Quantitative arithmetic geometry

丢芬图方程是一个多项式方程，人们感兴趣的是找到整数的解。自古以来，数学家们就对这样的方程着迷。通常简单的表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，虽然这些方程最初被认为只是一种好奇心，但在现代，它们在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解是否实际存在。这个问题本身是很困难的。如果有一个丢番图方程的“族”（比如通过改变方程的系数得到），事情会变得更有趣。在这种情况下，我们希望了解方程在族中的分布，并给出解。这是一个非常流行的现代话题，曼朱尔·巴尔加瓦（Manjul Bhargava）因在这类问题上的研究而在2014年获得了菲尔兹奖（Fields Medal）（这是数学家版的诺贝尔奖）。这个项目涉及这类问题。这里有一个由法国著名数学家Jean-Pierre Serre提出的关于丢番图方程在某些族中的分布并有解（即平面二次曲线）的猜想。我们将回答Serre问题的一些案例，并将Serre的原始框架扩展到更一般的问题。Erdos和Kac的一个著名定理也指出，一个“随机”整数n有大约log log n个素数因子（在精确的概率意义上）。我们将在丢芬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以模p不可解的素数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.