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    Muir String Quartet, December 1, 2003

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    This is the concert program of the Muir String Quartet performance on Monday, December 1, 2003 at 8:00 p.m., at the Tsai Performance Center, 685 Commonwealth Avenue, Boston, Massachusetts. Works performed were Quartet in A Major, Op. 18, No. 5 and Quartet in B-flat Major, Op. 130 by Ludwig van Beethoven. Digitization for Boston University Concert Programs was supported by the Boston University Center for the Humanities Library Endowed Fund

    Shtukas and the Taylor expansion of LL-functions (II)

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    For arithmetic applications, we extend and refine our results in \cite{YZ} to allow ramifications in a minimal way. Starting with a possibly ramified quadratic extension F/FF'/F of function fields over a finite field in odd characteristic, and a finite set of places Σ\Sigma of FF that are unramified in FF', we define a collection of Heegner--Drinfeld cycles on the moduli stack of PGL2\mathrm{PGL}_{2}-Shtukas with rr-modifications and Iwahori level structures at places of Σ\Sigma. For a cuspidal automorphic representation π\pi of PGL2(AF)\mathrm{PGL}_{2}(\mathbb{A}_{F}) with square-free level Σ\Sigma, and rZ0r\in\mathbb{Z}_{\ge0} whose parity matches the root number of πF\pi_{F'}, we prove a series of identities between: (1) The product of the central derivatives of the normalized LL-functions L(a)(π,1/2)L(ra)(πη,1/2)\mathcal{L}^{(a)}(\pi, 1/2)\mathcal{L}^{(r-a)}(\pi\otimes\eta, 1/2), where η\eta is the quadratic id\`ele class character attached to F/FF'/F, and 0ar0\le a\le r; (2) The self intersection number of a linear combination of Heegner--Drinfeld cycles. In particular, we can now obtain global LL-functions with odd vanishing orders. These identities are function-field analogues of the formulas of Waldspurger and Gross--Zagier for higher derivatives of LL-functions.Comment: 90 page
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