arXiv subjects
Publications and source records attributed to Niantao Tian.
This paper counts Frobenius-periodic vector bundles on elliptic curves over an algebraically closed field of characteristic $p>0$. By translating the problem into continuous representations of the étale fundamental group, it derives explicit generating functions and exact-period formulas, with separate treatments of the ordinary and supersingular cases.
Let $k$ be a field, $K/k$ a field extension, $X$ a connected scheme proper over $k$, $x_K\in X_K(K)$ lying over $x\in X(k)$, $\mathcal{C}_X$ and $\mathcal{C}_{X_K}$ the Tannakian categories whose objects consist of vector bundles on $X$ and $X_K$ respectively, $π(\mathcal{C}_X,x)$ and $π(\mathcal{C}_{X_K},x_K)$ the corresponding Tannaka group schemes respectively. We establish a unified criterion determining when the base change homomorphism $π(\mathcal{C}_{X_K},x_K)\rightarrow π(\mathcal{C}_X,x)_K$ is faithfully flat or an isomorphism. As applications, we recover and generalize base change results for the S, Nori, EN, F, EF, ét, Eét, Loc, ELoc, and unipotent-fundamental group schemes under different types of field extensions (e.g., separable, finite Galois, and algebraically closed extensions). Moreover, our approach provides a unified explanation for both positive and negative results, including previously known counterexamples.
Let $k$ be a field, $f:X\rightarrow S$ a proper morphism between connected schemes proper over $k$, $x\in X(k)$ lying over $s\in S(k)$, $X_s$ the fibre of $f$ over $s$, $\mathcal{C}_X$, $\mathcal{C}_{S}$, $\mathcal{C}_{X_s}$ Tannakian categories over $X,S,X_s$ respectively, $π(\mathcal{C}_X,x)$, $π(\mathcal{C}_S,s)$, $π(\mathcal{C}_{X_s},x)$ the Tannaka group schemes respectively. We give a unified criterion for the exactness of the homotopy sequence of Tannakian fundamental group schemes $π(\mathcal{C}_{X_s},x)\rightarrow π(\mathcal{C}_X,x)\rightarrow π(\mathcal{C}_S,s)\rightarrow 1$. In particular, we obtain the equivalent conditions for the Künneth formula of fundamental group schemes for the product $X\times_k Y$ of two connected schemes $X$ and $Y$ proper over $k$. As an application, we obtain the Künneth formula of certain fundamental group schemes over any field, such as S, N, EN, F, EF, ét, Eét, Loc, ELoc and uni-fundamental group schemes.
Let $k$ be a field, $X$ a connected scheme proper over $k$, $D\subsetneq X$ an ample effective connected divisor, $x\in D(k)$. For Tannakian categories $\mathcal{C}_X$ and $\mathcal{C}_D$ whose objects consist of vector bundles on $X$ and $D$ respectively, we establish general Tannakian criteria for the natural homomorphism \(π(\mathcal{C}_D,x)\to π(\mathcal{C}_X,x)\) to be faithfully flat, a closed immersion, or an isomorphism. As applications, under Langer type positivity assumptions, we prove that \(π^{\ast}(D,x)\longrightarrow π^{\ast}(X,x)\) is an isomorphism for $\ast\in\{S,N,EN,F, EF,Loc,ELoc,\acute{e}t,E\acute{e}t,uni\}$ over perfect fields.