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1900s–1930s: Branch Curves and the Italian School of Algebraic Geometry
In contrast to the separate beginnings during the nineteenth century presented in the previous chapter, the twentieth century saw the consolidation...
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New generalisation of Jacobi’s derivative formula
A stream of new theta relations is obtained. They follow from the general Thomae formula, which is a new result giving expressions for theta...
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Invariants for trees of non-archimedean polynomials and skeleta of superelliptic curves
In this paper we generalize the j -invariant criterion for the semistable reduction type of an elliptic curve to superelliptic curves X given by
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Bounding integral points on the Siegel modular variety \(A_2(2)\)
We determine an explicit upper bound for the stable Faltings height of principally polarised abelian surfaces over number fields corresponding to S -in...
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Hyperelliptically fibred surfaces with nodes
Using elementary methods of algebraic geometry, we present constructions of hyperelliptically fibred surfaces containing nodal fibres.
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Real algebraic curves with large finite number of real points
We address the problem of the maximal finite number of real points of a real algebraic curve (of a given degree and, sometimes, genus) in the...
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The fundamental group and extensions of motives of Jacobians of curves
In this paper, we construct extensions of mixed Hodge structure coming from the mixed Hodge structure on the graded quotients of the group ring of...
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Chapter 1 Hilbert’s 16th Problem and Its Weak Form
Usually, the maximum of the number of limit cycles is denoted by H(n), and is called the Hilbert number. We recall that a limit cycle of system (1.1)...
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Riemann surfaces and algebraic curves
A Riemann surface can be thought as the domain of definition of a holomorphic function f that has been continued analytically as far as such...
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Jacobian Varieties
A curve in this chapter means a complex smooth projective curve. To every curve C one can associate in a naturalway a principally polarized abelian...
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Computational Number Theory, Past, Present, and Future
A survey of computational number theory viewed through the prism of my own experience and the Pari/GP software.
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Quantum modular invariant and Hilbert class fields of real quadratic global function fields
This is the first of a series of two papers in which we present a solution to Manin’s Real Multiplication program (Manin in: Laudal and Piene (eds)...
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Generalized class polynomials
The Hilbert class polynomial has as roots the j -invariants of elliptic curves whose endomorphism ring is a given imaginary quadratic order. It can be...
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Haupt’s theorem for strata of abelian differentials
Let S be a closed topological surface. Haupt’s theorem provides necessary and sufficient conditions for a complex-valued character of the first...
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Main Examples of Abelian Varieties
In this chapter some of the most prominent examples of abelian varieties will be introduced, namely abelian surfaces, Picard and Albanese varieties,...