Bad Reduction of Curves and Abelian Varieties

算术几何中的一个主要问题是具有有理系数的多项式方程组的有理解集的确定。研究这类解的最成功的方法之一是将方程降模为a ' p，并首先研究后一类方程组的解。事实证明，在许多情况下，有可能描述一种正则化方程模化p的方法。例如，一个阿贝尔变量的正则化被称为Neron模型，这是Lorenzini第一个研究项目的研究对象。曲线的正则化简称为它的正则极小模型，这是洛伦齐尼第二个研究项目的研究对象。对于除了有限个素数p之外的所有素数，规范约简都是“好的”，顾名思义，这样的约简类型比有限个剩余素数的约简更容易理解。我们目前对规范约简中编码的信息的理解是不好的（而且不是半稳定的），这还远远不够完整。在研究小素数的模化时，出现了一些非常困难的问题。一个这样的困难可以描述如下。由Grothendieck等人提出的一个著名定理指出，存在一个有限域扩展L/K，使得在L上作为方程的初始方程的约简要么是好的，要么是半稳定的。换句话说，可以通过扩展初始域来改进缩减。当p较大时，扩展L/K的阶数已知，即为该阶数的唯一循环扩展。当p很小并除以L/K的阶时，对于给定的阶有无限多的扩展，并且对于改进初始方程的约简所需的特定扩展L/K几乎一无所知。Lorenzini提出的研究将更多地阐明当一个小素数p的约简模不“好”时出现的这种和其他特殊现象。几个世纪以来，人类一直着迷于解决丢番图方程，该方程以生活在公元三世纪的希腊数学家丢番图命名。丢芬图方程的领域在现代世界中具有更重要的意义，因为它在包括加密在内的各种领域都有应用。丢色图方程是由几个变量组成的数学表达式，比如x和y。该领域的核心问题是找到x和y都是整数或都是分数的所有可能解。例如，方程xy-10=0有许多解（例如，x = y =√10），但在这种情况下，整正数的解是10的除数，即(x,y) =(1,10),(2,5),（5,2）和（10,1）。虽然这样的方程非常简单，但稍加修改，例如将10替换为一个非常大的数字（例如，一个有150位数字），就会使新方程在实践中非常难以求解。要解出这样的方程是如此困难，而这正是目前许多最安全的军事密码和数据加密系统的关键所在。确定一个方程的所有整数或分数解的复杂性随着变量在方程中出现的次数的增加而增加。例如，方程“x ^ n + y ^ n = 1”在17世纪被推测为只有两个解，当n是大于1的任意奇数时。这两个解分别是(x,y)=(1,0)和(x,y)=（0,1）这个猜想被称为费马大定理，直到1994年才被证明。从希腊时代起，数学家们就发明了复杂的工具来帮助解方程。这位研究者已经开发了一些这样的数学工具，目前正在为这一领域的进一步贡献而努力。

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.