Quantitative arithmetic geometry

丢番图方程是一个多项式方程，其中一个人感兴趣的是在整数中寻找解。数学家自古以来就对这样的方程着迷。通常简单的陈述问题需要非常困难的工具来解决(最著名的例子之一是安德鲁·怀尔斯著名的费马大定理的证明)。此外，这种方程虽然最初只被视为一种好奇，但在现代信息安全和密码学中得到了重要的应用。给定丢番图方程，一个基本问题是确定是否确实存在解。这个问题本身就很难解决。如果一个人有一族丢番图方程(比如通过改变方程的系数来给出)，事情就会变得更加有趣。在这种情况下，人们想要了解方程在具有解的族中的分布。这是一个非常流行的现代话题，Manjul Bhargava因在这类问题上的工作而在2014年被授予菲尔兹奖(这是一种数学家版的诺贝尔奖)。该项目涉及这类问题。这里有法国著名数学家Jean-Pierre Serre关于丢番图方程在某些有解的族(即平面二次曲线)中的分布的猜想。我们将回答Serre问题的一些情况，并将Serre的原始框架扩展到更一般的问题。Erdos和Kac的一个著名定理还指出，“随机”整数n具有近似的logn素因数(在精确的概率意义下)。我们将在丢番图方程族的设置中得到类似的概率结果，其中我们要求给定方程不是模p可解的素数p的个数。

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.