Bad Reduction of Curves and Abelian Varieties

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One of the main problem in arithmetic geometry is the determination of the set of rational solutions of a system of polynomial equations with rational coefficients. One of the most successful technique in the study of such a set of solutions is to reduce the equations modulo a prime p and to study first the set of solutions of the latter system of equations. It turns out that in many situations, it is possible to describe a canonical way of reducing the equations modulo p. For instance, the canonical reduction of an abelian variety is called its Neron model, and is the object of study in Lorenzini's first research project. The canonical reduction of a curve is called its regular minimal model, and is the object of study in Lorenzini's second research project. For all but finitely many prime p, the canonical reduction is `good' and, as the name suggests, such a reduction type can be better understood than the reduction at the finitely many remaining primes. Our present understanding of the information encoded in the canonical reductions that are not good (and not semistable) is far from complete. Some very difficult problems arise when studying reductions modulo small primes. One such difficulty can be stated as follows. A famous theorem due to Grothendieck and others states that there exists a finite field extension L/K such that the reduction of the initial equations viewed as equations over L is either good or semistable. In other words, it is possible to improve the reduction by extending the initial field. When p is large, the extension L/K is totallyunderstood once its degree is known: it is the unique cyclic extension of thatdegree. When p is small and divides the degree of L/K, there are infinitely many extensions of that given degree, and almost nothing is known about the specific extension L/K needed to improve the reduction of the initial equations. Lorenzini's proposed research will shed more light on this and other special phenomena that arise when the reduction modulo a small prime p is not `good'.For centuries, human beings have been fascinated with solving diophantine equations, named after the Greek mathematician Diophantus who lived in the third century AD. The field of diophantine equations has taken onadded significance in the modern world as it finds applications in avariety of areas including, for example, encryption. A diophantine equation is a mathematical expression in several variables, say x and y. The central problem in the field is to find all possible solutions where x and y are both whole numbers or both fractions. For instance, the equation xy-10=0 has many solutions (e.g., x = y = square root of 10) but the solutions in whole positive numbers are in this case the divisors of 10, namely (x,y) = (1,10), (2,5), (5,2), and (10,1). While such an equation is very simple, a slight modification, such as replacing 10 by a very large number (for instance, one having 150 digits) renders the new equation extremely hard to solve in practice. It is this fact, that it is so hard to solve such equations, that is the key to many of the safest current military codes and data encryption systems. The complexity of the determination of all solutions in whole numbers or fractions of an equation increases with the power at whichthe variables appear in the equation. For instance, the equation `x to the power n plus y to the power n equals 1' was conjectured in the 17th century to have only two solutions when n is any odd number greater than 1. (Those two solutions are (x,y)=(1,0) and (x,y)= (0,1). This conjecture is called Fermat's Last Theorem and was proved only in 1994. Since the time of the Greeks, mathematicians have developed sophisticated tools to aid in solving equations. This investigator has developed some such mathematical tools and is currently working on further contributions to this field.