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bogomolov f geometric methods algebra number theory 2005

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Textbook in PDF format The transparency and power of geometric constructions has been a source of inspiration for generations of mathematicians. Their applications to problems in algebra and number theory go back to Diophantus, if not earlier. Naturally, the Greek techniques of intersecting lines and conics have given way to much more sophisticated and subtle constructions. What remains unchallenged is the beauty and persuasion of pictures, communicated in words or drawings. This volume contains a selection of articles exploring geometric approaches to problems in algebra, algebraic geometry and number theory. All papers are strongly inﬂuenced by geometric ideas and intuition. Several papers focus on algebraic curves: the themes range from the study of unramiﬁed curve covers (Bogomolov–Tschinkel), Jacobians of curves (Zarhin), moduli spaces of curves (Hassett) to modern problems inspired by physics (Hausel). The paper by Bogomolov–Tschinkel explores certain special aspects of the geometry of curves over number ﬁelds: there exist many more nontrivial correspondences between such curves than between curves deﬁned over larger ﬁelds. Zarhin studies the structure of Jacobians of cyclic covers of the projective line and provides an effective criterion for this Jacobian to be suffciently generic. Hassett applies the logarithmic minimal model program to moduli spaces of curves and describes it in complete detail in genus two. Hausel studies Hodge-type polynomials for mixed Hodge structure on moduli spaces of representations of the fundamental group of a complex projective curve into a reductive algebraic group. Explicit formulas are obtained by counting points over ﬁnite ﬁelds on these moduli spaces. Two contributions deal with surfaces: applying the structure theory of ﬁnite groups to the construction of interesting surfaces (Bauer–Catanese–Grunewald), and developing a conjecture about rational points of bounded height on cubic surfaces (Swinnerton-Dyer). Representation-theoretic and combinatorial aspects of higher-dimensional geometry are discussed in the papers by de Concini–Procesi and Tamvakis. The papers by Chai and Pink report on current active research exploring special points and special loci on Shimura varieties. Budur studies invariantsvi of higher-dimensional singular varieties. Spitzweck considers families of motives and describes an analog of limit mixed Hodge structures in the motivic setup. Cluckers–Loeser continue their foundational work on motivic integration. One of the immediate applications is the reduction of a central problem from the theory of automorphic forms (the Fundamental Lemma) from p-adic ﬁelds to function ﬁelds of positive characteristic, for large p. A different reduction to function ﬁelds of positive characteristic is shown in the paper by Ellenberg–Venkatesh: they ﬁnd a geometric interpretation, via Hurwitz schemes, of Malle’s conjectures about the asymptotic of number ﬁelds of bounded discriminant and ﬁxed Galois group and establish several upper bounds in this direction. Finally, Pineiro–Szpiro–Tucker relate algebraic dynamical systems on P1 to Arakelov theory on an arithmetic surface. They deﬁne heights associated to such dynamical systems and formulate an equidistribution conjecture in this context

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bogomolov f geometric methods algebra number theory 2005