We find two natural spherical functors associated to the Kummer surface and analyze how their induced twists fit with Bridgeland’s conjecture on the derived autoequivalence group of a complex algebraic K3 surface.

This is part IV of our series of articles on log abelian varieties. In this part, we study the algebraic theory of proper models of log abelian varieties.

We prove a McShane-type identity: a series, expressed in terms of geodesic lengths, that sums to $2\pi$ for any closed hyperbolic surface with one distinguished point. To do so, we prove a generalized Birman–Series theorem showing that the set of complete geodesics on a hyperbolic surface with large cone angles is sparse.

We address two aspects of finitely generated modules of finite projective dimension over local rings and their connection in between: embeddability and grade of order ideals of minimal generators of syzygies. We provide a solution of the embeddability problem and prove important reductions and special cases of the order ideal conjecture. In particular, we derive that, in any local ring $R$ of mixed characteristic $p>0$, where $p$ is a nonzero divisor, if $I$ is an ideal of finite projective dimension over $R$ and $p\in I$ or $p$ is a nonzero divisor on $R/I$, then every minimal generator of $I$ is a nonzero divisor. Hence, if $P$ is a prime ideal of finite projective dimension in a local ring $R$, then every minimal generator of $P$ is a nonzero divisor in $R$.

Tensor products usually have nonzero torsion. This is a central theme of Auslander’s 1961 paper; the theme continues in the work of Huneke and Wiegand in the 1990s. The main focus in this article is on tensor powers of a finitely generated module over a local ring. Also, we study torsion-free modules $N$ with the property that $M{\otimes }_{R}N$ has nonzero torsion unless $M$ is very special. An important example of such a module $N$ is the Frobenius power ${}^{p^e}R$ over a complete intersection domain $R$ of characteristic $p>0$.

A $\mathbb{D}$-semiclassical weight is one which satisfies a particular linear, first-order homogeneous equation in a divided-difference operator $\mathbb{D}$. It is known that the system of polynomials, orthogonal with respect to this weight, and the associated functions satisfy a linear, first-order homogeneous matrix equation in the divided-difference operator termed the spectral equation. Attached to the spectral equation is a structure which constitutes a number of relations such as those arising from compatibility with the three-term recurrence relation. Here this structure is elucidated in the general case of quadratic lattices. The simplest examples of the $\mathbb{D}$-semiclassical orthogonal polynomial systems are precisely those in the Askey table of hypergeometric and basic hypergeometric orthogonal polynomials. However within the $\mathbb{D}$-semiclassical class it is entirely natural to define a generalization of the Askey table weights which involve a deformation with respect to new deformation variables. We completely construct the analogous structures arising from such deformations and their relations with the other elements of the theory. As an example we treat the first nontrivial deformation of the Askey–Wilson orthogonal polynomial system defined by the $q$-quadratic divided-difference operator, the Askey–Wilson operator, and derive the coupled first-order divided-difference equations characterizing its evolution in the deformation variable. We show that this system is a member of a sequence of classical solutions to the $E_7^{\left(1\right)}$ $q$-Painlevé system.

We prove the linear and nonlinear instability of periodic traveling wave solutions for a generalized version of the symmetric regularized long wave (SRLW) equation. Using analytic and asymptotic perturbation theory, we establish sufficient conditions for the existence of exponentially growing solutions to the linearized problem and so the linear instability of periodic profiles is obtained. An application of this approach is made to obtain the linear/nonlinear instability of cnoidal wave solutions for the modified SRLW (mSRLW) equation. We also prove the stability of dnoidal wave solutions associated to the equation just mentioned.

Consider an elliptic curve defined over an imaginary quadratic field $K$ with good reduction at the primes above $p\ge 5$ and with complex multiplication by the full ring of integers ${\mathcal{O}}_K$ of $K$. In this paper, we construct $p$-adic analogues of the Eisenstein–Kronecker series for such an elliptic curve as Coleman functions on the elliptic curve. We then prove $p$-adic analogues of the first and second Kronecker limit formulas by using the distribution relation of the Kronecker theta function.