## Geometric Methods in Algebra and Number TheoryFedor Bogomolov, Yuri Tschinkel * Contains a selection of articles exploring geometric approaches to problems in algebra, algebraic geometry and number theory * The collection gives a representative sample of problems and most recent results in algebraic and arithmetic geometry * Text can serve as an intense introduction for graduate students and those wishing to pursue research in algebraic and arithmetic geometry |

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abelian variety action Ag,n algebraic applied associated assume automorphism bundle called canonical characteristic closed coefficients cohomology complex components computation conjecture connected consider construction contains corresponding covering curve defined definition denote dimension divisor elements equal equivalent example exists extension fact fiber field finite fixed follows formal formula function Galois geometric given gives Hecke orbit height hence Hodge implies induced integral intersection irreducible isomorphism Lemma Math maximal mixed modular moduli space morphism motivic multiplication natural Note obtain operator p-adic pair particular polynomial positive prime projective Proof properties Proposition prove quotient ramification rational reduced relations Remark representation respectively result ring satisfies Shimura simple smooth space stable statement subgroup subset subvariety Suppose surface symmetric Theorem theory unmixed Beauville valued write zero