Higher-order automorphic forms.

自同构形式是最初发明并用于研究整数（数论）的重要函数。它们封装了具有共同特征的整数的信息。然后通过对这些函数或相关的函数，称为l函数，进行微积分运算，我们通常能成功地获得所讨论的整数的信息。这些函数如此适合处理这类问题的原因之一是，它们满足一类恒等式，称为泛函方程。这些方法取得成功的一个显著例子是怀尔斯对费马大定理（FLT）的证明。对自同构形式的研究最近导致了更一般的泛函方程。我们把满足这些泛函方程的函数称为高阶自同构形式（简称hoaf），经典形式是hoaf的特殊例子。令人惊讶的是，在数论以外的学科，即数学物理中，也独立出现了hoaf。从数论的角度研究霍夫的第一个动机来自于一个由模符号组成的函数。模符号是一种数字，部分归功于上文提到的怀尔斯的工作，它帮助我们用自同构函数描述某些多项式方程及其整数解，这种自同构函数通常更容易处理。这反过来又会产生令人印象深刻的后果，正如FLT的证明所表明的那样。用模符号形成的模型已经成功地证明了模符号是按钟形曲线分布的。第二个激励问题（逆定理）与函数实际上是自同构形式的一个重要准则有关。一般来说，必须满足无限多个条件，但是，在最有趣的情况下，人们相信一个条件就足够了。最近的计算表明，这种说法可以用一个广义的术语来表述。有了这类函数的足够信息，我们就可以决定这种主张是否成立。这将是一个重要的结果，因为它将使我们更容易确定一个给定的函数是否是经典自同构形式。到目前为止，我和我的合作者已经建立了一些面包的基本性质。例如，我们已经找到了一般情况下汉堡的公式。虽然这些公式在许多情况下是足够的，但是当我们考虑更特殊类型（全纯）的函数时，它们就不够精确了。为了解决这一限制，我和我的合作者已经开始对这种特殊的面包进行表征。在数学物理领域，这种特征可能会产生有趣的影响，因为在数学物理领域，hoafs已经出现了。下一个自然的问题是，我们是否能想象出汉堡。经典的自同构形式可以用几何方式来描述，这是非常有用的（FLT的证明也是一个例子）。然而，这种描述不适用于更高阶。所以，一个人必须以一种不那么直接的方式工作，而前一个问题的答案已经对此有所帮助。图的可视化的许多应用之一是更好地理解附加到标准自同构形式上的l函数的重要方面，如前所述，这些函数通常提供关于整数的重要信息。这些问题的答案将具有独立的兴趣，并且与最初促使我研究hoafs的问题相关。特别是第二个激励问题（逆定理）很难，尽管相关的广义定理看起来与标准问题相似。为了解决这个问题，我们将需要根据上述两个主要计划目标尽可能多地提供有关hoafs的信息。这些是我在这个项目中感兴趣的一些问题。作为一门新学科，有许多有趣的问题，但我的指导是它们与激励问题的相关性。

Automorphic forms are important functions originally invented and used for the study of integer numbers (Number Theory). They encapsulate information about integers sharing a common characteristic. Then by doing Calculus on these functions or related ones, called L-functions, we often succeed in obtaining information about the integers in question. One of the reasons these functions are so suitable for dealing with such questions turns out to be that they satisfy a type of identity, called functional equation. A striking example of the successes of these methods is Wiles' proof of Fermat's last theorem (FLT).The study of automorphic forms has recently led to more general functional equations. We have called functions satisfying these functional equations higher-order automorphic forms (hoaf, for short) and the classical forms are special examples of hoafs. Surprisingly, hoafs have also appeared independently in subjects outside Number Theory, namely in Mathematical Physics.A first motivation for the study of hoafs from the point of view of Number theory came with a function formed with modular symbols. Modular symbols are numbers that, thanks partly to Wiles' work mentioned above, help us describe certain polynomial equations and their integers solutions using automorphic functions which are often easier to handle. This in turn can have impressive consequences, as the proof of FLT suggests. The hoaf formed with modular symbols has already been used successfully to prove that the modular symbols are distributed according to the bell curve. The second motivating problem ( converse theorem ) is related to an important criterion for a function to be actually an automorphic form. In general, infinitely many conditions must be satisfied, but, in the most interesting case, people believe that one condition suffices. Recent computations show that this claim can be formulated in terms of a generalised hoaf. Having enough information about functions of this kind will allow us to decide whether the claim holds. That would be an important result, because it will make it easier to decide whether a given function is a classical automorphic form. So far, my collaborators and I have established some of the basic properties of hoafs. For instance, we have found formulas for hoafs in a general setting. Although these formulas are sufficient for many purposes, they are not precise enough when we are considering functions of more special type (holomorphic). To address this limitation, my collaborators and I have begun the characterization of such special hoafs. This characterization may have, among other things, interesting implications in the area of Mathematical Physics where hoafs have appeared.A natural next question is if we can visualise hoafs. The classical automorphic forms can be described in a geometric way and this is very useful (the proof of FLT is again an example). However, this description does not work in higher orders. So, one must work in a less direct way and the answer to the previous question would already be helpful for that. One of the many applications of visualisations of hoafs is a better understanding of important aspects of L-functions attached to standard automorphic forms which, as mentioned, often give important information about integer numbers.The answers to such questions would be of independent interest and of relevance for the problems that motivated me to look at hoafs in the first place. The second motivating problem ( converse theorem ), in particular, is hard, even though the relevant generalised hoaf looks similar to the standard one. To resolve it we will need as much information about hoafs as possible according to the two main programme goals outlined above.These are some of the problems I am interested in pursuing in this project. As a new subject, there are many questions of interest, but my guide is their relevance for the motivating questions.