We study a function field analog of Chebyshev’s bias. Our results, as well as their proofs, are similar to those of Rubinstein and Sarnak in the case of the rational number field. Following Rubinstein and Sarnak, we introduce the grand simplicity hypothesis (GSH), a certain hypothesis on the inverse zeros of Dirichlet L-series of a polynomial ring over a finite field. Under this hypothesis, we investigate how primes, that is, irreducible monic polynomials in a polynomial ring over a finite field, are distributed in a given set of residue classes modulo a fixed monic polynomial. In particular, we prove under the GSH that, like the number field case, primes are biased toward quadratic nonresidues. Unlike the number field case, the GSH can be proved to hold in some cases and can be violated in some other cases. Also, under the GSH, we give the necessary and sufficient conditions for which primes are unbiased and describe certain central limit behaviors as the degree of modulus under consideration tends to infinity, all of which have been established in the number field case by Rubinstein and Sarnak.

We revisit recent work of Heath-Brown on the average order of the quantity r(L1(x))⋯r(L4(x)), for suitable binary linear forms L1,…,L4, as x=(x1,x2) ranges over quite general regions in ℤ2. In addition to improving the error term in Heath-Brown’s estimate, we generalise his result to cover a wider class of linear forms.

We present a simple description of moduli spaces of torsion-free 𝒟-modules (𝒟-bundles) on general smooth complex curves, generalizing the identification of the space of ideals in the Weyl algebra with Calogero–Moser quiver varieties. Namely, we show that the moduli of 𝒟-bundles form twisted cotangent bundles to moduli of torsion sheaves on X, answering a question of Ginzburg. The corresponding (untwisted) cotangent bundles are identified with moduli of perverse vector bundles on T*X, which contain as open subsets the moduli of framed torsion-free sheaves (the Hilbert schemes T*X[n] in the rank-one case). The proof is based on the description of the derived category of 𝒟-modules on X by a noncommutative version of the Beilinson transform on P1.

Let X be a smooth complex Fano variety. We study ‘quasi-elementary’ contractions of fiber type of X, which are a natural generalization of elementary contractions of fiber type. If f:X→Y is such a contraction, then the Picard numbers satisfy ρX≤ρY+ρF, where F is a general fiber of f. We show that, if dim Y ≤3 and ρY≥4, then Y is smooth and Fano; if moreover ρY≥6, then X is a product. This yields sharp bounds on ρX when dim X=4 and X has a quasi-elementary contraction of fiber type, and other applications in higher dimensions.

The enumerative geometry of rth roots of line bundles is crucial in the theory of r-spin curves and occurs in the calculation of Gromov–Witten invariants of orbifolds. It requires the definition of the suitable compact moduli stack and the generalization of the standard techniques from the theory of moduli of stable curves. In a previous paper, we constructed a compact moduli stack by describing the notion of stability in the context of twisted curves. In this paper, by working with stable twisted curves, we extend Mumford’s formula for the Chern character of the Hodge bundle to the direct image of the universal rth root in K-theory.

We prove that all monomials of κ-classes and ψ-classes are independent in for all . We also give a simple argument for in Rl(ℳg) for .

Let F be an arbitrary local field. Consider the standard embedding and the two-sided action of GLn(F)×GLn(F) on GLn+1(F). In this paper we show that any GLn(F)×GLn(F)-invariant distribution on GLn+1(F) is invariant with respect to transposition. We show that this implies that the pair (GLn+1(F), GLn(F)) is a Gelfand pair. Namely, for any irreducible admissible representation (π,E) of GLn+1(F), . For the proof in the archimedean case, we develop several tools to study invariant distributions on smooth manifolds.

We give an interpretation of the energy function and classically restricted one-dimensional sums associated to tensor products of level-zero fundamental representations of quantum affine algebras in terms of Lakshmibai–Seshadri paths of level-zero shape.

A notion of Hochschild cohomology HH*(𝒜) of an abelian category 𝒜 was defined by Lowen and Van den Bergh (Adv. Math. 198 (2005), 172–221). They also showed the existence of a characteristic morphism χ from the Hochschild cohomology of 𝒜 into the graded centre ℨ*(Db(𝒜)) of the bounded derived category of 𝒜. An element c∈HH2(𝒜) corresponds to a first-order deformation 𝒜c of 𝒜 (Lowen and Van den Bergh, Trans. Amer. Math. Soc. 358 (2006), 5441–5483). The problem of deforming an object M∈Db(𝒜) to Db(𝒜c) was treated by Lowen (Comm. Algebra 33 (2005), 3195–3223). In this paper we show that the element χ(c)M∈Ext𝒜2(M,M) is precisely the obstruction to deforming M to Db(𝒜c). Hence, this paper provides a missing link between the above works. Finally we discuss some implications of these facts in the direction of a ‘derived deformation theory’.

The chain complexes underlying Floer homology theories typically carry a real-valued filtration, allowing one to associate to each Floer homology class a spectral number defined as the infimum of the filtration levels of chains representing that class. These spectral numbers have been studied extensively in the case of Hamiltonian Floer homology by Oh, Schwarz and others. We prove that the spectral number associated to any nonzero Floer homology class is always finite, and that the infimum in the definition of the spectral number is always attained. In the Hamiltonian case, this implies that what is known as the ‘nondegenerate spectrality’ axiom holds on all closed symplectic manifolds. Our proofs are entirely algebraic and apply to any Floer-type theory (including Novikov homology) satisfying certain standard formal properties. The key ingredient is a theorem about the existence of best approximations of arbitrary elements of finitely generated free modules over Novikov rings by elements of prescribed submodules with respect to a certain family of non-Archimedean metrics.

The eta invariant of the Dirac operator over a non-compact cofinite quotient of PSL(2,ℝ) is defined through a regularized trace following Melrose. It reduces to the standard definition in terms of eigenvalues in the case of a totally non-trivial spin structure. When the S1-fibers are rescaled, the metric becomes of non-exact fibered-cusp type near the ends. We completely describe the continuous spectrum of the Dirac operator with respect to the rescaled metric and its dependence on the spin structure, and show that the adiabatic limit of the eta invariant is essentially the volume of the base hyperbolic Riemann surface with cusps, extending some of the results of Seade and Steer.