Volume 2021, Issue 773

We show that two properly embedded self-shrinkers in Euclidean space that are sufficiently separated at infinity must intersect at a finite point. The proof is based on a localized version of the Reilly formula applied to a suitable f -harmonic function with controlled gradient. In the immersed case, a new direct proof of the generalized half-space property is also presented.

Let M be a simply connected homogeneous three-manifold with isometry group of dimension 4, and let Σ be any compact surface of genus zero immersed in M whose mean, extrinsic and Gauss curvatures satisfy a smooth elliptic relation Φ ⁢ ( H , K e , K ) = 0 {\Phi(H,K_{e},K)=0} . In this paper we prove that Σ is a sphere of revolution, provided that the unique inextendible rotational surface S in M that satisfies this equation and touches its rotation axis orthogonally has bounded second fundamental form. In particular, we prove that: (i) Any elliptic Weingarten sphere immersed in ℍ 2 × ℝ {\mathbb{H}^{2}\times\mathbb{R}} is a rotational sphere. (ii) Any sphere of constant positive extrinsic curvature immersed in M is a rotational sphere. (iii) Any immersed sphere in M that satisfies an elliptic Weingarten equation H = ϕ ⁢ ( H 2 - K e ) ≥ a > 0 {H=\phi(H^{2}-K_{e})\geq a>0} with ϕ bounded, is a rotational sphere. As a very particular case of this last result, we recover the Abresch–Rosenberg classification of constant mean curvature spheres in M .

We give a formula of an asymptotic expansion of a function, provided in the form of a fiber integral around a critical value of a holomorphic map of toric type.

Elementary abelian groups are finite groups in the form of A = ( ℤ / p ⁢ ℤ ) r {A=(\mathbb{Z}/p\mathbb{Z})^{r}} for a prime number p . For every integer ℓ > 1 {\ell>1} and r > 1 {r>1} , we prove a non-trivial upper bound on the ℓ {\ell} -torsion in class groups of every A -extension. Our results are pointwise and unconditional. This establishes the first case where for some Galois group G , the ℓ {\ell} -torsion in class groups are bounded non-trivially for every G -extension and every integer ℓ > 1 {\ell>1} . When r is large enough, the unconditional pointwise bound we obtain also breaks the previously best known bound shown by Ellenberg and Venkatesh under GRH.

The determinantal variety Σ p ⁢ q {\Sigma_{pq}} is defined to be the set of all p × q {p\times q} real matrices with p ≥ q {p\geq q} whose ranks are strictly smaller than q . It is proved that Σ p ⁢ q {\Sigma_{pq}} is a minimal cone in ℝ p ⁢ q {\mathbb{R}^{pq}} and all its strata are regular minimal submanifolds.

We develop a theory of Frobenius functors for symmetric tensor categories (STC) 𝒞 {\mathcal{C}} over a field 𝒌 {\boldsymbol{k}} of characteristic p , and give its applications to classification of such categories. Namely, we define a twisted-linear symmetric monoidal functor F : 𝒞 → 𝒞 ⊠ Ver p {F:\mathcal{C}\to\mathcal{C}\boxtimes{\rm Ver}_{p}} , where Ver p {{\rm Ver}_{p}} is the Verlinde category (the semisimplification of Rep 𝐤 ( ℤ / p ) {\mathop{\mathrm{Rep}}\nolimits_{\mathbf{k}}(\mathbb{Z}/p)} ); a similar construction of the underlying additive functor appeared independently in [K. Coulembier, Tannakian categories in positive characteristic, preprint 2019]. This generalizes the usual Frobenius twist functor in modular representation theory and also the one defined in [V. Ostrik, On symmetric fusion categories in positive characteristic, Selecta Math. (N.S.) 26 2020, 3, Paper No. 36], where it is used to show that if 𝒞 {\mathcal{C}} is finite and semisimple, then it admits a fiber functor to Ver p {{\rm Ver}_{p}} . The main new feature is that when 𝒞 {\mathcal{C}} is not semisimple, F need not be left or right exact, and in fact this lack of exactness is the main obstruction to the existence of a fiber functor 𝒞 → Ver p {\mathcal{C}\to{\rm Ver}_{p}} . We show, however, that there is a 6-periodic long exact sequence which is a replacement for the exactness of F , and use it to show that for categories with finitely many simple objects F does not increase the Frobenius–Perron dimension. We also define the notion of a Frobenius exact category, which is a STC on which F is exact, and define the canonical maximal Frobenius exact subcategory 𝒞 ex {\mathcal{C}_{\rm ex}} inside any STC 𝒞 {\mathcal{C}} with finitely many simple objects. Namely, this is the subcategory of all objects whose Frobenius–Perron dimension is preserved by F . One of our main results is that a finite STC is Frobenius exact if and only if it admits a (necessarily unique) fiber functor to Ver p {{\rm Ver}_{p}} . This is the strongest currently available characteristic p version of Deligne’s theorem (stating that a STC of moderate growth in characteristic zero is the representation category of a supergroup). We also show that a sufficiently large power of F lands in 𝒞 ex {\mathcal{C}_{\rm ex}} . Also, in characteristic 2 we introduce a slightly weaker notion of an almost Frobenius exact category (namely, one having a fiber functor into the category of representations of the triangular Hopf algebra 𝒌 ⁢ [ d ] / d 2 {\boldsymbol{k}[d]/d^{2}} with d primitive and R -matrix R = 1 ⊗ 1 + d ⊗ d {R=1\otimes 1+d\otimes d} ), and show that a STC with Chevalley property is (almost) Frobenius exact. Finally, as a by-product, we resolve Question 2.15 of [P. Etingof and S. Gelaki, Exact sequences of tensor categories with respect to a module category, Adv. Math. 308 2017, 1187–1208].

This is an investigation into the possible existence and consequences of a Birch–Swinnerton-Dyer-type formula for L -functions of elliptic curves twisted by Artin representations. We translate expected properties of L -functions into purely arithmetic predictions for elliptic curves, and show that these force some peculiar properties of the Tate–Shafarevich group, which do not appear to be tractable by traditional Selmer group techniques. In particular, we exhibit settings where the different p -primary components of the Tate–Shafarevich group do not behave independently of one another. We also give examples of “arithmetically identical” settings for elliptic curves twisted by Artin representations, where the associated L -values can nonetheless differ, in contrast to the classical Birch–Swinnerton-Dyer conjecture.

Anderson generating functions have received a growing attention in function field arithmetic in the last years. Despite their introduction by Anderson in the 1980s where they were at the heart of comparison isomorphisms, further important applications, e.g., to transcendence theory have only been discovered recently. The Anderson–Thakur special function interpolates L -values via Pellarin-type identities, and its values at algebraic elements recover Gauss–Thakur sums, as shown by Anglès and Pellarin. For Drinfeld–Hayes modules, generalizations of Anderson generating functions have been introduced by Green–Papanikolas and – under the name of “special functions” – by Anglès, Ngo Dac and Tavares Ribeiro. In this article, we provide a general construction of special functions attached to any Anderson A -module. We show direct links of the space of special functions to the period lattice, and to the Betti cohomology of the A -motive. We also undertake the study of Gauss–Thakur sums for Anderson A -modules, and show that the result of Anglès–Pellarin relating values of the special functions to Gauss–Thakur sums holds in this generality.