Bad Reduction of Curves and Abelian Varieties

曲线和阿贝尔簇的不良约简

• 批准号: 0070522
• 负责人: Dino Lorenzini
• 金额: $ 7.65万
• 依托单位: University of Georgia Research Foundation Inc
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070522&HistoricalAwards=false
• 关键词: Bad; Reduction; Curves; Abelian; Varieties

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.

确定有理系数多项式方程组的有理解集是算术几何中的主要问题之一。在研究这样一组解的过程中，最成功的方法之一是将方程简化为素数p的模，并首先研究后者方程组的解。事实证明，在许多情况下，它是可能的，以描述一个规范的方式减少方程模p.例如，规范减少的阿贝尔品种被称为其Neron模型，是研究的对象，在洛仑兹尼的第一个研究项目。曲线的正则化约被称为其正则极小模型，这是Lorenzini第二个研究项目的研究对象。对于除了1000个素数p之外的所有素数，规范化约简都是“好的”，正如其名称所暗示的，这样的约简类型可以比在1000个剩余素数上的约简更好地理解。我们目前对编码在不好的（也不是半稳定的）规范约简中的信息的理解还远远没有完成。在研究模小素数约简时出现了一些非常困难的问题。一个这样的困难可以陈述如下。一个著名的定理，由于格罗滕迪克和其他国家，存在一个有限的域扩展L/K，使减少的初始方程被视为方程在L是好的或半稳定的。换句话说，可以通过扩展初始场来改善减少。当p很大时，扩张L/K一旦知道它的次数就完全被理解了：它是那个次数的唯一循环扩张。当p很小并且整除L/K的次数时，给定的次数有无穷多个扩展，并且几乎不知道需要改进初始方程的约化的特定扩展L/K。Lorenzini提出的研究将为这一问题以及其他一些特殊现象提供更多的线索，这些特殊现象是在对小素数p取模进行归约并不“好”时出现的。几个世纪以来，人类一直着迷于求解丢番图方程，该方程以生活在公元世纪的希腊数学家丢番图的名字命名。丢番图方程的领域在现代世界中具有更重要的意义，因为它在包括例如加密在内的各种领域中都有应用。丢番图方程是一个包含多个变量的数学表达式，比如x和y。 该领域的核心问题是找到所有可能的解决方案，其中x和y都是整数或分数。例如，方程xy-10=0有许多解（例如，x = y = 10的平方根），但在这种情况下，整数正数的解是10的约数，即（x，y）=（1，10），（2，5），（5，2）和（10，1）。虽然这样的方程非常简单，但稍微修改，例如用一个非常大的数字（例如，一个有150位数的数字）代替10，会使新方程在实践中非常难以求解。这是一个事实，它是如此难以解决这样的方程，这是目前许多最安全的军事代码和数据加密系统的关键。确定一个方程的整数或分数形式的所有解的复杂性随变量在方程中出现的幂次的增加而增加。例如，方程“x的n次幂加上y的n次幂等于1”在世纪被证明，当n是大于1的任何奇数时，只有两个解。（这两个解是（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。自希腊时代以来，数学家们已经开发了复杂的工具来帮助解方程。这位研究员已经开发了一些这样的数学工具，目前正在努力对这一领域作出进一步的贡献。