We compute all the automorphisms in $\operatorname{GL}_2(\mathbb Z)$ of an integral binary quadratic form. Two tables at the end summarize the results. This note is part of [1].
We prove a strong decay bound for the Harish-Chandra inverse transform under a similar assumption for the candidate function. This is essentially done in Section 4.2 of [2].
In high school, I spent some time generating fractales, mostly using Javascript and the web worker API. Some of the results are in the gallery and around the website. I based myself on a pdf by Draves and Reckase. Their technique, called fractale flame, gives beautiful fractales with very nice colors. It is based on a classical iterated function system, where the functions are allowed to be non linear. The vibrant colors are due to a simple trick. The brightness of a point is given by the log-density of amount of times it was visited instead of a linear density or none. Under are 3 different examples.
I also coded at that time a small online app, where you can generate various fractales with this density, available here. You may have trouble launching it if your browser does not allow web workers. You may also have trouble with french, but there isn’t that much to understand. It is based on transformation of the view into a smaller quadrilateral via interpolation of the sides. For example, the following screenshot consists of 3 different squares set to red, green and blue.
After clicking “Lancer”, waiting a moment and “Rendu”, we get the following image which is the Sierpiński triangle.
It is possible to set precise coordinates to get a more aligned image and to use gamma correction to make it look brighter. It is not possible to get all the images in the gallery using this app but one can get Barnsley’s fern and a lot of other nice images, when playing a bit.
Here are a few codes that you can input under “Charger” and see how it looks like:
This one is in the gallery: 0:0.28:1:0.8425:1920:1080:100000:1:1,1;1;1:0.32;0.782:0.762;0.78:0.578;0.252:0.578;0.252,1;1;1:0.198;0.334:0.448;0.552:0.41;0.398:0.28;0.544,1;1;1:0.002;0.992:0.992;0.006:0.998;0.994:0.01;0.016,1;1;1:0.204;0.95:0.92;0.954:0.944;0.044:0.924;0.606,1;1;1:0.786;0.96:0.006;0.95:0.018;0.448:0.016;0.05
Here a list of some plugins I used to render equations, add citations, etc. I will keep this list updated
LaTex2HTML: compile LaTeX into images. Still testing if this is the best plugin for this purpose.
teachPress: create a bibliographic database. Can import BibTeX files. I changed 2 things in the source code: 1) Made the references’ list bigger in the shortcode tpref. 2) Removed the exponent in the shortcode tpcite.
Note: I will come back to this post to add more details.
I uploaded my first paper on arXiv at the end of August [3]. Have a look here. This is the main project in my PhD thesis. It is a continuation of a paper by Valentin Blomer (my advisor) and Andy Corbett [4]. This is part of the general topic of equidistributions of Laplace eigenfunctions and modular forms. There are various ways to study this, for example comparing various $L^p$-norms of the functions or Quantum Unique Ergodicity (QUE).
I considered the $L^2$-restriction norm of a form. That is, instead of asking for equidistribution on the whole space, I looked at a subspace of lower dimension. Here, QUE does not give us information since the integral can’t see that subspace. More precisely, I considered the $L^2$-norm of a form along the imaginary axis. This is reminiscent of the Mellin transform. We can therefore connect this norm to a period formula for the corresponding $L$-function.
Now there is two conjectures in the picture. On one side, we can find a version of QUE for this subspace. We can imagine that the measure associated to the form still converges to the uniform one in that case. This gives an asymptotic for the size of the norm. On the other side, the Lindelöf hypothesis gives an upper bound for the norm.
In my paper, I considered Siegel modular forms of degree 2. After averaging over a basis of the weight $k$ and over the weight in a dyadic interval, I obtained an asymptotic formula for the norm on average. This formula is coherent with QUE. More precisely, since the subspace has infinite volume, the norm diverges to infinity. But an heuristic argument tells us how fast it should grow. This is also a strong version of Lindelöf on average.
