3,685 research outputs found

    Graded structures and differential operators on nearly holomorphic and quasimodular forms on classical groups

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    We wish to use graded structures [KrVu87], [Vu01] on dffierential operators and quasimodular forms on classical groups and show that these structures provide a tool to construct p-adic measures and p-adic L-functions on the corresponding non-archimedean weight spaces. An approach to constructions of automorphic L-functions on uni-tary groups and their p-adic avatars is presented. For an algebraic group G over a number eld K these L functions are certain Euler products L(s, π\pi, r, χ\chi). In particular, our constructions cover the L-functions in [Shi00] via the doubling method of Piatetski-Shapiro and Rallis. A p-adic analogue of L(s, π\pi, r, χ\chi) is a p-adic analytic function L p (s, π\pi, r, χ\chi) of p-adic arguments s ∈\in Z p , χ\chi mod p rPresented in a talk for the INTERNATIONAL SCIENTIFIC CONFERENCE "GRADED STRUCTURES IN ALGEBRA AND THEIR APPLICATIONS" dedicated to the memory of Prof. Marc Krasner on Friday, September 23, 2016, International University Centre (IUC), Dubrovnik, Croatia

    On the simplest sextic fields and related Thue equations

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    We consider the parametric family of sextic Thue equations x6−2mx5y−5(m+3)x4y2−20x3y3+5mx2y4+2(m+3)xy5+y6=λ x^6-2mx^5y-5(m+3)x^4y^2-20x^3y^3+5mx^2y^4+2(m+3)xy^5+y^6=\lambda where m∈Zm\in\mathbb{Z} is an integer and λ\lambda is a divisor of 27(m2+3m+9)27(m^2+3m+9). We show that the only solutions to the equations are the trivial ones with xy(x+y)(x−y)(x+2y)(2x+y)=0xy(x+y)(x-y)(x+2y)(2x+y)=0.Comment: 12 pages, 2 table

    Numerical Ricci-flat metrics on K3

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    We develop numerical algorithms for solving the Einstein equation on Calabi-Yau manifolds at arbitrary values of their complex structure and Kahler parameters. We show that Kahler geometry can be exploited for significant gains in computational efficiency. As a proof of principle, we apply our methods to a one-parameter family of K3 surfaces constructed as blow-ups of the T^4/Z_2 orbifold with many discrete symmetries. High-resolution metrics may be obtained on a time scale of days using a desktop computer. We compute various geometric and spectral quantities from our numerical metrics. Using similar resources we expect our methods to practically extend to Calabi-Yau three-folds with a high degree of discrete symmetry, although we expect the general three-fold to remain a challenge due to memory requirements.Comment: 38 pages, 10 figures; program code and animations of figures downloadable from http://schwinger.harvard.edu/~wiseman/K3/ ; v2 minor corrections, references adde
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