Volume 2013, Issue 676

We present a Serre-type conjecture on the modularity of four-dimensional symplectic mod   p Galois representations. We assume that the Galois representation is irreducible and odd (in the symplectic sense). The modularity condition is formulated using the étale and the algebraic de Rham cohomology of Siegel modular varieties of level prime to p . We concentrate on the case when the Galois representation is ordinary at p and we give a corresponding list of Serre weights. When the representation is moreover tamely ramified at p , we conjecture that all weights of this list are modular, otherwise we describe a subset of weights on the list that should be modular. We propose a construction of de Rham cohomology classes using the dual BGG complex, which should realise some of these weights.

Suppose that X is an analytic subvariety of a Stein manifold M and that φ is a plurisubharmonic (psh) function on X which is dominated by a continuous psh exhaustion function u of M . Given any number c  > 1, we show that φ admits a psh extension to M which is dominated by cu + on M . We use this result to prove that any ω -psh function on a subvariety of the complex projective space is the restriction of a global ω -psh function, where ω is the Fubini–Study Kähler form.

A well-known property of the signature of closed oriented 4 n -dimensional manifolds is Novikov additivity, which states that if a manifold is split into two manifolds with boundary along an oriented smooth hypersurface, then the signature of the original manifold equals the sum of the signatures of the resulting manifolds with boundary. Wall showed that this property is not true of signatures on manifolds with boundary and that the difference from additivity could be described as a certain Maslov triple index. Perverse signatures are signatures defined for any oriented stratified pseudomanifold, using the intersection homology groups of Goresky and MacPherson. In the case of Witt spaces, the middle perverse signature is the same as the Witt signature. This paper proves a generalization to perverse signatures of Wall's non-additivity theorem for signatures of manifolds with boundary. Under certain topological conditions on the dividing hypersurface, Novikov additivity for perverse signatures may be deduced as a corollary. In particular, Siegel's version of Novikov additivity for Witt signatures is a special case of this corollary.

We investigate a relationship between MacMahon's generalized sum-of-divisors functions and Chebyshev polynomials of the first kind. This determines a recurrence relation to compute these functions, as well as proving a conjecture of MacMahon about their general form by relating them to quasi-modular forms. These functions arise as solutions to a curve-counting problem on Abelian surfaces.

Chebyshev was the first to observe a bias in the distribution of primes in residue classes. The general phenomenon is that if a is a nonsquare (mod  q ) and b is a square (mod  q ), then there tend to be more primes congruent to a (mod  q ) than b (mod  q ) in initial intervals of the positive integers; more succinctly, there is a tendency for π ( x ;  q ,  a ) to exceed π ( x ;  q ,  b ). Rubinstein and Sarnak defined δ ( q ;  a ,  b ) to be the logarithmic density of the set of positive real numbers x for which this inequality holds; intuitively, δ ( q ;  a ,  b ) is the “probability” that π ( x ;  q ,  a ) >  π ( x ;  q ,  b ) when x is “chosen randomly”. In this paper, we establish an asymptotic series for δ ( q ;  a ,  b ) that can be instantiated with an error term smaller than any negative power of q . This asymptotic formula is written in terms of a variance V ( q ;  a ,  b ) that is originally defined as an infinite sum over all nontrivial zeros of Dirichlet L -functions corresponding to characters (mod  q ); we show how V ( q ;  a ,  b ) can be evaluated exactly as a finite expression. In addition to providing the exact rate at which δ ( q ;  a ,  b ) converges to 1/2 as q grows, these evaluations allow us to compare the various density values δ ( q ;  a ,  b ) as a and b vary modulo q ; by analyzing the resulting formulas, we can explain and predict which of these densities will be larger or smaller, based on arithmetic properties of the residue classes a and b (mod  q ). For example, we show that if a is a prime power and a ′ is not, then δ ( q ;  a , 1) <  δ ( q ;  a ′, 1) for all but finitely many moduli q for which both a and a ′ are nonsquares. Finally, we establish rigorous numerical bounds for these densities δ ( q ;  a ,  b ) and report on extensive calculations of them, including for example the determination of all 117 density values that exceed 9/10.

The quotient field of a generalized Krull domain of dimension exceeding one is Hilbertian by a theorem of Weissauer. Building on work of R. Klein we generalize this criterion for Hilbertianity to a wider class of domains. This allows us to extend recent results on Hilbertianity of fields of power series and obtain new Hilbertian fields.

As part of the general investigation of Ricci flow on complete surfaces with finite total curvature, we study this flow for surfaces with asymptotically conical (which includes as a special case asymptotically Euclidean) geometries. After establishing long-time existence, and in particular the fact that the flow preserves the asymptotically conic geometry, we prove that the solution metric g ( t ) expands at a locally uniform linear rate; moreover, the rescaled family of metrics t −1 g ( t ) exhibits a transition at infinite time inasmuch as it converges locally uniformly to a complete, finite area hyperbolic metric which is the unique uniformizing metric in the conformal class of the initial metric g 0 .