New Results in modularity, Christmas Update II

Just like last year, once again saint Nick has brought us a bounty of treasures related to Galois representations and automorphic forms in the final week of the year.

First there was this paper by Newton and Thorne, proving, among other things, the modularity of symmetric powers for a large range of holomorphic modular forms, including \(\Delta\) and any newform associated to a semistable elliptic curve. There is a lot to enjoy about this paper, not least of which is the nice application of an old computation of Buzzard and Kilford. But there are also some very nice new results on Selmer groups and reducible modularity lifting proved in the substantial related papers by Newton-Thorne and Allen-Newton-Thorne respectively. (Added: It was remiss of me not to also mention this paper by Thorne and Christos Anastassiades as well!) It’s often hard for the non-specialist to appreciate “technical” improvements on previous theorems, but in this case, they are all wrapped up neatly with a bow by such a clean application: \(\mathrm{Sym}^n(\Delta)\) is modular!

Moving on, we have this paper (monograph?) by Liu, Tian, Xiao, Zhang, and Zhu on the Bloch-Kato conjecture for a very general class of motives associated to Rankin-Selberg convolutions of forms on \(\mathrm{GL}_n\) and \(\mathrm{GL}_{n-1}\). I remember a few years ago talking to Yifeng during his interview at Northwestern (reader, we hired him) about this beautiful paper, giving a totally new argument to study questions of Selmer groups using cycles and level raising congruences. The current paper seems to be not only a version of that on steroids but also with a nice hot cup of tea with 3 lumps of potassium. It’s an amazing achievement which pulls together a lot of wonderful ideas, including Xiao-Zhu’s work on the Tate conjecture, not to mention all the previous work on the Gan-Gross-Prasad conjecture.

Well done to both groups of authors!

(In different times I would have given more details as to what these papers actually do, but as my free time nowadays consists of brief moments like this at 5:00AM in the morning you will have to forgive me, and anyway, these papers all seem to be very well written with nice introductions. That said, there will be some more technical mathematics posts coming up, not least of which relates to work of my own students. Stay tuned, Persiflage intends to keep posting!)

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6 Responses to New Results in modularity, Christmas Update II

  1. JC says:

    Happy New Year, Persliflage!

  2. David Loeffler says:

    Happy new year from me as well. Just a remark on the paper by Liu et al: the “level-raising congruences” approach to bounding Selmer groups didn’t emerge entirely from nowhere in Liu’s 2015 paper — it is recognisably a generalisation of earlier work of Bertolini and Darmon (Annals #162, 2005), which does a version of this for Heegner points on modular curves.

    Liu was the first to make this idea work in a higher-dimensional case, and now he and his coauthors have pushed this far beyond anything that one could have dreamed possible in 2005 or even in 2015 — it’s a magnificent piece of work.

    • Persiflage says:

      Of course you are completely correct! And I even knew this (and no doubt Yifeng told me so as well, and probably also Henri, but of course Henri has nothing but the highest praise for Yifeng’s work). Then again, as I’m sure you would agree, very few papers in mathematics can be truly considered sui generis.

  3. Edison says:

    Recently liu has proved part of the beilinson bloch conjecture!

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