Let $D$ be an odd fundamental discriminant, with $D=1$ permitted, and let $r\geq 1$ be fixed. We prove that, for every sufficiently large integer $\ell$ satisfying $(-1)^\ell D>0$, the first $r$ traced diagonal Rankin--Cohen brackets \[ \mathrm{Tr}_1^{|D|}[G_{\ell-2e,D},G_{\ell-2e,D}]_{2e}, \qquad 1\leq e\leq r, \] are linearly independent in $S_{2\ell}(SL_2(\mathbb Z))$. Here $G_{k,D}$ is the Eisenstein series of weight $k$, level $|D|$, and nebentypus $\chi_D$. The Petersson formula of Kayath, Lane, Neifeld, Ni, and Xue then implies that at least $r$ normalized Hecke eigenforms $f\in S_{2\ell}(SL_2(\mathbb Z))$ satisfy $L(f\otimes\chi_D,\ell)\neq 0$. For $D=1$, this gives, for every fixed $r$ and every sufficiently large $K\equiv 0\pmod 4$, at least $r$ level-one eigenforms of weight $K$ with nonzero central value.
For every negative fundamental discriminant $-f$ and every $k\geq1$, we construct a rational function $R_{f,2k}\in\mathbb{Q}(x_1,\ldots,x_{2k})$ and a constant $r_{f,2k}\in\mathbb{Q}^\times$ such that \[ m(R_{f,2k})=r_{f,2k}L'(\chi_{-f},1-2k), \] where $m$ denotes the logarithmic Mahler measure and $\chi_{-f}$ is the q...
Let $p$ and $\ell$ be distinct odd primes. For a finite extension $F/\mathbb{Q}_p$, the local Langlands correspondence states that there is a canonical bijection between irreducible, smooth representations of $\text{GL}_n(F)$ and $n$-dimensional, $\Phi$-semisimple Weil--Deligne representations of the Weil group $W_F$....
Let $\varphi_{d}(|{\,\cdot\,}|)$ be the radial extremal function in the problem for the sharp Nikolskii constant $\mathcal C_d^{-1}=\inf \|f\|_{1}$ over functions $f\in\mathit{PW}\,_{1}^{1}(\mathbb R^{d})$, $f(0)=1$. For every odd dimension, we construct an entire function $\Phi$ of exponential type $1/2$ such that $\v...
Let $K$ be an algebraically closed field of odd characteristic $p$, let $G= {\rm SL}_2(K)$, and let $V$ be the natural representation of $G$. Let $b_k$ denote the number of $G$-indecomposable factors of $V^{\otimes k}$, counted with multiplicity, and let $\delta_p=1-\log_{p^2}\!\bigl(\tfrac{p+1}{2}\bigr)$. Then there e...
Let $k$ be a finite field of characteristic $p\neq 3$, let $E/k$ be the Fermat cubic curve, and let $\ell\neq p$ be a prime. If $p\equiv1\pmod3$, assume moreover that $\ell>3$. Then $H^3_{\mathrm{nr}}(k(E^3)/k,\mathbb{Q}_\ell/\mathbb{Z}_\ell(2))=0$ and the cycle map \[CH^2(E^3)_{\mathbb{Z}_\ell}\longrightarrow H^4(E^3,...
Let $X=\mathbb{P}^3$ and let \[ \alpha_n=(0,1,-\tfrac12,\tfrac16-n)\in H^{\mathrm{even}}(X,\mathbb{Q}) \] with respect to the basis $1,H,H^2,H^3$, where $H=c_1(\mathcal{O}_X(1))$. We prove that every Gieseker semistable sheaf on $X$ with Chern character $\alpha_n$ is stable and uniquely of the form $\iota_{P*}\mathcal{...
R. Anderson· Journal of Geometry and Phys...· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
