We continue our study on the elliptic curve discrete logarithm problem over finite extension fields. We show, among others, the following results:

For sequences of prime powers ${\left(q_i\right)}_{i\in \mathbb{N}}$ and natural numbers ${\left(n_i\right)}_{i\in \mathbb{N}}$ with $n_i\to \infty$ and $n_i∕log{\left(q_i\right)}^2\to 0$ for $i\to \infty$, the discrete logarithm problem in the groups of rational points of elliptic curves over the fields ${\mathbb{F}}_{q_i^{n_i}}$ can be solved in subexponential expected time ${\left(q_i^{n_i}\right)}^{o\left(1\right)}$.

Let $a$, $b>0$ be fixed. Then the problem over fields ${\mathbb{F}}_{q^n}$, where $q$ is a prime power and $n$ a natural number with $a\cdot log{\left(q\right)}^{1∕3}\le n\le b\cdot log\left(q\right)$, can be solved in an expected time of $e^{\mathcal{O}\left(log{\left(q^n\right)}^{3∕4}\right)}$.

We present a method to determine Frobenius elements in arbitrary Galois extensions of global fields, which may be seen as a generalisation of Euler’s criterion. It is a part of the general question how to compare splitting fields and identify conjugacy classes in Galois groups, which we will discuss as well.

We address the problem of weak approximation for general cubic hypersurfaces defined over number fields with arbitrary singular locus. In particular, weak approximation is established for the smooth locus of projective, geometrically integral, nonconical cubic hypersurfaces of dimension at least 17. The proof utilises the Hardy–Littlewood circle method and the fibration method.

For a finite braided tensor category $\mathcal{C}$ we introduce its Picard crossed module $\mathfrak{P}\left(\mathcal{C}\right)$ consisting of the group of invertible $\mathcal{C}$-module categories and the group of braided tensor autoequivalences of $\mathcal{C}$. We describe $\mathfrak{P}\left(\mathcal{C}\right)$ in terms of braided autoequivalences of the Drinfeld center of $\mathcal{C}$. As an illustration, we compute the Picard crossed module of a braided pointed fusion category.

We prove a higher dimensional generalization of Gross and Zagier’s theorem on the factorization of differences of singular moduli. Their result is proved by giving a counting formula for the number of isomorphisms between elliptic curves with complex multiplication by two different imaginary quadratic fields $K$ and $K^{\prime }$ when the curves are reduced modulo a supersingular prime and its powers. Equivalently, the Gross–Zagier formula counts optimal embeddings of the ring of integers of an imaginary quadratic field into particular maximal orders in $B_{p,\infty }$, the definite quaternion algebra over $\mathbb{Q}$ ramified only at $p$ and infinity. Our work gives an analogous counting formula for the number of simultaneous embeddings of the rings of integers of primitive CM fields into superspecial orders in definite quaternion algebras over totally real fields of strict class number $1$. Our results can also be viewed as a counting formula for the number of isomorphisms modulo $\mathfrak{p}\mid p$ between abelian varieties with CM by different fields. Our counting formula can also be used to determine which superspecial primes appear in the factorizations of differences of values of Siegel modular functions at CM points associated to two different CM fields and to give a bound on those supersingular primes that can appear. In the special case of Jacobians of genus-$2$ curves, this provides information about the factorizations of numerators of Igusa invariants and so is also relevant to the problem of constructing genus-$2$ curves for use in cryptography.

A conjecture of Manin predicts the distribution of $K$-rational points on certain algebraic varieties defined over a number field $K$. In recent years, a method using universal torsors has been successfully applied to several hard special cases of Manin’s conjecture over the field $\mathbb{Q}$. Combining this method with techniques developed by Schanuel, we give a proof of Manin’s conjecture over arbitrary number fields for the singular cubic surface $S$ given by the equation $x_0^3=x_1{x}_2{x}_3$.

Let $Y$ be a smooth rational surface, and let $D$ be a cycle of rational curves on $Y$ that is an anticanonical divisor, i.e., an element of $|-K_Y|$. Looijenga studied the geometry of such surfaces $Y$ in case $D$ has at most five components and identified a geometrically significant subset $R$ of the divisor classes of square $-2$ orthogonal to the components of $D$. Motivated by recent work of Gross, Hacking, and Keel on the global Torelli theorem for pairs $\left(Y,D\right)$, we attempt to generalize some of Looijenga’s results in case $D$ has more than five components. In particular, given an integral isometry $f$ of $H^2\left(Y\right)$ that preserves the classes of the components of $D$, we investigate the relationship between the condition that $f$ preserves the “generic” ample cone of $Y$ and the condition that $f$ preserves the set $R$.

Let ${\sigma }_1$ and ${\sigma }_2$ be commuting involutions of a connected reductive algebraic group $G$ with $\mathfrak{g}=Lie\left(G\right)$. Let

$$\mathfrak{g}=\underset{i,j=0,1}{\oplus }{\mathfrak{g}}_{@ij}$$

be the corresponding ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-grading. If $\left\{\alpha ,\beta ,\gamma \right\}=\left\{01,10,11\right\}$, then $\left[,\right]$ maps ${\mathfrak{g}}_{@\alpha }\times {\mathfrak{g}}_{\beta }$ into ${\mathfrak{g}}_{\gamma }$, and the zero fiber of this bracket is called a $\overrightarrow{\sigma }$-commuting variety. The commuting variety of $\mathfrak{g}$ and commuting varieties related to one involution are particular cases of this construction. We develop a general theory of such varieties and point out some cases, when they have especially good properties. If $G∕G^{{\sigma }_1}$ is a Hermitian symmetric space of tube type, then one can find three conjugate pairwise commuting involutions ${\sigma }_1$, ${\sigma }_2$, and ${\sigma }_3={\sigma }_1{\sigma }_2$. In this case, any $\overrightarrow{\sigma }$-commuting variety is isomorphic to the commuting variety of the simple Jordan algebra associated with ${\sigma }_1$. As an application, we show that if $\mathcal{J}$ is the Jordan algebra of symmetric matrices, then the product map $\mathcal{J}\times \mathcal{J}\to \mathcal{J}$ is equidimensional, while for all other simple Jordan algebras equidimensionality fails.