We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
In this work, we develop an integral representation for the partial L-function of a pair 
$\pi \times \tau $ of genuine irreducible cuspidal automorphic representations, 
$\pi $ of the m-fold covering of Matsumoto of the symplectic group 
$\operatorname {\mathrm {Sp}}_{2n}$ and 
$\tau $ of a certain covering group of 
$\operatorname {\mathrm {GL}}_k$, with arbitrary m, n and k. Our construction is based on the recent extension by Cai, Friedberg, Ginzburg and the author, of the classical doubling method of Piatetski-Shapiro and Rallis, from rank-
$1$ twists to arbitrary rank twists. We prove a basic global identity for the integral and compute the local integrals with unramified data. Our global results are subject to certain conjectures, but when 
$k=1$ they are unconditional for all m. One possible future application of this work will be a Shimura-type lift of representations from covering groups to general linear groups. In a recent work, we used the present results in order to provide an analytic definition of local factors for representations of the m-fold covering of 
$\operatorname {\mathrm {Sp}}_{2n}$.
Let 
$\mathcal {O}$ be a maximal order in the quaternion algebra over 
$\mathbb Q$ ramified at p and 
$\infty $. We prove two theorems that allow us to recover the structure of 
$\mathcal {O}$ from limited information. The first says that for any infinite set S of integers coprime to p, 
$\mathcal {O}$ is spanned as a 
${\mathbb {Z}}$-module by elements with norm in S. The second says that 
$\mathcal {O}$ is determined up to isomorphism by its theta function.
We study the low-temperature 
$(2+1)$D solid-on-solid model on 
with zero boundary conditions and nonnegative heights (a floor at height 
$0$). Caputo et al. (2016) established that this random surface typically admits either 
$\mathfrak h $ or 
$\mathfrak h+1$ many nested macroscopic level line loops 
$\{\mathcal L_i\}_{i\geq 0}$ for an explicit 
$\mathfrak h\asymp \log L$, and its top loop 
$\mathcal L_0$ has cube-root fluctuations: For example, if 
$\rho (x)$ is the vertical displacement of 
$\mathcal L_0$ from the bottom boundary point 
$(x,0)$, then 
$\max \rho (x) = L^{1/3+o(1)}$ over 
. It is believed that rescaling 
$\rho $ by 
$L^{1/3}$ and 
$I_0$ by 
$L^{2/3}$ would yield a limit law of a diffusion on 
$[-1,1]$. However, no nontrivial lower bound was known on 
$\rho (x)$ for a fixed 
$x\in I_0$ (e.g., 
$x=\frac L2$), let alone on 
$\min \rho (x)$ in 
$I_0$, to complement the bound on 
$\max \rho (x)$. Here, we show a lower bound of the predicted order 
$L^{1/3}$: For every 
$\epsilon>0$, there exists 
$\delta>0$ such that 
$\min _{x\in I_0} \rho (x) \geq \delta L^{1/3}$ with probability at least 
$1-\epsilon $. The proof relies on the Ornstein–Zernike machinery due to Campanino–Ioffe–Velenik and a result of Ioffe, Shlosman and Toninelli (2015) that rules out pinning in Ising polymers with modified interactions along the boundary. En route, we refine the latter result into a Brownian excursion limit law, which may be of independent interest. We further show that in a 
$ K L^{2/3}\times K L^{2/3}$ box with boundary conditions 
$\mathfrak h-1,\mathfrak h,\mathfrak h,\mathfrak h$ (i.e., 
$\mathfrak h-1$ on the bottom side and 
$\mathfrak h$ elsewhere), the limit of 
$\rho (x)$ as 
$K,L\to \infty $ is a Ferrari–Spohn diffusion.
$K3^{[n]}$ type are algebraic
Let 
$X$ and 
$Y$ be compact hyper-Kähler manifolds deformation equivalent to the Hilbert scheme of length 
$n$ subschemes of a 
$K3$ surface. A class in 
$H^{p,p}(X\times Y,{\mathbb {Q}})$ is an analytic correspondence, if it belongs to the subring generated by Chern classes of coherent analytic sheaves. Let 
$f:H^2(X,{\mathbb {Q}})\rightarrow H^2(Y,{\mathbb {Q}})$ be a rational Hodge isometry with respect to the Beauville–Bogomolov–Fujiki pairings. We prove that 
$f$ is induced by an analytic correspondence. We furthermore lift 
$f$ to an analytic correspondence 
$\tilde {f}: H^*(X,{\mathbb {Q}})[2n]\rightarrow H^*(Y,{\mathbb {Q}})[2n]$, which is a Hodge isometry with respect to the Mukai pairings and which preserves the gradings up to sign. When 
$X$ and 
$Y$ are projective, the correspondences 
$f$ and 
$\tilde {f}$ are algebraic.
