42,635 research outputs found

    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 Fβ€²/FF'/F of function fields over a finite field in odd characteristic, and a finite set of places Ξ£\Sigma of FF that are unramified in Fβ€²F', 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 Ξ£\Sigma. For a cuspidal automorphic representation Ο€\pi of PGL2(AF)\mathrm{PGL}_{2}(\mathbb{A}_{F}) with square-free level Ξ£\Sigma, and r∈Zβ‰₯0r\in\mathbb{Z}_{\ge0} whose parity matches the root number of Ο€Fβ€²\pi_{F'}, we prove a series of identities between: (1) The product of the central derivatives of the normalized LL-functions L(a)(Ο€,1/2)L(rβˆ’a)(Ο€βŠ—Ξ·,1/2)\mathcal{L}^{(a)}(\pi, 1/2)\mathcal{L}^{(r-a)}(\pi\otimes\eta, 1/2), where Ξ·\eta is the quadratic id\`ele class character attached to Fβ€²/FF'/F, and 0≀a≀r0\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

    Constraints on inflation revisited: An analysis including the latest local measurement of the Hubble constant

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    We revisit the constraints on inflation models by using the current cosmological observations involving the latest local measurement of the Hubble constant (H0=73.00Β±1.75H_{0} = 73.00\pm 1.75 km s βˆ’1^{-1} Mpcβˆ’1^{-1}). We constrain the primordial power spectra of both scalar and tensor perturbations with the observational data including the Planck 2015 CMB full data, the BICEP2 and Keck Array CMB B-mode data, the BAO data, and the direct measurement of H0H_0. In order to relieve the tension between the local determination of the Hubble constant and the other astrophysical observations, we consider the additional parameter NeffN_{\rm eff} in the cosmological model. We find that, for the Ξ›\LambdaCDM+rr+NeffN_{\rm eff} model, the scale invariance is only excluded at the 3.3Οƒ\sigma level, and Ξ”Neff>0\Delta N_{\rm eff}>0 is favored at the 1.6Οƒ\sigma level. Comparing the obtained 1Οƒ\sigma and 2Οƒ\sigma contours of (ns,r)(n_s,r) with the theoretical predictions of selected inflation models, we find that both the convex and concave potentials are favored at 2Οƒ\sigma level, the natural inflation model is excluded at more than 2Οƒ\sigma level, the Starobinsky R2R^2 inflation model is only favored at around 2Οƒ\sigma level, and the spontaneously broken SUSY inflation model is now the most favored model.Comment: 10 pages, 6 figure

    Preheating in Bubble Collisions

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    In a landscape with metastable minima, the bubbles will inevitably nucleate. We show that during the bubbles collide, due to the dramatically oscillating of the field at the collision region, the energy deposited in the bubble walls can be efficiently released by the explosive production of the particles. In this sense, the collision of bubbles is actually highly inelastic. The cosmological implications of this result are discussed.Comment: 5 pages, 4 figs, discussion improved, refs. added, and Fig.3 revised, to publish in PR
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