We prove a direct image theorem stating that the direct image of a Galois formula by a morphism of difference schemes is equivalent to a Galois formula over fields with powers of Frobenius. As a consequence, we obtain an effective quantifier elimination procedure and a precise algebraic–geometric description of definable sets over fields with Frobenii in terms of twisted Galois formulas associated with finite Galois covers of difference schemes.

Let $I$ and $J$ be homogeneous ideals in a standard graded polynomial ring. We study upper bounds of the Hilbert function of the intersection of $I$ and $g(J)$, where $g$ is a general change of coordinates. Our main result gives a generalization of Green’s hyperplane section theorem.

In analogy with the complex analytic case, Mustaţă constructed (a family of) Bernstein–Sato polynomials for the structure sheaf ${\mathcal{O}}_{X}$ and a hypersurface $(f=0)$ in $X$, where $X$ is a regular variety over an $F$-finite field of positive characteristic (see Mustaţă, Bernstein–Sato polynomials in positive characteristic, J. Algebra 321(1) (2009), 128–151). He shows that the suitably interpreted zeros of his Bernstein–Sato polynomials correspond to the $F$-jumping numbers of the test ideal filtration ${\it\tau}(X,f^{t})$. In the present paper we generalize Mustaţă’s construction replacing ${\mathcal{O}}_{X}$ by an arbitrary $F$-regular Cartier module $M$ on $X$ and show an analogous correspondence of the zeros of our Bernstein–Sato polynomials with the jumping numbers of the associated filtration of test modules ${\it\tau}(M,f^{t})$ provided that $f$ is a nonzero divisor on $M$.

To any Nash germ invariant under right composition with a linear action of a finite group, we associate its equivariant zeta functions, inspired from motivic zeta functions, using the equivariant virtual Poincaré series as a motivic measure. We show Denef–Loeser formulas for the equivariant zeta functions and prove that they are invariants for equivariant blow-Nash equivalence via equivariant blow-Nash isomorphisms. Equivariant blow-Nash equivalence between invariant Nash germs is defined as a generalization involving equivariant data of the blow-Nash equivalence.

The aim of this paper is to carry out an explicit construction of CAP representations of $\text{GL}(2)$ over a division quaternion algebra with discriminant two. We first construct cusp forms on such a group explicitly by lifting from Maass cusp forms for the congruence subgroup ${\rm\Gamma}_{0}(2)$. We show that this lifting is nonzero and Hecke-equivariant. This allows us to determine each local component of a cuspidal representation generated by such a lifting. We then show that our cuspidal representations provide examples of CAP (cuspidal representation associated to a parabolic subgroup) representations, and, in fact, counterexamples to the Ramanujan conjecture.

Let $M$ be a Fano manifold. We call a Kähler metric ${\it\omega}\in c_{1}(M)$ a Kähler–Ricci soliton if it satisfies the equation $\text{Ric}({\it\omega})-{\it\omega}=L_{V}{\it\omega}$ for some holomorphic vector field $V$ on $M$. It is known that a necessary condition for the existence of Kähler–Ricci solitons is the vanishing of the modified Futaki invariant introduced by Tian and Zhu. In a recent work of Berman and Nyström, it was generalized for (possibly singular) Fano varieties, and the notion of algebrogeometric stability of the pair $(M,V)$ was introduced. In this paper, we propose a method of computing the modified Futaki invariant for Fano complete intersections in projective spaces.