211.1596. In the process of the proof, we get some new automorphic sheaves for GL_2 in the ramified setting. We also formulate stronger conjectures about Waldspurger periods and geometric theta-lifting for the dual pair (SL_2, Mp_2).Comment: 59 pages, final version, to appear in Representation theory (electronic J. of AMS

    Periods implying almost all periods, trees with snowflakes, and zero entropy maps

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    Let XX be a compact tree, ff be a continuous map from XX to itself, End(X)End(X) be the number of endpoints and Edg(X)Edg(X) be the number of edges of XX. We show that if n>1n>1 has no prime divisors less than End(X)+1End(X)+1 and ff has a cycle of period nn, then ff has cycles of all periods greater than 2End(X)(n1)2End(X)(n-1) and topological entropy h(f)>0h(f)>0; so if pp is the least prime number greater than End(X)End(X) and ff has cycles of all periods from 1 to 2End(X)(p1)2End(X)(p-1), then ff has cycles of all periods (this verifies a conjecture of Misiurewicz for tree maps). Together with the spectral decomposition theorem for graph maps it implies that h(f)>0h(f)>0 iff there exists nn such that ff has a cycle of period mnmn for any mm. We also define {\it snowflakes} for tree maps and show that h(f)=0h(f)=0 iff every cycle of ff is a snowflake or iff the period of every cycle of ff is of form 2lm2^lm where mEdg(X)m\le Edg(X) is an odd integer with prime divisors less than End(X)+1End(X)+1

    Notes on Motivic Periods

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    The second part of a set of notes based on lectures given at the IHES in 2015 on Feynman amplitudes and motivic periods