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1. Restricted spaces of holomorphic sections vanishing along subvarieties

   https://doi.org/10.1007/s00209-025-03706-w  

   Coman, Dan; Marinescu, George; Nguyên, Viêt-Anh (May 2025, Mathematische Zeitschrift)

   Abstract LetXbe a compact normal complex space of dimensionnandLbe a holomorphic line bundle onX. Suppose that$$\Sigma =(\Sigma _1,\ldots ,\Sigma _\ell )$$ $\Sigma =({\Sigma }_1,\dots ,{\Sigma }_{\ell })$ is an$$\ell $$ $\ell$ -tuple of distinct irreducible proper analytic subsets ofX,$$\tau =(\tau _1,\ldots ,\tau _\ell )$$ $\tau =({\tau }_1,\dots ,{\tau }_{\ell })$ is an$$\ell $$ $\ell$ -tuple of positive real numbers, and let$$H^0_0(X,L^p)$$ $H_0^0(X,L^p)$ be the space of holomorphic sections of$$L^p:=L^{\otimes p}$$ $L^p:=L^{\otimes p}$ that vanish to order at least$$\tau _jp$$ ${\tau }_{j}p$ along$$\Sigma _j$$ ${\Sigma }_j$ ,$$1\le j\le \ell $$ $1\le j\le \ell$ . If$$Y\subset X$$ $Y\subset X$ is an irreducible analytic subset of dimensionm, we consider the space$$H^0_0 (X|Y, L^p)$$ $H_0^0\left(X\right|Y,L^p)$ of holomorphic sections of$$L^p|_Y$$ $L^p{|}_Y$ that extend to global holomorphic sections in$$H^0_0(X,L^p)$$ $H_0^0(X,L^p)$ . Assuming that the triplet$$(L,\Sigma ,\tau )$$ $(L,\Sigma ,\tau )$ is big in the sense that$$\dim H^0_0(X,L^p)\sim p^n$$ $dim{H}_0^0(X,L^p)\sim p^n$ , we give a general condition onYto ensure that$$\dim H^0_0(X|Y,L^p)\sim p^m$$ $dim{H}_0^0\left(X\right|Y,L^p)\sim p^m$ . WhenLis endowed with a continuous Hermitian metric, we show that the Fubini-Study currents of the spaces$$H^0_0(X|Y,L^p)$$ $H_0^0\left(X\right|Y,L^p)$ converge to a certain equilibrium current onY. We apply this to the study of the equidistribution of zeros inYof random holomorphic sections in$$H^0_0(X|Y,L^p)$$ $H_0^0\left(X\right|Y,L^p)$ as$$p\rightarrow \infty $$ $p\to \infty$ . 

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2. Spin density matrix elements in exclusive $$\rho ^0$$ meson muoproduction

   https://doi.org/10.1140/epjc/s10052-023-11359-4  

   Alexeev, G D; Alexeev, M G; Alice, C; Amoroso, A; Andrieux, V; Anosov, V; Augsten, K; Augustyniak, W; Azevedo, C_D R; Badelek, B; et al (October 2023, The European Physical Journal C)

   Abstract We report on a measurement of Spin Density Matrix Elements (SDMEs) in hard exclusive$$\rho ^0$$ ${\rho }^0$ meson muoproduction at COMPASS using 160 GeV/cpolarised$$ \mu ^{+}$$ ${\mu }^{+}$ and$$ \mu ^{-}$$ ${\mu }^{-}$ beams impinging on a liquid hydrogen target. The measurement covers the kinematic range 5.0 GeV/$$c^2$$ $c^2$ $$< W<$$ $<W<$ 17.0 GeV/$$c^2$$ $c^2$ , 1.0 (GeV/c)$$^2$$ ${}^2$ $$< Q^2<$$ $<Q^2<$ 10.0 (GeV/c)$$^2$$ ${}^2$ and 0.01 (GeV/c)$$^2$$ ${}^2$ $$< p_{\textrm{T}}^2<$$ $<p_{\text{T}}^2<$ 0.5 (GeV/c)$$^2$$ ${}^2$ . Here,Wdenotes the mass of the final hadronic system,$$Q^2$$ $Q^2$ the virtuality of the exchanged photon, and$$p_{\textrm{T}}$$ $p_{\text{T}}$ the transverse momentum of the$$\rho ^0$$ ${\rho }^0$ meson with respect to the virtual-photon direction. The measured non-zero SDMEs for the transitions of transversely polarised virtual photons to longitudinally polarised vector mesons ($$\gamma ^*_T \rightarrow V^{ }_L$$ ${\gamma }_T^{\ast }\to V_L^{}$ ) indicate a violation ofs-channel helicity conservation. Additionally, we observe a dominant contribution of natural-parity-exchange transitions and a very small contribution of unnatural-parity-exchange transitions, which is compatible with zero within experimental uncertainties. The results provide important input for modelling Generalised Parton Distributions (GPDs). In particular, they may allow one to evaluate in a model-dependent way the role of parton helicity-flip GPDs in exclusive$$\rho ^0$$ ${\rho }^0$ production. 

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3. Riesz Transforms, Function Spaces, and Their Applications on Infinite Dimensional Compact Groups

   https://doi.org/10.1007/s11118-026-10288-7  

   Bendikov, Alexander; Chen, Li; Saloff-Coste, Laurent (April 2026, Potential Analysis)

   Abstract On a compact connected groupG, consider the infinitesimal generator$$-L$$ $-L$ of a central symmetric Gaussian convolution semigroup$$(\mu _t)_{t>0}$$ ${\left({\mu }_t\right)}_{t>0}$ . We establish several regularity results of the solution to the Poisson equation$$LU=F$$ $LU=F$ , both in strong and weak senses. To this end, we introduce two classes of Lipschitz spaces for$$1\le p\le \infty $$ $1\le p\le \infty$ :$$\Lambda _{\theta }^p$$ ${\Lambda }_{\theta }^p$ , defined via the associated Markov semigroup, and$$\mathrm L_{\theta }^p$$ $L_{\theta }^p$ , defined via the intrinsic distance. In the strong sense, we prove a priori Sobolev regularity and Lipschitz regularity in the class of$$\Lambda _{\theta }^p$$ ${\Lambda }_{\theta }^p$ space. In the distributional sense, we further show local regularity in the class of$$\mathrm L_{\theta }^{\infty }$$ $L_{\theta }^{\infty }$ space. These results require some strong assumptions on$$-L$$ $-L$ . Our main techniques build on the differentiability of the associated semigroup, explicit dimension-free$$L^p$$ $L^p$ ($$1 $1<p<\infty$ ) boundedness of first and second order Riesz transforms, and a comparison between the two Lipschitz norms. 

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4. Relating elliptic curve point-counting and solutions of quadratic forms with congruence conditions

   https://doi.org/10.1007/s11139-026-01341-5  

   Mondal, Koustav (March 2026, The Ramanujan Journal)

   Abstract In this paper, we analyze the theta series associated to the quadratic form$$Q(\vec {x}) :=x_1^2+x_2^2+x_3^2+x_4^2$$ $Q\left(\overrightarrow{x}\right):=x_1^2+x_2^2+x_3^2+x_4^2$ with congruence conditions on$$x_i$$ $x_i$ modulo 2, 3, 4 and 6. By employing special operators on modular, non-holomorphic Eisenstein series of weight 2, we construct a basis for Eisenstein space for levels$$2^k, k\le 7$$ $2^k,k\le 7$ ,$$3^{\ell }, \ell \le 3$$ $3^{\ell },\ell \le 3$ andp, for odd primep. Using the relation between the trace of Frobenius on an elliptic curve and the Fourier coefficients of the cusp form part of theta series corresponding toQ, we establish relation between the number of integer solutions to the equation$$Q(\vec {x}) = p$$ $Q\left(\overrightarrow{x}\right)=p$ and the number of$$\mathbb {F}_p$$ $F_p$ -rational points on the associated elliptic curve under certain congruence conditions onp. 

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5. More accurate $$ \sigma \left(\mathcal{GG}\to h\right),\Gamma \left(h\to \mathcal{GG},\mathcal{AA},\overline{\Psi}\Psi \right) $$ and Higgs width results via the geoSMEFT

   https://doi.org/10.1007/JHEP01(2024)170  

   Martin, Adam; Trott, Michael (January 2024, Journal of High Energy Physics)

   A<sc>bstract</sc> We develop Standard Model Effective Field Theory (SMEFT) predictions ofσ($$ \mathcal{GG} $$ $\mathrm{GG}$ →h), Γ(h→$$ \mathcal{GG} $$ $\mathrm{GG}$ ), Γ(h→$$ \mathcal{AA} $$ $\mathrm{AA}$ ) to incorporate full two loop Standard Model results at the amplitude level, in conjunction with dimension eight SMEFT corrections. We simultaneously report consistent Γ(h→$$ \overline{\Psi}\Psi $$ $\overline{\Psi }\Psi$ ) results including leading QCD corrections and dimension eight SMEFT corrections. This extends the predictions of the former processes Γ, σto a full set of corrections at$$ \mathcal{O}\left({\overline{v}}_T^2/{\varLambda}^2{\left(16{\pi}^2\right)}^2\right) $$ $O\left({\overline{v}}_T^2/{\Lambda }^2{\left(16{\pi }^2\right)}^2\right)$ and$$ \mathcal{O}\left({\overline{v}}_T^4/{\Lambda}^4\right) $$ $O\left({\overline{v}}_T^4/{\Lambda }^4\right)$ , where$$ {\overline{v}}_T $$ ${\overline{v}}_T$ is the electroweak scale vacuum expectation value and Λ is the cut off scale of the SMEFT. Throughout, cross consistency between the operator and loop expansions is maintained by the use of the geometric SMEFT formalism. For Γ(h→$$ \overline{\Psi}\Psi $$ $\overline{\Psi }\Psi$ ), we include results at$$ \mathcal{O}\left({\overline{v}}_T^2/{\Lambda}^2\left(16{\pi}^2\right)\right) $$ $O\left({\overline{v}}_T^2/{\Lambda }^2\left(16{\pi }^2\right)\right)$ in the limit where subleadingm_Ψ→ 0 corrections are neglected. We clarify how gauge invariant SMEFT renormalization counterterms combine with the Standard Model counter terms in higher order SMEFT calculations when the Background Field Method is used. We also update the prediction of the total Higgs width in the SMEFT to consistently include some of these higher order perturbative effects. 

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