{"id":3335,"date":"2012-08-15T23:48:32","date_gmt":"2012-08-15T21:48:32","guid":{"rendered":"https:\/\/wpethzprd.ethz.ch\/kowalski\/?p=3335"},"modified":"2024-02-20T13:40:35","modified_gmt":"2024-02-20T11:40:35","slug":"algebraic-twists-of-modular-forms-iii","status":"publish","type":"post","link":"https:\/\/blogs.ethz.ch\/kowalski\/2012\/08\/15\/algebraic-twists-of-modular-forms-iii\/","title":{"rendered":"Algebraic twists of modular forms, III"},"content":{"rendered":"

After a small break while traveling, I continue the discussion of my paper with Fouvry and Michel, continuing from here<\/a>. But first of all, I’m happy to mention two proofs of the counting identities mentioned in my introductory post<\/a>: one that D. Zywina has sent me, a nice proof based on explicit computations of modular forms and functions, and another that is in fact just an application of a more general result of Deligne and Flicker on local systems on the projective line minus four points (see Section 7 of this preprint<\/a>).<\/p>\n

\n

Now, if memory serves, at the end of the last post, we saw that analytic techniques reduce the study of our sums<\/p>\n

$latex S(f;K)=\\sum_{n}\\rho_f(n)K(n)V(n\/p)$<\/p>\n

to the study of correlation sums<\/p>\n

$latex C(K;\\gamma)=\\sum_{z}\\overline{\\hat{K}(\\gamma\\cdot z)}\\hat{K}(z),$<\/p>\n

where $latex \\hat{K}$ is the normalized Fourier transform modulo $latex p$.<\/p>\n

For these, we need to show that “most” are small in order to conclude. Fixing a parameter $latex M\\geq 1$, let us therefore define $latex G_{K,M}$ to be the set of matrices $latex \\gamma$ in $latex \\mathrm{PGL}_2(\\mathbf{F}_p)$ such that<\/p>\n

$latex |C(K;\\gamma)|>Mp^{1\/2}.$<\/p>\n

The discussion in the previous post shows that if, for some $latex M\\geq 1$, all matrices in $latex G_{K,M}$ are upper-triangular, we get<\/p>\n

$latex S(f;K)\\ll p^{1-1\/8+\\epsilon}$<\/p>\n

for any $latex \\epsilon>0$. But we can deal with some more exceptions. More precisely, we show that this estimate is still valid, with an implied constant depending only on $latex M$, if the matrices in $latex G_{K,M}$ are all either<\/p>\n

    \n
  1. parabolic (they have a single fixed point in $latex \\mathbf{P}^1$), or<\/li>\n
  2. upper-triangular, or<\/li>\n
  3. if they either fix or permute two distinct points taken in a finite list, say containing at most $latex M$ pairs $latex \\{x,y\\}$.<\/li>\n<\/ol>\n

    Compared with the previous discussion, the subtlety here is that it can indeed happen that such matrices appear in the transformations we apply to $latex S(f;K)$, and proving that their contributions remain under control involves some rather fun analysis.<\/p>\n

    This summarizes, in a rather hurried fashion, the first part of our paper. Logically, we obtain statements which are self-contained, but which are only applicable directly in a few cases (the case $latex K(n)=\\chi(n)$ is an excellent example where this can be done). To go further, we need to use algebro-geometric tools.<\/p>\n

    And here comes a dilemma well-known to anyone who has had to present research involving two relatively distant areas of mathematics, so that specialists of one may not know the other. I can give a concise definition of the class of weights $latex K(n)$ that we consider, and it will be immediately familiar and natural to algebraic geometers — but not to most analytic number theorists.<\/p>\n

    So, instead, I will just say (very fast) that these weights, which we call “irreducible trace weights”, are trace functions of geometrically irreducible $latex \\ell$-adic middle-extension Fourier sheaves pointwise pure of weight $latex 0$ on $latex \\mathbf{A}^1\/\\mathbf{F}_p$<\/em>. Then I will defer to a later post a more leisurely run-through this definition, together with more examples of weights and their formalisms, and with some further analytic properties which are of independent interest.<\/p>\n

    The reason this class of weight “works” can however be quickly summarized in a rather miraculous property, which is essentially a consequence of the Riemann Hypothesis: assuming $latex K$ has small complexity (in a precise sense, saying basically that it comes from a sheaf with small rank and small ramification), the correlation sum $latex C(K;\\gamma)$ is either $latex \\ll p^{1\/2}$, where the implied constant depends only on the numerical invariant measuring the complexity of the weight, or we have an equality<\/em><\/p>\n

    $latex \\hat{K}(z)=\\varepsilon(\\gamma) \\hat{K}(\\gamma\\cdot z)$<\/p>\n

    for all $latex z$ and some fixed complex number $latex \\varepsilon(\\gamma)$ of modulus $latex 1$. From this second part, we see that, for a suitable $latex M$, the set<\/em> $latex G_{K,M}$ discussed above is contained in the group<\/em> $latex \\mathbf{G}_K$ of all $latex \\gamma$ (viewed in $latex \\mathrm{PGL}_2(\\mathbf{F}_p)$) for which there exists $latex \\varepsilon(\\gamma)$ (of modulus $latex 1$) such that the identity above holds. This provides us with enough structure on the set of “bad” matrices (with large correlation sums), from which the bound<\/p>\n

    $latex S(f;K)\\ll p^{1-1\/8+\\epsilon}$<\/p>\n

    can fairly simply be deduced for irreducible trace weights using the automorphic part of our paper. <\/p>\n

    Indeed, we distinguish two cases, depending on the structure of the subgroup $latex \\mathbf{G}_K\\subset \\mathrm{PGL}_2(\\mathbf{F}_p)$:<\/p>\n