Note: I will come back to this post to add more details.

I uploaded my first paper on arXiv at the end of August [1]. 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 [2]. 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.

References

1. Gilles Felber (2023): A Restriction Norm Problem for Siegel Modular Forms. In: ArXiv, 2023.
2. Valentin Blomer and Andrew Corbett (2021): A symplectic restriction problem. In: Mathematische Annalen, vol. 382, no. 3-4, pp. 1323–1424, 2021.