We consider the linear system $|2{\Theta }_0|$ of second order theta functions over the Jacobian $JC$ of a non-hyperelliptic curve $C$. A result by J.Fay says that a divisor $D\in |2{\Theta }_0|$ contains the origin $𝒪\in JC$ with multiplicity $4$ if and only if $D$ contains the surface $C-C=\left\{𝒪\right(p-q)\mid p,q\in C\}\subset JC$. In this paper we generalize Fay’s result and some previous work by R.C.Gunning. More precisely, we describe the relationship between divisors containing $𝒪$ with multiplicity $6$, divisors containing the fourfold $C_2-C_2=\{𝒪(p+q-r-s)\mid p,q,r,s\in C\}$, and divisors singular along $C-C$, using...

Let $(M,\theta )$ be a compact CR manifold of dimension $2n+1$ with a contact form $\theta$, and $L=(2+2/n){\Delta }_b+R$ its associated CR conformal laplacien. The CR Yamabe conjecture states that there is a contact form $\tilde{\theta }$ on $M$ conformal to $\theta$ which has a constant Webster curvature. This problem is equivalent to the existence of a function $u$ such that $Lu=u^{1+2/n}$, $u>0$ on $M$. D. Jerison and J. M. Lee solved the CR Yamabe problem in the case where $n\ge 2$ and $(M,\theta )$ is not locally CR equivalent to the sphere $S^{2n+1}$ of ${𝐂}^n$. In a join work with R. Yacoub, the CR Yamabe...

The notion of $\beta$-normality was introduced and studied by Arhangel’skii, Ludwig in 2001. Recently, almost $\beta$-normal spaces, which is a simultaneous generalization of $\beta$-normal and almost normal spaces, were introduced by Das, Bhat and Tartir. We introduce a new generalization of normality, namely weak $\beta$-normality, in terms of $\theta$-closed sets, which turns out to be a simultaneous generalization of $\beta$-normality and $\theta$-normality. A space $X$ is said to be weakly $\beta$-normal (w$\beta$-normal$)$ if for every...

We find necessary and sufficient conditions under which the norms of the interpolation spaces ${(N₀,N₁)}_{\theta ,q}$ and ${(X₀,X₁)}_{\theta ,q}$ are equivalent on N, where N is the kernel of a nonzero functional ψ ∈ (X₀ ∩ X₁)* and $N_i$ is the normed space N with the norm inherited from $X_i$ (i = 0,1). Our proof is based on reducing the problem to its partial case studied by Ivanov and Kalton, where ψ is bounded on one of the endpoint spaces. As an application we completely resolve the problem of when the range of the operator $T_{\theta }=S-2^{\theta }I$ (S...

We prove that the global geometric theta-lifting functor for the dual pair $(H,G)$ is compatible with the Whittaker functors, where $(H,G)$ is one of the pairs $({\mathrm{S}𝕆}_{2n},{𝕊p}_{2n})$, $({𝕊p}_{2n},{\mathrm{S}𝕆}_{2n+2})$ or $({𝔾\mathrm{L}}_n,{𝔾\mathrm{L}}_{n+1})$. That is, the composition of the theta-lifting functor from $H$ to $G$ with the Whittaker functor for $G$ is isomorphic to the Whittaker functor for $H$.

Let $X$ be a smooth projective curve over an algebraically closed field of characteristic $\>2$. Consider the dual pair $H={\mathrm{SO}}_{2m},G={\mathrm{Sp}}_{2n}$ over $X$ with $H$ split. Write ${\mathrm{Bun}}_G$ and ${\mathrm{Bun}}_H$ for the stacks of $G$-torsors and $H$-torsors on $X$. The theta-kernel ${\mathrm{Aut}}_{G,H}$ on ${\mathrm{Bun}}_G\times {\mathrm{Bun}}_H$ yields theta-lifting functors $F_G:\mathrm{D}\left({\mathrm{Bun}}_H\right)\to \mathrm{D}\left({\mathrm{Bun}}_G\right)$ and $F_H:\mathrm{D}\left({\mathrm{Bun}}_G\right)\to \mathrm{D}\left({\mathrm{Bun}}_H\right)$ between the corresponding derived categories. We describe the relation of these functors with Hecke operators. In two particular cases these functors realize the geometric Langlands functoriality for the above pair (in the non...

The extremal functions  $f_0\left(z\right)$  realizing the maxima of some functionals (e.g. $max|a_3|$, and  $maxarg{f}^{{}^{\text{'}}}\left(z\right)$) within the so-called universal linearly invariant family $U_{\alpha }$ (in the sense of Pommerenke [10]) have such a form that $f_0^{{}^{\text{'}}}\left(z\right)$  looks similar to generating function for Meixner-Pollaczek (MP) polynomials [2], [8]. This fact gives motivation for the definition and study of the generalized Meixner-Pollaczek (GMP) polynomials $P_n^{\lambda }(x;\theta ,\psi )$ of a real variable $x$ as coefficients of $$G^{\lambda }(x;\theta ,\psi ;z)=\frac{1}{{(1-z{e}^{i\theta })}^{\lambda -ix}{(1-z{e}^{i\psi })}^{\lambda +ix}}=\sum _{n=0}^{\infty }{P}_n^{\lambda }(x;\theta ,\psi )z^n,\left|z\right|<1,$$ where the parameters $\lambda$, $\theta$, $\psi$ satisfy the conditions:...

Let $G$ be a finite group and $p$ a prime number. We prove that if $G$ is a finite group of order $\left|\mathrm{PSL}\right(2,p^2\left)\right|$ such that $G$ has an irreducible character of degree $p^2$ and we know that $G$ has no irreducible character $\theta$ such that $2p\mid \theta \left(1\right)$, then $G$ is isomorphic to $\mathrm{PSL}(2,p^2)$. As a consequence of our result we prove that $\mathrm{PSL}(2,p^2)$ is uniquely determined by the structure of its complex group algebra.

Let $t$ be any integer and fix an odd prime $\ell$. Let $\Phi \left(x\right)=T_{\ell }^n\left(x\right)-t$ denote the $n$-fold composition of the Chebyshev polynomial of degree $\ell$ shifted by $t$. If this polynomial is irreducible, let $K=\mathbb{Q}\left(\theta \right)$, where $\theta$ is a root of $\Phi$. We use a theorem of Dedekind in conjunction with previous results of the author to give conditions on $t$ that ensure $K$ is monogenic. For other values of $t$, we apply a result of Guàrdia, Montes, and Nart to obtain a formula for the discriminant of $K$ and compute an integral basis for the ring...

Let $C$ be a hyperelliptic curve of genus $g\ge 1$ over a number field $K$ with good reduction outside a finite set of places $S$ of $K$. We prove that $C$ has a Weierstrass model over the ring of integers of $K$ with height effectively bounded only in terms of $g$, $S$ and $K$. In particular, we obtain that for any given number field $K$, finite set of places $S$ of $K$ and integer $g\ge 1$ one can in principle determine the set of $K$-isomorphism classes of hyperelliptic curves over $K$ of genus $g$ with good reduction outside...

We show that any ${\Sigma }_s$-product of at most $𝔠$-many $L\Sigma (\le \omega )$-spaces has the $L\Sigma (\le \omega )$-property. This result generalizes some known results about $L\Sigma (\le \omega )$-spaces. On the other hand, we prove that every ${\Sigma }_s$-product of monotonically monolithic spaces is monotonically monolithic, and in a similar form, we show that every ${\Sigma }_s$-product of Collins-Roscoe spaces has the Collins-Roscoe property. These results generalize some known results about the Collins-Roscoe spaces and answer some questions due to Tkachuk [Lifting the Collins-Roscoe...

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...