42,635 research outputs found

### Shtukas and the Taylor expansion of $L$-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

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

We revisit the constraints on inflation models by using the current
cosmological observations involving the latest local measurement of the Hubble
constant ($H_{0} = 73.00\pm 1.75$ km s $^{-1}$ Mpc$^{-1}$). 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 $H_0$. In
order to relieve the tension between the local determination of the Hubble
constant and the other astrophysical observations, we consider the additional
parameter $N_{\rm eff}$ in the cosmological model. We find that, for the
$\Lambda$CDM+$r$+$N_{\rm eff}$ model, the scale invariance is only excluded at
the 3.3$\sigma$ level, and $\Delta N_{\rm eff}>0$ is favored at the 1.6$\sigma$
level. Comparing the obtained 1$\sigma$ and 2$\sigma$ contours of $(n_s,r)$
with the theoretical predictions of selected inflation models, we find that
both the convex and concave potentials are favored at 2$\sigma$ level, the
natural inflation model is excluded at more than 2$\sigma$ level, the
Starobinsky $R^2$ inflation model is only favored at around 2$\sigma$ level,
and the spontaneously broken SUSY inflation model is now the most favored
model.Comment: 10 pages, 6 figure

### Preheating in Bubble Collisions

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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