Let $(R,\mathfrak{𝔪})$ be a Noetherian local ring of prime characteristic $p$. We define the F-rational signature of $R$, denoted by $\text{r}(R)$, as the infimum, taken over pairs of ideals $I⊊J$ such that $I$ is generated by a system of parameters and $J$ is a strictly larger ideal, of the drops ${\text{e}}_{\text{HK}}(I,R))-{\text{e}}_{\text{HK}}(J,R)$ in the Hilbert–Kunz multiplicity. If $R$ is excellent, then $R$ is F-rational if and only if $\text{r}(R)>0$. The proof of this fact depends on the following result in the sequel: Given an $\mathfrak{𝔪}$-primary ideal $I$ in $R$, there exists a positive ${\delta }_I\in {ℝ}^{+}$ such that, for any ideal $J⊋I$, ${\text{e}}_{\text{HK}}(I,R)-{\text{e}}_{\text{HK}}(J,R)$ is either $0$ or at least ${\delta }_I$. We study how the F-rational signature behaves under deformation, flat local ring extension, and localization.

We use birational geometry to show that the existence of rational points on proper rationally connected varieties over fields of characteristic $0$ is a consequence of the existence of rational points on terminal Fano varieties. We discuss several consequences of this result, especially in relation to the $C_1$-conjecture. We also provide evidence that supports the conjecture in dimension 3 for $C_1$ fields of characteristic $0$.

This work addresses the prime geodesic theorem for the Picard manifold $\mathcal{ℳ}={\text{PSL}}_2(\mathbb{Z}[i])\setminus {\mathfrak{𝔥}}^3$, which asks for the asymptotic evaluation of a counting function for the closed geodesics on $\mathcal{ℳ}$. Let $E_{\mathrm{\Gamma }}(X)$ be the error term in the prime geodesic theorem. We establish that $E_{\mathrm{\Gamma }}(X)=O_{\varepsilon }(X^{3∕2+\varepsilon })$ on average as well as many pointwise bounds. The second moment bound parallels an analogous result for $\mathrm{\Gamma }={\text{PSL}}_2(\mathbb{Z})$ due to Balog et al. and our innovation features the delicate analysis of sums of Kloosterman sums with an explicit manipulation of oscillatory integrals. The proof of the pointwise bounds requires Weyl-strength subconvexity for quadratic Dirichlet $L$-functions over $\mathbb{Q}(i)$. Moreover, an asymptotic formula for a spectral exponential sum in the spectral aspect for a cofinite Kleinian group $\mathrm{\Gamma }$ is given. Our numerical experiments visualise in particular that $E_{\mathrm{\Gamma }}(X)$ obeys a conjectural bound of size $O_{\varepsilon }(X^{1+\varepsilon })$.

We construct blocks of finite groups with arbitrarily large Morita Frobenius numbers, an invariant which determines the size of the minimal field of definition of the associated basic algebra. This answers a question of Benson and Kessar. This also improves upon a result of the second author where arbitrarily large $\mathcal{𝒪}$-Morita Frobenius numbers are constructed.

We give explicit descriptions of some Noether–Lefschetz divisors in the moduli space of Gushel–Mukai fourfolds. As a consequence we obtain that their Kodaira dimension is negative.

We study odd-dimensional modular tensor categories and maximally nonself dual (MNSD) modular tensor categories of low rank. We give lower bounds for the ranks of modular tensor categories in terms of the rank of the adjoint subcategory and the order of the group of invertible objects. As an application of these results, we prove that odd-dimensional modular tensor categories of ranks 13 and 15 are pointed. In addition, we show that odd-dimensional tensor categories of ranks 17, 19, 21 and 23 are either pointed or perfect.

We extend the work of Salberger; Walsh; Castryck, Cluckers, Dittmann and Nguyen; and Vermeulen to prove the uniform dimension growth conjecture of Heath-Brown and Serre for varieties of degree at least $4$ over global fields. As an intermediate step, we generalize the bounds of Bombieri and Pila to curves over global fields and in doing so we improve the $B^{𝜀}$ factor by a $\text{log}(B)$ factor.

This is an erratum to our note “Lefschetz theorem for abelian fundamental group with modulus”. One proof is not correct and has to be replaced.