Quantitative arithmetic geometry

丢番图方程是一个多项式方程，其中人们感兴趣的是在整数中找到解决方案。自古以来，数学家们就对这类方程着迷。通常，简单的问题需要非常困难的工具来解决（其中一个最著名的例子是安德鲁·怀尔斯对费马大定理的著名证明）。此外，这样的方程，而最初被视为只是一个好奇心，发现在现代的信息安全和密码学的重要应用。给定一个丢番图方程，一个基本的问题是要确定是否有一个解决方案实际存在。这个问题本身就很难。如果有一个丢番图方程族（比如说，通过改变方程的系数来给出），事情就变得更加有趣了。在这种情况下，人们希望了解方程族中有解的分布。这是一个非常受欢迎的现代话题，Manjul Bhargava因其在此类问题上的工作而于2014年获得菲尔兹奖（这是数学家版的诺贝尔奖）。这里有一个猜想，由于让皮埃尔塞尔，一位著名的法国数学家，对分布的丢番图方程在某些家庭的解决方案（即平面二次曲线）。我们将回答Serre问题的一些情况，以及将Serre的原始框架扩展到更一般的问题。Erdos和Kac的一个著名定理也指出，一个“随机”整数n大约有log log n个素因子（在精确的概率意义上）。我们将得到类似的概率结果设置家庭的丢番图方程，在那里我们要求的数量素数p的一个给定的方程是不可溶模p。

A Diophantine equation is a polynomial equation where one is interested in finding solutions in the whole numbers. Mathematicians have been fascinated by such equations since antiquity. Often simple to state problems require very difficult tools to solve (one of the most famous examples of this being Andrew Wiles's celebrated proof of Fermat's last theorem). Moreover such equations, whilst originally viewed as nothing but a curiosity, have found important applications in modern times to information security and cryptography.Given a Diophantine equation, a fundamental problem is to determine whether a solution actually exists. This problem in itself is very difficult. Things get even more interesting if one has a *family* of Diophantine equations (given by varying the coefficients of the equations, say). In this case one would like to understand the distribution of equations in the family with a solution. This is a very popular modern topic, with Manjul Bhargava being awarded the Fields Medal in 2014 for his work on such problems (this is a kind of mathematician's version of the Nobel prize).The project concerns problems of this type. Here there is a conjecture due to Jean-Pierre Serre, a famous French mathematician, on the distribution of Diophantine equations in certain families with a solution (namely plane conics). We will answer some cases of Serre's problem, as well as extending Serre's original framework to more general problems.A famous theorem of Erdos and Kac also states that a "random" integer n has approximately log log n prime factors (in a precise probabilistic sense). We will obtain analogues of this probabilistic result in the setting of families of Diophantine equations, where we ask for the number of primes p for which a given equation is not soluble modulo p.