For a smooth projective variety
Fix a positive integer
 NSFPAR ID:
 10371268
 Publisher / Repository:
 Springer Science + Business Media
 Date Published:
 Journal Name:
 Mathematische Annalen
 Volume:
 387
 Issue:
 12
 ISSN:
 00255831
 Page Range / eLocation ID:
 p. 615687
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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Abstract X over an algebraic number fieldk a conjecture of Bloch and Beilinson predicts that the kernel of the Albanese map ofX is a torsion group. In this article we consider a product of smooth projective curves and show that if the conjecture is true for any subproduct of two curves, then it is true for$$X=C_1\times \cdots \times C_d$$ $X={C}_{1}\times \cdots \times {C}_{d}$X . For a product of two curves over$$X=C_1\times C_2$$ $X={C}_{1}\times {C}_{2}$ with positive genus we construct many nontrivial examples that satisfy the weaker property that the image of the natural map$$\mathbb {Q} $$ $Q$ is finite, where$$J_1(\mathbb {Q})\otimes J_2(\mathbb {Q})\xrightarrow {\varepsilon }{{\,\textrm{CH}\,}}_0(C_1\times C_2)$$ ${J}_{1}\left(Q\right)\otimes {J}_{2}\left(Q\right)\stackrel{\epsilon}{\to}{\phantom{\rule{0ex}{0ex}}\text{CH}\phantom{\rule{0ex}{0ex}}}_{0}({C}_{1}\times {C}_{2})$ is the Jacobian variety of$$J_i$$ ${J}_{i}$ . Our constructions include many new examples of nonisogenous pairs of elliptic curves$$C_i$$ ${C}_{i}$ with positive rank, including the first known examples of rank greater than 1. Combining these constructions with our previous result, we obtain infinitely many nontrivial products$$E_1, E_2$$ ${E}_{1},{E}_{2}$ for which the analogous map$$X=C_1\times \cdots \times C_d$$ $X={C}_{1}\times \cdots \times {C}_{d}$ has finite image.$$\varepsilon $$ $\epsilon $ 
Abstract We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loopensemble
for$$\hbox {CLE}_{\kappa '}$$ ${\text{CLE}}_{{\kappa}^{\prime}}$ in (4, 8) that is drawn on an independent$$\kappa '$$ ${\kappa}^{\prime}$ LQG surface for$$\gamma $$ $\gamma $ . The results are similar in flavor to the ones from our companion paper dealing with$$\gamma ^2=16/\kappa '$$ ${\gamma}^{2}=16/{\kappa}^{\prime}$ for$$\hbox {CLE}_{\kappa }$$ ${\text{CLE}}_{\kappa}$ in (8/3, 4), where the loops of the CLE are disjoint and simple. In particular, we encode the combined structure of the LQG surface and the$$\kappa $$ $\kappa $ in terms of stable growthfragmentation trees or their variants, which also appear in the asymptotic study of peeling processes on decorated planar maps. This has consequences for questions that do a priori not involve LQG surfaces: In our paper entitled “$$\hbox {CLE}_{\kappa '}$$ ${\text{CLE}}_{{\kappa}^{\prime}}$CLE Percolations ” described the law of interfaces obtained when coloring the loops of a independently into two colors with respective probabilities$$\hbox {CLE}_{\kappa '}$$ ${\text{CLE}}_{{\kappa}^{\prime}}$p and . This description was complete up to one missing parameter$$1p$$ $1p$ . The results of the present paper about CLE on LQG allow us to determine its value in terms of$$\rho $$ $\rho $p and . It shows in particular that$$\kappa '$$ ${\kappa}^{\prime}$ and$$\hbox {CLE}_{\kappa '}$$ ${\text{CLE}}_{{\kappa}^{\prime}}$ are related via a continuum analog of the EdwardsSokal coupling between$$\hbox {CLE}_{16/\kappa '}$$ ${\text{CLE}}_{16/{\kappa}^{\prime}}$ percolation and the$$\hbox {FK}_q$$ ${\text{FK}}_{q}$q state Potts model (which makes sense even for nonintegerq between 1 and 4) if and only if . This provides further evidence for the longstanding belief that$$q=4\cos ^2(4\pi / \kappa ')$$ $q=4{cos}^{2}(4\pi /{\kappa}^{\prime})$ and$$\hbox {CLE}_{\kappa '}$$ ${\text{CLE}}_{{\kappa}^{\prime}}$ represent the scaling limits of$$\hbox {CLE}_{16/\kappa '}$$ ${\text{CLE}}_{16/{\kappa}^{\prime}}$ percolation and the$$\hbox {FK}_q$$ ${\text{FK}}_{q}$q Potts model whenq and are related in this way. Another consequence of the formula for$$\kappa '$$ ${\kappa}^{\prime}$ is the value of halfplane arm exponents for such divideandcolor models (a.k.a. fuzzy Potts models) that turn out to take a somewhat different form than the usual critical exponents for twodimensional models.$$\rho (p,\kappa ')$$ $\rho (p,{\kappa}^{\prime})$ 
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) with a mass of 125$$\text {H}$$ $\text{H}$ to a pair of light pseudoscalars$$\,\text {Ge}\hspace{.08em}\text {V}$$ $\phantom{\rule{0ex}{0ex}}\text{Ge}\phantom{\rule{0ex}{0ex}}\text{V}$ is performed in final states where one pseudoscalar decays to two$$\text {a}_{1} $$ ${\text{a}}_{1}$ quarks and the other to a pair of muons or$${\textrm{b}}$$ $\text{b}$ leptons. A data sample of proton–proton collisions at$$\tau $$ $\tau $ corresponding to an integrated luminosity of 138$$\sqrt{s}=13\,\text {Te}\hspace{.08em}\text {V} $$ $\sqrt{s}=13\phantom{\rule{0ex}{0ex}}\text{Te}\phantom{\rule{0ex}{0ex}}\text{V}$ recorded with the CMS detector is analyzed. No statistically significant excess is observed over the standard model backgrounds. Upper limits are set at 95% confidence level ($$\,\text {fb}^{1}$$ $\phantom{\rule{0ex}{0ex}}{\text{fb}}^{1}$ ) on the Higgs boson branching fraction to$$\text {CL}$$ $\text{CL}$ and to$$\upmu \upmu \text{ b } \text{ b } $$ $\mu \mu \phantom{\rule{0ex}{0ex}}\text{b}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\text{b}\phantom{\rule{0ex}{0ex}}$ via a pair of$$\uptau \uptau \text{ b } \text{ b },$$ $\tau \tau \phantom{\rule{0ex}{0ex}}\text{b}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\text{b}\phantom{\rule{0ex}{0ex}},$ s. The limits depend on the pseudoscalar mass$$\text {a}_{1} $$ ${\text{a}}_{1}$ and are observed to be in the range (0.17–3.3)$$m_{\text {a}_{1}}$$ ${m}_{{\text{a}}_{1}}$ and (1.7–7.7)$$\times 10^{4}$$ $\times {10}^{4}$ in the$$\times 10^{2}$$ $\times {10}^{2}$ and$$\upmu \upmu \text{ b } \text{ b } $$ $\mu \mu \phantom{\rule{0ex}{0ex}}\text{b}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\text{b}\phantom{\rule{0ex}{0ex}}$ final states, respectively. In the framework of models with two Higgs doublets and a complex scalar singlet (2HDM+S), the results of the two final states are combined to determine upper limits on the branching fraction$$\uptau \uptau \text{ b } \text{ b } $$ $\tau \tau \phantom{\rule{0ex}{0ex}}\text{b}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\text{b}\phantom{\rule{0ex}{0ex}}$ at 95%$${\mathcal {B}}(\text {H} \rightarrow \text {a}_{1} \text {a}_{1} \rightarrow \ell \ell \text{ b } \text{ b})$$ $B(\text{H}\to {\text{a}}_{1}{\text{a}}_{1}\to \ell \ell \phantom{\rule{0ex}{0ex}}\text{b}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\text{b})$ , with$$\text {CL}$$ $\text{CL}$ being a muon or a$$\ell $$ $\ell $ lepton. For different types of 2HDM+S, upper bounds on the branching fraction$$\uptau $$ $\tau $ are extracted from the combination of the two channels. In most of the Type II 2HDM+S parameter space,$${\mathcal {B}}(\text {H} \rightarrow \text {a}_{1} \text {a}_{1} )$$ $B(\text{H}\to {\text{a}}_{1}{\text{a}}_{1})$ values above 0.23 are excluded at 95%$${\mathcal {B}}(\text {H} \rightarrow \text {a}_{1} \text {a}_{1} )$$ $B(\text{H}\to {\text{a}}_{1}{\text{a}}_{1})$ for$$\text {CL}$$ $\text{CL}$ values between 15 and 60$$m_{\text {a}_{1}}$$ ${m}_{{\text{a}}_{1}}$ .$$\,\text {Ge}\hspace{.08em}\text {V}$$ $\phantom{\rule{0ex}{0ex}}\text{Ge}\phantom{\rule{0ex}{0ex}}\text{V}$