5,789 research outputs found
Two lectures on the arithmetic of K3 surfaces
In these lecture notes we review different aspects of the arithmetic of K3
surfaces. Topics include rational points, Picard number and Tate conjecture,
zeta functions and modularity.Comment: 26 pages; v4: typos corrected, references update
Counting curves over finite fields
This is a survey on recent results on counting of curves over finite fields.
It reviews various results on the maximum number of points on a curve of genus
g over a finite field of cardinality q, but the main emphasis is on results on
the Euler characteristic of the cohomology of local systems on moduli spaces of
curves of low genus and its implications for modular forms.Comment: 25 pages, to appear in Finite Fields and their Application
Crystal melting on toric surfaces
We study the relationship between the statistical mechanics of crystal
melting and instanton counting in N=4 supersymmetric U(1) gauge theory on toric
surfaces. We argue that, in contrast to their six-dimensional cousins, the two
problems are related but not identical. We develop a vertex formalism for the
crystal partition function, which calculates a generating function for the
dimension 0 and 1 subschemes of the toric surface, and describe the
modifications required to obtain the corresponding gauge theory partition
function.Comment: 30 pages; v2: references adde
Prepotentials from Symmetric Products
We investigate the prepotential that describes certain F^4 couplings in eight
dimensional string compactifications, and show how they can be computed from
the solutions of inhomogenous differential equations. These appear to have the
form of the Picard-Fuchs equations of a fibration of Sym^2(K3) over P^1. Our
findings give support to the conjecture that the relevant geometry which
underlies these couplings is given by a five-fold.Comment: 19p, harvmac; One sign in eq. (A.2) change
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