{"id":3270,"date":"2012-07-07T21:03:44","date_gmt":"2012-07-07T19:03:44","guid":{"rendered":"https:\/\/wpethzprd.ethz.ch\/kowalski\/?p=3270"},"modified":"2024-02-20T13:40:35","modified_gmt":"2024-02-20T11:40:35","slug":"algebraic-twists-of-modular-forms-ii","status":"publish","type":"post","link":"https:\/\/blogs.ethz.ch\/kowalski\/2012\/07\/07\/algebraic-twists-of-modular-forms-ii\/","title":{"rendered":"Algebraic twists of modular forms, II"},"content":{"rendered":"<p>I continue here the discussion, begun in <a href=\"https:\/\/blogs.ethz.ch\/kowalski\/2012\/07\/02\/algebraic-twists-of-modular-forms-i\/\">my previous posts<\/a>, of <a href=\"http:\/\/www.math.ethz.ch\/~kowalski\/twists.pdf\">my recent work with Fouvry and Michel<\/a>. So, recall that we want to estimate a sum of the type<\/p>\n<p>$latex S(f,K)=\\sum_{n\\geq 1}{\\rho_f(n)K(n)V(n\/p)}$<\/p>\n<p>were $latex \\rho_f(n)$ denotes the Fourier coefficients of a fixed modular form $latex f$, $latex K(n)$ is some complex-valued function defined modulo a prime $latex p$ and extended to all integers by $latex p$-periodicity, and $latex V$ is a test function, compactly supported in $latex ]0,1[$<\/p>\n<p>I will present the strategy rather roughly first, and then refine it. For the moment, nothing will be assumed of $latex K(n)$.<\/p>\n<ul>\n<li><b>[Analytic step].<\/b> Using (rather sophisticated) techniques from the analytic theory of modular forms (in particular, amplification and the Kuznetsov formula), we show that one can bound from above the sum $latex S(f,K)$ using some auxiliary sums. Precisely, we first need the discrete Fourier transform of $latex K$, which we normalize by putting\n<p>$latex \\hat{K}(t)=\\frac{1}{\\sqrt{p}}\\sum_{x\\in\\mathbf{F}_p}{K(x)e\\Bigl(\\frac{tx}{p}\\Bigr)}$<\/p>\n<p>for $latex t$ modulo $latex p$ (a unitary normalization). Then, for $latex \\gamma\\in \\mathrm{PGL}_2(\\mathbf{F}_p)$, we let<\/p>\n<p>$latex C(K;\\gamma)=\\sum_{z\\in\\mathbf{F}_p}{\\overline{\\hat{K}(\\gamma\\cdot z)}\\hat{K}(z)}.$<\/p>\n<p>We call these sums <em>correlation sums<\/em>.  Using an auxiliary parameter $latex L\\geq 1$, their relation to our problem can be first expressed by<\/p>\n<p>$latex |S(f,K)|^2 \\ll p^2L^{-1}+pL\\times \\max_{\\gamma\\in \\mathrm{PGL}_2(\\mathbf{F}_p)} |C(K;\\gamma)|$<\/p>\n<p>(up to terms like $latex p^{\\varepsilon}$ and logarithms). In fact, we do not write this down, because one can see that it useless as is: for $latex \\gamma=1$, the correlation sum $latex C(K;1)$ is typically as large as $latex p$, so the right-hand side is at least of size $latex p^2$, which is essentially the trivial bound (that one can get from Rankin-Selberg theory, for $latex K$ bounded)&#8230;<\/p>\n<\/li>\n<li>\n<b>[Pruning].<\/b> This first objection to the method is however not significant. Indeed, it is clear from the proof that not all correlations sums $latex C(K;\\gamma)$ play a role when estimating $latex |S(f,K)|^2$. In particular, provided $latex L$ is not too large (say $latex L&lt;p^{1\/2}$), it is essentially immediate that one can replace the bound above with<\/p>\n<p>$latex |S(f,K)|^2 \\ll p^2L^{-1}+pL\\times \\max_{\\gamma\\notin B_p} |C(K;\\gamma)|$<\/p>\n<p>where $latex B_p$ is the stabilizer of $latex \\infty$ in the action of $latex G_p=\\mathrm{PGL}_2(\\mathbf{F}_p)$ on the projective line $latex \\mathbf{P}^1(\\mathbf{F}_p)$, i.e.<\/p>\n<p>$latex B_p=\\Bigl\\{\\begin{pmatrix}a&amp; b\\\\\\ 0&amp;d\\end{pmatrix}\\Bigr\\}\\subset G_p.$<\/p>\n<p>This means that we do not mind if some correlation sums are big, provided these &#8220;bad&#8221; sums are of specific shape, here, upper-triangular.  <\/p>\n<\/li>\n<li>\n<b>[Estimating].<\/b> Suppose we want to use the previous bound heuristically first for a weight $latex K(n)$ which is generic (or random), and bounded by $latex 1$. Then we can expect that the unitary Fourier transform will be basically bounded (by the philosophy of <a href=\"https:\/\/blogs.ethz.ch\/kowalski\/2008\/07\/07\/is-there-a-combinatorial-square-root-cancellation-philosophy\/\">square root cancellation<\/a> &#8212; for which, by the way, I would have expected that there existed a better informal introduction on the internet than in this four-year-old post), and will also be rather random. But then the correlation sum is also an oscillating sum of length $latex p$, and optimistically, should be of size $latex \\sqrt{p}$ for $latex \\gamma\\not=1$.  We can therefore hope to get<\/p>\n<p>$latex |S(f,K)|^2\\ll p^2L^{-1}+p^{3\/2}L.$<\/p>\n<p>The optimal choice of $latex L$ is $latex L=p^{1\/4}$, and this gives<br \/>\n$latex |S(f,K)|^2 \\ll p^{2-1\/4},\\quad\\quad\\text{ or } \\quad\\quad S(f,K)\\ll p^{1-1\/8}.$<\/p>\n<p>This is the limit of the method.\n<\/li>\n<\/ul>\n<p>But can we really apply this approach to concrete functions $latex K(n)$?  The reader will see that the quality of the bound here is exactly (up to epsilons) what I stated in the last post for cases like<\/p>\n<p>$latex K_1(n)=\\chi(n),\\quad\\quad K_2(n)=e\\left(\\frac{\\bar{n}}{p}\\right).$<\/p>\n<p>However, these two cases are in fact rather different.  In the second case, spelling out the definition of $latex C(K_2,\\gamma)$, it is easy to see that the correlations sums are, up to a factor $latex 1\/p$, additive exponential sums in three variables (when $latex \\gamma$ is anti-diagonal, it is a Kloosterman sum in three variables).  For these, we show that there exists $latex M\\geq 1$ such that<\/p>\n<p>$latex |C(K_2,\\gamma)|\\leq Mp^{1\/2}$<\/p>\n<p>for all $latex \\gamma\\not=1$, so that the argument above applies (this is a case of the Riemann Hypothesis over finite fields with optimal cancellation for sums in three variables; when $latex \\gamma$ is restricted to anti-diagonal matrices, the fundamental result of Deligne on hyper-Kloosterman sums shows that one can take $latex M=4$).<\/p>\n<p>But for $latex K_1(n)$, things are different. Here it is easy to analyze the correlation sums because the Fourier transform of $latex \\chi(n)$, for a non-trivial character modulo $latex p$, is a multiple of $latex \\bar{\\chi}(t)$. Hence $latex C(K_1,\\gamma)$ is a one-variable multiplicative character sum, and can be analyzed using Weil&#8217;s methods. One finds that there exists an absolute constant $latex M\\geq 1$ such that <\/p>\n<p>$latex |C(K_1,\\gamma)|\\leq Mp^{1\/2}$<\/p>\n<p>except if either $latex \\chi$ is non-real, and $latex \\gamma$ is diagonal, or if $latex \\chi$ is real, and $latex \\gamma$ is either diagonal or anti-diagonal (and these are genuine exceptions, if $latex M$ is supposed to be an absolute constant).  <\/p>\n<p>The first case lies within the realm of the previous discussion, but when $latex \\chi$ is a real character, this is not true anymore.<\/p>\n<p>Similarly, if<\/p>\n<p>$latex K_3(n)=\\frac{S(n,n;p)^2}{p}-1,$<\/p>\n<p>as at the end of the previous post, one finds that the Fourier transform $latex \\hat{K}_3(t)$ is exactly the coefficient $latex a_p(t)$ such that<\/p>\n<p>$latex |E_t(\\mathbf{F}_p)|=p+1-a_p(t),$<\/p>\n<p>where $latex E_t$ is the elliptic curve which was discussed in <a href=\"https:\/\/blogs.ethz.ch\/kowalski\/2012\/06\/29\/a-bijective-challenge\/\">the first post in this series<\/a>. The relations between $latex |E_t(\\mathbf{F}_p)|$ and $latex |E_{\\gamma\\cdot t}(\\mathbf{F}_p)|$ mentioned there imply that the correlation sums $latex C(K_3,\\gamma)$ are of size $latex p$ for $latex \\gamma$ in a certain dihedral group $latex D$ of order $latex 8$, namely the setwise stabilizer of $latex \\{-4,4,0,\\infty\\}$ in $latex \\mathrm{PGL}_2$. Furthermore, one shows that there exists an absolute constant $latex M\\geq 1$ such that<\/p>\n<p>$latex |C(K_3,\\gamma)|\\leq Mp^{1\/2}$<\/p>\n<p>for all $latex \\gamma\\notin D$. But $latex D$ is not upper-triangular, and hence the first argument we sketched is also insufficient here&#8230;<\/p>\n<p>I will stop here for now; in the next post, we have two obvious questions to discuss: (1) how does one deal with the complications which arose for $latex K_2(n)$ and $latex K_3(n)$? and (2) even in the most favorable case, how exactly can one show that the correlation sums are small?  As can be expected already from today&#8217;s discussion, it is the Riemann Hypothesis over finite fields which is crucial.  In fact, to attain the generality of our results, we need to apply it <em>twice<\/em>; in both cases we require the very deep statements proved by Deligne in the amazing achievement which people call <a href=\"http:\/\/numdam.org\/numdam-bin\/fitem?id=PMIHES_1980__52__137_0\"><em>Weil 2<\/em><\/a>, but in one case, this is encapsulated in the <a href=\"http:\/\/en.wikipedia.org\/wiki\/Fourier%E2%80%93Deligne_transform\">Deligne<\/a>&#8211;<a href=\"http:\/\/numdam.org\/numdam-bin\/fitem?id=PMIHES_1985__62__145_0\">Katz<\/a>&#8211;<a href=\"http:\/\/numdam.org\/numdam-bin\/fitem?id=PMIHES_1987__65__131_0\">Laumon<\/a> theory of the $latex \\ell$-adic Fourier transform&#8230;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>I continue here the discussion, begun in my previous posts, of my recent work with Fouvry and Michel. So, recall that we want to estimate a sum of the type $latex S(f,K)=\\sum_{n\\geq 1}{\\rho_f(n)K(n)V(n\/p)}$ were $latex \\rho_f(n)$ denotes the Fourier coefficients of a fixed modular form $latex f$, $latex K(n)$ is some complex-valued function defined modulo &hellip; <a href=\"https:\/\/blogs.ethz.ch\/kowalski\/2012\/07\/07\/algebraic-twists-of-modular-forms-ii\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Algebraic twists of modular forms, II<\/span><\/a><\/p>\n","protected":false},"author":625,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[],"class_list":["post-3270","post","type-post","status-publish","format-standard","hentry","category-blogroll"],"_links":{"self":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts\/3270","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/users\/625"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/comments?post=3270"}],"version-history":[{"count":0,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts\/3270\/revisions"}],"wp:attachment":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/media?parent=3270"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/categories?post=3270"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/tags?post=3270"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}