L-invariants and p-adic automorphic forms in ranks 2 and 3
Final Report Abstract
During the duration of the Walter Benjamin Fellowship, we were able to make significant progress on the questions outlined at the start of the project. While the original question concerning the relation between the ghost conjecture and valuations of L-invariants was mostly answered in the preprint by Jiawei An, we found new and extremely interesting data supporting a relationship between valuations of L-invariants and deep congruences between newforms, which became the focal point of our research. We were able to use our data to formulate a precise, yet completely unexpected conjecture that drew strong interest within the Number Theory community and is the basis of ongoing research both by us and other researchers. Since congruences between modular forms have always been of great arithmetic interest, the fact that we are able to use our conjecture together with the recent preprint [10] to predict such very deep congruences in a general setup is very remarkable. We made progress towards proving our conjecture in special cases and expect to publish a preprint containing all of our results in the near future. The data and code underlying our computations is publicly available for other researchers to use. We explored various approaches towards proving a stronger non-criticality statement for automorphic forms for GL3 , but we have not yet succeeded in obtaining the desired result. We hope that recent computational data together with the exploration of new approaches will help gain insights into the situation and lead to a resolution of the question. Concerning the final project constructing a Teitelbaum-style L-invariant in rank 3 and for arbitrary weight, we were able to construct one of the two maps needed and began exploring its properties. For the other map, the preprint opened up a different approach toward its construction – also in light of the unexpected developments in the first project, we are planning to compare the two approaches and complete the construction of this map in future work.
Publications
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Predict valuations of L-invariants. Sagemath software package
P. M. Gräf
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L-invariants and deep congruences 1. Magma software package
P. M. Gräf
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Finite polynomial cohomology with coefficients. Rendiconti del Seminario Matematico della Università di Padova, 155, 193-263.
Huang, Ting-Han & Wu, Ju-Feng
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L-invariants and deep congruences 2. Sagemath software package
P. M. Gräf
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New phenomena arising from L-invariants of modular forms.
J. Bergdall & R. Pollack
