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

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

Bell Inequality in the Holographic EPR Pair

We study the Bell inequality in a holographic model of the casually disconnected Einstein-Podolsky-Rosen (EPR) pair. The Clauser-Horne-Shimony-Holt(CHSH) form of Bell inequality is constructed using holographic Schwinger-Keldysh (SK) correlators. We show that the manifestation of quantum correlation in Bell inequality can be holographically reproduced from the classical fluctuations of dual accelerating string in the bulk gravity. The violation of this holographic Bell inequality supports the essential quantum property of this holographic model of an EPR pair.Comment: 8 pages, 2 figures; references and texts added; v3: matches published versio