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Unformatted text preview: NOTES ON THE ARITHMETIC OF ALGEBRAIC CURVES PETE L. CLARK 1. What’s Nice About Algebraic Curves In this course we are going to study algebraic curves – i.e., onedimensional alge braic varieties – and how they vary in oneparameter families. In fact curves are by far the most intensively studied class of algebraic varieties (and this is true pretty much across the board, from complex algebraic geome ters to arithmetic geometers). 1 Let us begin by recalling some nice properties that curves have and most higherdimensional varieties lack. 1. Topological classification: Here let us work, for simplicity, over the complex numbers (although with suitable technology, one can establish quite similar results in positive characteristic). Any smooth, projective connected algebraic variety over C determines a complex manifold, whose complex dimension is equal to the alge braic dimension of the variety and therefore whose dimension as a real manifold is equal to twice that. Therefore a complex algebraic curve gives rise to a compact topological surface, whereas a complex surface gives rise to a compact topological 4manifold. But the jump in topological complexity between 2dimensional mani folds and 4dimensional manifolds is extreme. Every (orientable, as all Cmanifolds are) compact surface is homeomorphic to a gholed torus (a 0holed torus being a sphere). Thus there is a single topological invariant, the genus g . Moreover the topology of a genus g surface is wellunderstood: in particular its fundamental group is known to be Π( g ) = h a 1 ,b 1 ,...,a g ,b g  [ a 1 ,b 1 ] ··· [ a g ,b g ] = 1 i . On the other hand, every finitely presented group is the fundamental group of some compact topological 4manifold. Being the fundamental group of an alge braic complex manifold brings some additional restrictions: for instance, Hodge theory implies that its abelianization is of the form Z 2 n ⊕ T where T is a finite abelian group – i.e., its first Betti number must be even. In fact it can be shown that a finitely presented group is the fundamental group of an algebraic surface iff it is the fundamental group of any smooth algebraic variety (iff it is the fundamental group of a compact K¨ ahler manifold). Exactly what class of groups this might be is a fascinating open problem: indeed there is a book called Fundamental Groups of Compact K¨ ahler Manifolds (which does not come close to solving the prob lem, but presents lots of interesting results). This observation has important geometric consequences: when one studies fam ilies of algebraic varieties – roughly speaking, moduli spaces – in order to for the 1 Probably second place goes to abelian varieties, in part because of their close relationship with curves....
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 Fall '10
 Clark
 Algebra, Algebraic geometry, Elliptic Curve, pete l. clark

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