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zTp`*}!C1i%C>WHVUjh^~1Ezr>E`86uw0t1X9LjT104_6da&~mFFm`dWbTxN%wluLc zGjw(`GBI~_HFhyIH*>U8AgqKJXpv89UV2G}f{8IC?WkVd=`Wfn!up{)ZQsL#$-xue zGWxRwIzIh5AvCPe)gjC^AaV(#le5!{H5ooW9vT`75x@8~R6+!#q>?nJT@w)E5?U0L zaBktxr*mfKznebO(AfI>nIH$Aw3o+B9NbPsf9mp?;ArG`Px-{H2M@2Zgw3xosCO}T z-%>JVkDBwGN}u}t6V^fk?59ocN-J@0@REsyOt`*IZ#+pBna3a8LjJsd9&-e=h&Ra_?vQ(&Zl(i`V~>{Buuv>GKca z_1aqx|1Vtjh3VhJg+c5wueyTNW6WY1Zs=E5SaRC6>;~!`M diff --git a/number-theory/langlands-functoriality/ref.bib b/number-theory/langlands-functoriality/ref.bib index 640736e..7883ab2 100644 --- a/number-theory/langlands-functoriality/ref.bib +++ b/number-theory/langlands-functoriality/ref.bib @@ -275,4 +275,45 @@ @article{fite2014sato pages = {543--585}, year = {2014}, publisher = {Mathematical Sciences Publishers} +} + +@inproceedings{satake1966spherical, + title = {Spherical functions and Ramanujan conjecture}, + author = {Satake, Ichir{\^o}}, + booktitle = {Proc. sympos. pure math}, + volume = {9}, + pages = {258--264}, + year = {1966} +} + +@article{kim2003refined, + title = {Refined estimates towards the Ramanujan and Selberg conjectures}, + author = {Kim, Henry and Sarnak, Peter}, + journal = {J. Amer. Math. Soc}, + volume = {16}, + number = {1}, + pages = {175--181}, + year = {2003} +} + +@article{kim1999langlands, + title = {Langlands-Shahidi method and poles of automorphic L-functions: application to exterior square L-functions}, + author = {Kim, Henry H}, + journal = {Canadian Journal of Mathematics}, + volume = {51}, + number = {4}, + pages = {835--849}, + year = {1999}, + publisher = {Cambridge University Press} +} + +@article{shahidi1988ramanujan, + title = {On the Ramanujan conjecture and finiteness of poles for certain L-functions}, + author = {Shahidi, Freydoon}, + journal = {Annals of Mathematics}, + volume = {127}, + number = {3}, + pages = {547--584}, + year = {1988}, + publisher = {JSTOR} } \ No newline at end of file diff --git a/number-theory/langlands-functoriality/symmetricpower.tex b/number-theory/langlands-functoriality/symmetricpower.tex index 3fa4a39..bf858cc 100644 --- a/number-theory/langlands-functoriality/symmetricpower.tex +++ b/number-theory/langlands-functoriality/symmetricpower.tex @@ -370,6 +370,44 @@ \subsection{Ramanujan's conjecture and Selberg's 1/4 conjecture} The importance of symmetric power lifting is due to it's application on Ramanujan's conjecture and Selberg's eigenvalue conjecture. +In 1916, Ramanujan conjectured that the Fourier coefficients $\tau(n)$ of disciminant function +$$ +\Delta(z) = q\prod_{n\geq 1}(1 - q^{n})^{24} = \sum_{n\geq 1}\tau(n)q^{n}, \qquad q = e^{2\pi i z} +$$ +(which is a weight 12 and level 1 holomorphic cusp form) satisfy +\begin{enumerate} + \item $\tau(mn) = \tau(m)\tau(n)$ for $(m, n) =1$ (i.e. $\tau$ is multiplicative), + \item $\tau(p^{k+1}) = \tau(p)\tau(p^{k}) - p^{11}\tau(p^{k-1})$ for prime $p$ and $k\geq 1$, + \item $|\tau(p)|\leq p^{11/2}$. +\end{enumerate} +The first two statements were proved by Mordell in 1917, and Hecke showed that +similar phenomena can be found in other modular forms, by defining so-called \emph{Hecke operators}. +Third statement remained as a conjecture until 1974 when it was proved by Deligne +as a consequence of his proof of Weil's conjecture. +Weil's conjecture, which is an analogue of Riemann's hypothesis for varieties over finite fields, +states that the \emph{Zeta function} of a curve is a rational function which can be written as +an alternating product of certain polynomials, and the zeros of each polynomial have the same norm. + +In \cite{satake1966spherical}, Satake formulated the Ramanujan's conjecture +in terms of automorphic representations. +He conjectured that, for any automorphic representation $\pi = \otimes_{v}\pi_{v}$ of $\GL(2)$, +the local components $\pi_{v}$ are \emph{tempered}, i.e. the matrix coefficients +of $\pi_{v}$ are in $L^{2+\epsilon}(\GL(2, F_{v}))$ for all $\epsilon > 0$. +For example, let's consider the automorphic representation $\pi = \pi_{\Delta}$ associated +to the disciminant function $\Delta$. +For $p\neq 2, 3$, $\pi_{p}$ is an unramified principal series representation, which is +\begin{align*} +\pi(\chi_{1}, \chi_{2}) &= \mathrm{Ind}_{B(\mathbb{Q}_{p})}^{\GL(2, \mathbb{Q}_{p})}(\chi_{1}\boxtimes \chi_{2})\\ +&= \left\{f: \GL(2, F_{v})\to\mathbb{C}:f(bg) = \bigg|\frac{b_{1}}{b_{2}}\bigg|^{\frac{1}{2}}\chi_{1}(b_{1})\chi_{2}(b_{2})f(g),b = \begin{pmatrix}b_{1}&*\\&b_{2}\end{pmatrix}\right\} +\end{align*} +where $\chi_{1}, \chi_{2}: \mathbb{Q}_{p}^{\times}\to \mathbb{C}^{\times}$ are unramified characters ($\mathbb{Z}_{p}^{\times}\subseteq \ker\chi_{i}$) +with $\chi_{i}(p) = \alpha_{i}$ where $\alpha_{1}, \alpha_{2}$ are zeros of the polynomial $x^{2} - \tau(p)x + p^{11}$. + + +Several authors found counterexamples in some groups like $\mathrm{U}(2, 1)$ and $\mathrm{Sp}(4)$ +by constructing corresponding automorphic forms that are non-tempered almost everywhere. + + \begin{conjecture}[Selberg's 1/4 conjecture] For any Maass form on a congruence subgroup $\Gamma \subseteq \mathrm{SL}(2, \mathbb{Z})$, its eigenvalue is at least $1/4$. \end{conjecture} @@ -408,7 +446,6 @@ \subsection{Ramanujan's conjecture and Selberg's 1/4 conjecture} \end{proof} Combined with Theorem \ref{quarticpow}, Proposition \ref{powerselberg} gives the current best bound for the Selberg's conjecture. - \begin{corollary} Eigenvalus of Maass forms on a congruence subgroup is at least $$ @@ -416,6 +453,46 @@ \subsection{Ramanujan's conjecture and Selberg's 1/4 conjecture} $$ \end{corollary} +The current best known bound for $\GL(2, \mathbb{A}_{\mathbb{Q}})$ is the following result from Kim and Sarnak \cite{kim2003refined}. + +\begin{theorem} +Let $\pi = \otimes_{p}'\pi_{p}$ be an automorphic cusp form on $\GL(2, \mathbb{A}_{\mathbb{Q}})$. +If $\pi$ is unramified at $p$, then its Satake parameter $\alpha_{1, p}, \alpha_{2, p}$ satisfies +$$ +|\log_{p}|\alpha_{j, p}|| \leq \frac{7}{64},\,\, j = 1, 2. +$$ +If $\pi_{\infty}$ is unramified, then +$$ +|\Re(\mu_{j, \infty})|\leq \frac{7}{64},\,\,j = 1, 2. +$$ +\end{theorem} +In terms of classical Masss forms, this implies: if $f$ is a classical Maass form, +on a congruence subgroup $\Gamma \subseteq \SL(2, \mathbb{Z})$, +then its' Fourier coefficient satisfies +$$ +|a_{p}| \leq p^{7/64} + p^{-7/64} +$$ +and its eigenvalue is at least +$$ +\frac{1}{4} - \left(\frac{7}{64}\right)^{2} = \frac{975}{4096} \approx 0.238037\cdots. +$$ +\begin{proof} +When $\Pi$ is a non self-dual $\GL(n)$ automorphic form, it is known that the symmetric square $L$-function $L(s, \Pi, \Sym^{2})$ +satisfies desired analytic properties by Kim and Shahidi \cite{kim1999langlands,shahidi1988ramanujan}. +Using this, the authors proved that the Satake parameters of unramified $\Pi_{p}$ satisfy +$$ +|\log_{p}|\alpha_{j, p}|| \leq \frac{1}{2} - \frac{1}{\frac{n(n+1)}{2} + 1},\,\, 1\leq j \leq n. +$$ +for Satake parameters of unramified $\Pi_{p}$ with $p <\infty$, and +$$ +|\Re(\mu_{j, \infty})|\leq \frac{1}{2} - \frac{1}{\frac{n(n+1)}{2} + 1},\,\, 1\leq j \leq n. +$$ +for Satake parameters $\mu_{j, \infty}$ when $\Pi_{\infty}$ is unramified. +By Theorem \ref{quarticpow}, $\Pi = \Sym^{4}(\pi)$ is automorphic +and applying the above bound for $\Pi$ gives the desired bound. +\end{proof} + + \subsection{Sato-Tate conjecture} Another consequence of symmetric power lifting is Sato-Tate conjecture. @@ -438,9 +515,9 @@ \subsection{Sato-Tate conjecture} \label{sato-tate} Let $E/\mathbb{Q}$ be an elliptic curve without CM. The sequence of Frobenius angles $\{\theta_{p}\}$ is uniformly distributed on the -interval $[0, \pi]$. In terms of traces $\{x_{p}\}$, for every subinterval $[a, b]$ of $[-2, 2]$, +interval $[0, \pi]$. In terms of traces $\{t_{p}\}$, for every subinterval $[a, b]$ of $[-2, 2]$, $$ -\lim_{B\to \infty} \frac{\#\{p\leq B\,:\, x_{p} \in [a, b]\}}{\#\{p\leq B\}} = \int_{a}^{b} \frac{1}{2\pi} \sqrt{4 - t^{2}} dt. +\lim_{B\to \infty} \frac{\#\{p\leq B\,:\, t_{p} \in [a, b]\}}{\#\{p\leq B\}} = \int_{a}^{b} \frac{1}{2\pi} \sqrt{4 - t^{2}} dt. $$ \end{theorem} From e52739594334563ff115cc6209aa1f4e7ec5716d Mon Sep 17 00:00:00 2001 From: Seewoo Lee Date: Sat, 5 Mar 2022 18:55:14 +0900 Subject: [PATCH 2/4] more about ramanujan conj, need to add explanation about temperedness --- .../langlands-functoriality/main.pdf | Bin 348962 -> 361132 bytes number-theory/langlands-functoriality/ref.bib | 109 +++++++++++++++++- .../symmetricpower.tex | 107 +++++++++++------ 3 files changed, 181 insertions(+), 35 deletions(-) diff --git a/number-theory/langlands-functoriality/main.pdf b/number-theory/langlands-functoriality/main.pdf index 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b/number-theory/langlands-functoriality/ref.bib @@ -168,7 +168,7 @@ @article{ramakrishnan2000modularity } @article{labesse1979indistinguishability, - title = {L-indistinguishability for SL (2)}, + title = {L-indistinguishability for $\mathrm{SL}(2)$}, author = {Labesse, Jean-Pierre and Langlands, Robert P}, journal = {Canadian Journal of Mathematics}, volume = {31}, @@ -316,4 +316,111 @@ @article{shahidi1988ramanujan pages = {547--584}, year = {1988}, publisher = {JSTOR} +} + +@article{jacquet1981euler, + title = {On Euler products and the classification of automorphic representations I}, + author = {Jacquet, Herv{\'e} and Shalika, Joseph A}, + journal = {American Journal of Mathematics}, + volume = {103}, + number = {3}, + pages = {499--558}, + year = {1981}, + publisher = {JSTOR} +} + +@article{winnie2020ramanujan, + title = {The Ramanujan conjecture and its applications}, + author = {Winnie Li, Wen-Ching}, + journal = {Philosophical Transactions of the Royal Society A}, + volume = {378}, + number = {2163}, + pages = {20180441}, + year = {2020}, + publisher = {The Royal Society Publishing} +} + +@article{cohen1988q, + title = {q-identities for Maass waveforms}, + author = {Cohen, Henri}, + journal = {Inventiones mathematicae}, + volume = {91}, + number = {3}, + pages = {409--422}, + year = {1988}, + publisher = {Springer} +} + +@misc{buzzard2012explicit, + title = {Explicit Maass forms}, + author = {Buzzard, Kevin}, + howpublished = {\url{https://www.ma.imperial.ac.uk/~buzzard/maths/research/notes/explicit_maass_forms.pdf}}, +} + +@article{jacquet1979automorphic1, + title = {Automorphic forms on $\mathrm{GL}(3)$ I}, + author = {Jacquet, Herv{\'e} and Piatetski-Shapiro, Ilja Iosifovitch and Shalika, Joseph}, + journal = {Annals of Mathematics}, + volume = {109}, + number = {1}, + pages = {169--212}, + year = {1979}, + publisher = {JSTOR} +} + +@article{jacquet1979automorphic2, + title = {Automorphic forms on $\mathrm{GL}(3)$ II}, + author = {Jacquet, Herv{\'e} and Piatetski-Shapiro, Ilja Iosifovitch and Shalika, Joseph}, + journal = {Annals of Mathematics}, + volume = {109}, + number = {2}, + pages = {213--258}, + year = {1979}, + publisher = {JSTOR} +} + +@article{kim2002cuspidality, + title = {Cuspidality of symmetric powers with applications}, + author = {Kim, Henry H and Shahidi, Freydoon}, + journal = {Duke Mathematical Journal}, + volume = {112}, + number = {1}, + pages = {177--197}, + year = {2002}, + publisher = {Duke University Press} +} + +@article{blomer2013role, + title = {The role of the Ramanujan conjecture in analytic number theory}, + author = {Blomer, Valentin and Brumley, Farrell}, + journal = {Bulletin of the American Mathematical Society}, + volume = {50}, + number = {2}, + pages = {267--320}, + year = {2013} +} + +@article{blomer2011ramanujan, + title = {On the Ramanujan conjecture over number fields}, + author = {Blomer, Valentin and Brumley, Farrell}, + journal = {Annals of Mathematics}, + pages = {581--605}, + year = {2011}, + publisher = {JSTOR} +} + +@inproceedings{deligne1974formes, + title = {Formes modulaires de poids $1$}, + author = {Deligne, Pierre and Serre, Jean-Pierre}, + booktitle = {Annales scientifiques de l'{\'E}cole Normale Sup{\'e}rieure}, + volume = {7}, + pages = {507--530}, + year = {1974} +} + +@inproceedings{deligne1971formes, + title = {Formes modulaires et repr{\'e}sentations $\ell$-adiques}, + author = {Deligne, Pierre}, + booktitle = {S{\'e}minaire Bourbaki. Vol. 1968/69: Expos{\'e}s 347--363}, + year = {1971} } \ No newline at end of file diff --git a/number-theory/langlands-functoriality/symmetricpower.tex b/number-theory/langlands-functoriality/symmetricpower.tex index bf858cc..e020c03 100644 --- a/number-theory/langlands-functoriality/symmetricpower.tex +++ b/number-theory/langlands-functoriality/symmetricpower.tex @@ -203,7 +203,7 @@ \subsection{Automorphic forms on $\mathrm{GL}(3, \mathbb{R})$} Let $\mathcal{H} = \mathbb{Z}[\Gamma \backslash G / \Gamma]$ be a $\mathbb{Z}$-algebra of double cosets where $G = \GL(3, \mathbb{R})$ and $\Gamma = \GL(3, \mathbb{Z})$, which is called \emph{Hecke algebra}. It decomposes as a (internal) tensor product $$ -\mathcal{H} = \bigotimes_{p} \mathcal{H}_{p} +\mathcal{H} = \bigotimes_{p}' \mathcal{H}_{p} $$ where $\mathcal{H}_{p}$ is a subalgebra of $\mathcal{H}$ corresponding to the double cosets whose elementary divisors are powers of a given prime $p$. For each prime $p$, $\mathcal{H}_{p}$ is generated by three elements @@ -299,7 +299,7 @@ \subsection{Symmetric square lifting by Gelbart-Jacquet} $$ Now we will interpret the situation in the context of representation theory. -Let $\pi = \otimes_{v} \pi_{v}$ be an automorphic representation of $\GL_{2}(\mathbb{A})$, and let $\varphi_{v}$ be the 2-dimensional representations of the Weil-Deligne group $W_{v}:=W_{F_{v}}$ of $F_{v}$ attached to $\pi_{v}$ via Local Langlands correspondence. +Let $\pi = \otimes_{v}' \pi_{v}$ be an automorphic representation of $\GL_{2}(\mathbb{A})$, and let $\varphi_{v}$ be the 2-dimensional representations of the Weil-Deligne group $W_{v}:=W_{F_{v}}$ of $F_{v}$ attached to $\pi_{v}$ via Local Langlands correspondence. The symmetric square representation of $\GL(2, \mathbb{C})$ \begin{align*} \Sym^{2}: \GL(2, \mathbb{C}) \to \GL(3, \mathbb{C}), \qquad @@ -316,8 +316,15 @@ \subsection{Symmetric square lifting by Gelbart-Jacquet} % It is cuspidal unless $\pi$ is monomial, i.e. $\pi \not\simeq \pi \otimes \eta$ where $\eta\neq 1$ is a gr\"ossencharacter of $F$. \end{theorem} -% Here we give a sketch of proof. -% Basically, the proof uses converse theorem for $\GL_{3}$ proven by +The idea of the proof is based on Shimura's proof of analytic continutation and functional equation +for symmetric square $L$-function $L(s, \Sym^{2}f)$ of modular forms \cite{shimura1973onmodular}, +where he used Rankin-Selberg convolution of $f$ with theta series of half-integral weight. +In the context of representation theory, theta functions correspond to the automorphic representations +on \emph{metaplectic groups}, which are double cover of symplectic groups. +(See Chapter 7 for more details about half-integral modular forms.) +This proves analytic properties of $L(s, \Sym^{2}(\pi))$ (and its twists), and $\GL(3)$ converse theorem by +Jacquet, Piatetski-Shapiro and Shalika \cite{jacquet1979automorphic1,jacquet1979automorphic2} +concludes the proof. In \cite{ramakrishnan2014exercise}, Ramakrishnan proved the following converse of the Gelbart-Jacquet, by using $L$-functions. \begin{theorem}[Ramakrishnan, \cite{ramakrishnan2014exercise}] @@ -359,7 +366,8 @@ \subsection{Higher symmetric power} \wedge^{2}(\Sym^{3}(\pi) \otimes \omega_{\pi}^{-1}) = (\Sym^{4}(\pi) \otimes \omega_{\pi}^{-1}) \boxplus \omega_{\pi}. $$ -Recently, it is proved that the functoriality holds for arbitrary power when $\pi$ is a \emph{regular algebraic cuspidal} representation, which corresponds to twists of cuspidal modular forms. +Recently, it was proven that the functoriality holds for arbitrary power when $\pi$ is a \emph{regular algebraic cuspidal} representation, +which corresponds to twists of cuspidal modular forms. \begin{theorem}[Newton-Thorne \cite{newton2021symmetric, newton2021symmetric2}] Let $\pi$ be a regular algebraic cuspidal representation of $\GL(2, \mathbb{A}_{\mathbb{Q}})$ of level 1, or without complex multiplication. For any $n\geq 1$, $\mathrm{Sym}^{n}(f)$ is a regular algebraic cuspidal representation of $\GL(n + 1, \mathbb{A})$. @@ -380,17 +388,36 @@ \subsection{Ramanujan's conjecture and Selberg's 1/4 conjecture} \item $\tau(p^{k+1}) = \tau(p)\tau(p^{k}) - p^{11}\tau(p^{k-1})$ for prime $p$ and $k\geq 1$, \item $|\tau(p)|\leq p^{11/2}$. \end{enumerate} -The first two statements were proved by Mordell in 1917, and Hecke showed that +The first two statements were first proved by Mordell in 1917, and Hecke showed that similar phenomena can be found in other modular forms, by defining so-called \emph{Hecke operators}. Third statement remained as a conjecture until 1974 when it was proved by Deligne -as a consequence of his proof of Weil's conjecture. -Weil's conjecture, which is an analogue of Riemann's hypothesis for varieties over finite fields, -states that the \emph{Zeta function} of a curve is a rational function which can be written as +as a consequence of his proof of Weil's conjecture, an analogue of Riemann's hypothesis for varieties over finite fields +which states that the \emph{Zeta function} of a variety is a rational function which can be written as an alternating product of certain polynomials, and the zeros of each polynomial have the same norm. +In fact, he proved that every normalized eigenforms have similar bound for their Fourier coefficients: +$|a_{p}| \leq 2p^{\frac{k-1}{2}}$ for weight $k$ modular forms. +Deligne himself proved it for $k\geq 2$ \cite{deligne1971formes}, +and weight 1 case was proven by Deligne and Serre \cite{deligne1974formes}. -In \cite{satake1966spherical}, Satake formulated the Ramanujan's conjecture + +For Maass forms, there is a well-known conjecture on eigenvalues by Selberg. +\begin{conjecture}[Selberg's 1/4 conjecture] +For any Maass form on a congruence subgroup $\Gamma \subseteq \mathrm{SL}(2, \mathbb{Z})$, its eigenvalue is at least $1/4$. +\end{conjecture} +Selberg himself proved a weaker bound $3/16$ for $\Gamma = \Gamma(N)$ in \cite{selberg1965estimation}. +The conjecture is false for non-congruence subgroups (See \cite{sarnak1995selberg}). +Also, it is widely believed that the Maass forms with eigenvalue $1/4$ are \emph{algebraic}, in a sense that they comes from \emph{even} 2-dimensional Galois representations. +For example, Cohen constructed an explicit Maass form of level 4 with eigenvalue $\frac{1}{4}$ +that is originated from one of $q$-series in Ramanujan's lost notebook (this is actually the first explicitly known +example of Maass cusp form), +and proved that its $L$-function equals to certain \emph{even} 2-dimensional Galois representation +induced from a Hecke character of $\mathbb{Q}(\sqrt{6})$ \cite{cohen1988q}. +Also, once can find additional computational examples from Buzzard's note \cite{buzzard2012explicit}. + + +In \cite{satake1966spherical}, Satake formulated both Ramanujan's conjecture and Selberg's conjecture in terms of automorphic representations. -He conjectured that, for any automorphic representation $\pi = \otimes_{v}\pi_{v}$ of $\GL(2)$, +He conjectured that, for any automorphic representation $\pi = \otimes_{v}'\pi_{v}$ of $\GL(2)$, the local components $\pi_{v}$ are \emph{tempered}, i.e. the matrix coefficients of $\pi_{v}$ are in $L^{2+\epsilon}(\GL(2, F_{v}))$ for all $\epsilon > 0$. For example, let's consider the automorphic representation $\pi = \pi_{\Delta}$ associated @@ -406,28 +433,30 @@ \subsection{Ramanujan's conjecture and Selberg's 1/4 conjecture} Several authors found counterexamples in some groups like $\mathrm{U}(2, 1)$ and $\mathrm{Sp}(4)$ by constructing corresponding automorphic forms that are non-tempered almost everywhere. +In this case, the correct version of Ramanujan's conjecture is for \emph{generic} +representations, i.e. automorphic representations which has a Whittaker model. +(Note that all the automorphic representations of $\GL(n)$ are generic.) -\begin{conjecture}[Selberg's 1/4 conjecture] -For any Maass form on a congruence subgroup $\Gamma \subseteq \mathrm{SL}(2, \mathbb{Z})$, its eigenvalue is at least $1/4$. -\end{conjecture} -It is known that the conjecture is false for non-congruence subgroups (See \cite{sarnak1995selberg} for Sarnak's argument). -Also, it is widely believed that the Maass forms with eigenvalue $1/4$ are \emph{algebraic}, in a sense that they comes from \emph{even} 2-dimensional Galois representations. -Selberg himself proved a weaker bound $3/16$ for $\Gamma = \Gamma(N)$ in \cite{selberg1965estimation}, - \begin{proposition} \label{powerselberg} Assume that symmetric power lifting holds for arbitrary power, i.e. for any cuspidal automorphic representation $\pi$ on $\GL(2,\mathbb{A})$, $\Sym^{r}(\pi)$ is an automorphic representation of $\GL(r + 1, \mathbb{A})$ for any $r$. Then the Ramanujan's conjecture and Selberg's conjecture are true. \end{proposition} \begin{proof} -By Jacquet-Shalika [REFERENCE], it is proven that the Satake parameters of automorphic forms of $\GL(n, \mathbb{A})$ satisfy +By Jacquet-Shalika \cite{jacquet1981euler}, it was proven that the Satake parameters of automorphic forms of $\GL(n, \mathbb{A})$ satisfy +$$ +q_{v}^{-1/2} < |\alpha_{j,v}| < q_{v}^{1/2} $$ -q_{v}^{-1/2} < |\alpha_{i,v}| < q_{v}^{1/2} +for all $1\leq j\leq n$ and finite places $v$ where $\pi_{v}$ is unramified. +Similarly, one has $$ -for all $1\leq i\leq n$ and unramified places $v$ (including archimedean places). +|\Re(\mu_{j, v})| < \frac{1}{2} +$$ +for unramified archimedean places $v$. +This follows from non-vanishing result of Rankin-Selberg $L$-function $L(s, \pi \times \pi^{\vee})$ proved by the authors in \cite{jacquet1981euler}. Now, assume that symmetric power lifting holds for arbitrary power. -If $\Pi = \otimes_{v}\Pi_{v} = \Sym^{r}(\pi)$ is the corresponding representation, then the Satake parameters at place $v$ are given as +If $\Pi = \otimes_{v}'\Pi_{v} = \Sym^{r}(\pi)$ is the corresponding representation, then the Satake parameters at place $v$ are given as $$ \begin{pmatrix} \alpha_{1, v}^{r} & & & & \\ @@ -439,17 +468,25 @@ \subsection{Ramanujan's conjecture and Selberg's 1/4 conjecture} $$ and Jacquet-Shalika's bound gives $$ -q_{v}^{-1/2} < |\alpha_{i, v}^{r}| < q_{v}^{1/2} \Longleftrightarrow q_{v}^{-1/2r} < |\alpha_{i, v}| < q_{v}^{1/2r} +q_{v}^{-1/2} < |\alpha_{j, v}^{r}| < q_{v}^{1/2} \Longleftrightarrow q_{v}^{-1/2r} < |\alpha_{j, v}| < q_{v}^{1/2r} +$$ +for unramified finite places $v$ and Similarly $$ -for all $r\geq 1$. Now taking the limit $r\to \infty$ proves both conjecture. -% For details, see \cite{sarnak2005notes}. +|\Re(\mu_{j, v})| < \frac{1}{2r} +$$ +for unramified archimedean places $v$. +Now taking the limit $r\to \infty$ proves both conjecture. \end{proof} -Combined with Theorem \ref{quarticpow}, Proposition \ref{powerselberg} gives the current best bound for the Selberg's conjecture. +Combined with Theorem \ref{quarticpow}, Proposition \ref{powerselberg} gives the following bound. \begin{corollary} -Eigenvalus of Maass forms on a congruence subgroup is at least +Let $f$ be a Maass form. Its' Fourier coefficient is bounded by +$$ +|a_{p}| \leq p^{1/8} + p^{-1/8}. $$ -\frac{1}{4} - \left(\frac{7}{64}\right)^{2} = \frac{975}{4096} \approx 0.238037\dots +Also, eigenvalue of $f$ is at least +$$ +\frac{1}{4} - \left(\frac{1}{8}\right)^{2} = \frac{15}{64} = 0.234375 $$ \end{corollary} @@ -466,9 +503,9 @@ \subsection{Ramanujan's conjecture and Selberg's 1/4 conjecture} |\Re(\mu_{j, \infty})|\leq \frac{7}{64},\,\,j = 1, 2. $$ \end{theorem} -In terms of classical Masss forms, this implies: if $f$ is a classical Maass form, +If $f$ is a classical Maass form on a congruence subgroup $\Gamma \subseteq \SL(2, \mathbb{Z})$, -then its' Fourier coefficient satisfies +then the above inequality gives the following bound on Fourier coefficients $$ |a_{p}| \leq p^{7/64} + p^{-7/64} $$ @@ -477,9 +514,9 @@ \subsection{Ramanujan's conjecture and Selberg's 1/4 conjecture} \frac{1}{4} - \left(\frac{7}{64}\right)^{2} = \frac{975}{4096} \approx 0.238037\cdots. $$ \begin{proof} -When $\Pi$ is a non self-dual $\GL(n)$ automorphic form, it is known that the symmetric square $L$-function $L(s, \Pi, \Sym^{2})$ +When $\Pi$ is a $\GL(n)$ automorphic form, it is known that the symmetric square $L$-function $L(s, \Sym^{2}(\Pi))$ satisfies desired analytic properties by Kim and Shahidi \cite{kim1999langlands,shahidi1988ramanujan}. -Using this, the authors proved that the Satake parameters of unramified $\Pi_{p}$ satisfy +Using this, the authors proved $$ |\log_{p}|\alpha_{j, p}|| \leq \frac{1}{2} - \frac{1}{\frac{n(n+1)}{2} + 1},\,\, 1\leq j \leq n. $$ @@ -489,9 +526,11 @@ \subsection{Ramanujan's conjecture and Selberg's 1/4 conjecture} $$ for Satake parameters $\mu_{j, \infty}$ when $\Pi_{\infty}$ is unramified. By Theorem \ref{quarticpow}, $\Pi = \Sym^{4}(\pi)$ is automorphic -and applying the above bound for $\Pi$ gives the desired bound. +and applying the above bound for $\Pi$ with $n = 5$ gives the desired results. \end{proof} - +The same $7/64$ bound also holds over general number fields, due to Blomer and Brumley \cite{blomer2011ramanujan}. +We refer to \cite{sarnak2005notes,blomer2013role,winnie2020ramanujan} for further discussion +about Ramanujan's conjecture. \subsection{Sato-Tate conjecture} From 67027e65e61077a76760e9c382249b85ff7cf03d Mon Sep 17 00:00:00 2001 From: Seewoo Lee Date: Fri, 26 Aug 2022 09:15:31 -0700 Subject: [PATCH 3/4] WIP: jacquet-langlands, basis problem --- .../jacquetlanglands.tex | 61 +++++++++++++++++- .../langlands-functoriality/main.pdf | Bin 361132 -> 374719 bytes .../langlands-functoriality/main.tex | 1 + number-theory/langlands-functoriality/ref.bib | 30 +++++++++ 4 files changed, 89 insertions(+), 3 deletions(-) diff --git a/number-theory/langlands-functoriality/jacquetlanglands.tex b/number-theory/langlands-functoriality/jacquetlanglands.tex index 5aac07b..91e7529 100644 --- a/number-theory/langlands-functoriality/jacquetlanglands.tex +++ b/number-theory/langlands-functoriality/jacquetlanglands.tex @@ -1,7 +1,62 @@ \newpage \section{Jacquet-Langlands correspondence} -\subsection{Quaternionic modular forms and Basis problem} -\cite{martin2020basis} +\subsection{Basis problem and quaternionic modular forms} -\subsection{Jacquet-Langlands correspondence} \ No newline at end of file +Here's a fundamental question that we can ask about modular forms. + +\begin{question} +Find a basis of a space $S_{k}(N)$ for given $k, N$. +\end{question} + +For $N = 1$, it is possible to construct a basis using the Eisenstein series $E_{4}$ and $E_{6}$. +In particular, they are algebraically independent over $\mathbb{C}$ (i.e. there's no nonzero polynomial $p(x, y) \in \mathbb{C}[x, y]$) +such that $p(E_{4}, E_{6}) = 0$ identically), and they generate the space of modular forms of level 1 (with trivial characters). +We have $\Delta = \frac{1}{1728}(E_{4}^{3} - E_{6}^{2})$, where $\Delta(z) = e^{2\pi i z}\prod_{n\geq 1}(1 - e^{2\pi i n z})^{24}$ is a +discriminant function, and multiplying $\Delta$ induces an isomorphism $M_{k}(1)$ to $S_{k+6}(1)$. +With $S_{k}(1) = 0$ for $k = 2, 4, 6, 8, 10$, we can inductively construct the basis of $S_{k}(1)$. +(See \cite{serre2012course} or Zagier's article in \cite{bruinier20081} for details.) + +For general $N$, Hecke \cite{Hecke1940AnalytischeAD} conjectured that one can construct a basis using \emph{theta series associated to +orders in certain quaternion algebra}. +Before we give definition of such theta series, let's introduce some notations. +For a prime $p$, let $B_{p}$ be a quaternion algebra over $\mathbb{Q}$ which is ramified at $p$ and infinity. +It known that quaternion algebra over $\mathbb{Q}$ is uniquely determined by the set of primes that ramify. + +\cite{eichler1973basis} + + +\subsection{Jacquet-Langlands correspondence} + +\subsection{Basis problem by means of Jacquet-Langlands correspondence} +In this section, we will give a solution to the basis problem by using Jacquet-Langlands correspondence. +Our goal is to prove the following theorem. +\begin{theorem}[Kimball, \cite{martin2020basis}] + Let $F$ be a totally real number field of degree $d = [F:\mathbb{Q}]$ and $B$ be a quaternion algebra over $F$ with discriminant $\mathfrak{D}$. + Let $\mathcal{O}$ be a special order of level $\mathfrak{N}$. + This means that $\mathcal{O}_{\mathfrak{p}}$ is an Eichler order for $\mathfrak{p}\nmid \mathfrak{D}$ + and $\mathcal{O}_{\mathfrak{p}}$ contains the ring of integers of a quadratic extension + $E_{\mathfrak{p}} / F_{\mathfrak{p}}$ for $\mathfrak{p} | \mathfrak{D}$, + and that the product of levels of the local orders is $\mathfrak{N}$. + Let $\mathbf{k} = (k_{1}, \dots, k_{d}) \in (\mathbb{Z}_{\geq 0})^{d}$ and $S_{\mathbf{k}}(\mathcal{O})$ + be the space of quaternionic cusp forms of level $\mathbf{k}$. + + There is a Hecke-module homomorphism + $$\mathrm{JL}: S_{\mathbf{k}}(\mathcal{O}) \to S_{\mathbf{k} + \mathbf{2}}(\mathfrak{N})$$ + such that + \begin{enumerate} + \item any newform $f \in S_{\mathbf{k} + \mathbf{2}}(\mathfrak{N})$ which is $\mathfrak{p}$-primitive + for $\mathfrak{p}|\mathfrak{D}$ is contained in the image; + \item if $v_{\mathfrak{p}}(\mathfrak{N})$ is odd for all $\mathfrak{p}|\mathfrak{D}$, + then $\mathrm{JL}$ is injective and yields an isomorphism + $$ + S_{\mathbf{k}}(\mathcal{O}) \simeq \bigoplus_{\mathfrak{d}} S_{\mathbf{k} + \mathbf{2}}^{\mathfrak{d}- new}(\mathfrak{dM}) + $$ + where $\mathfrak{d}$ runs over all divisors of $\mathfrak{N}'$ such that $v_{\mathfrak{p}}(\mathfrak{d})$ + is odd for all $\mathfrak{p} | \mathfrak{D}$, and $\mathfrak{M}$ is $\mathfrak{D}$-prime part of $\mathfrak{N}$. + Here $S_{\mathbf{k} + \mathbf{2}}^{\mathfrak{d}-new}$ is the subspace of $S_{\mathbf{k} + \mathbf{2}}(\mathfrak{N})$consisting of forms that are $\mathfrak{p}$-new + for all $\mathfrak{p} | \mathfrak{d}$. + \end{enumerate} + +\end{theorem} +This section closely follow Kimball's article \cite{martin2020basis}. \ No newline at end of file diff --git a/number-theory/langlands-functoriality/main.pdf b/number-theory/langlands-functoriality/main.pdf index df3d629516b13b3ebbbd2f19f160cdb99c5343e4..4129281755c303b335865c7410c4933a9406380f 100644 GIT binary patch delta 112402 zcmV(xK7O7?c47Ddcisvp^b4BLd6FvyDBs0RgJBML#(K1Z+w3 zC$oY;Py_bHi?VzCX|l3SeXIr?TS-Eb2?h&iVci3Rf5ugET5v(C1(>Q?cd;~}v6~IK z)?`+%z#LK`Q-+6wr1TwLcrbG&13-S;Z97kWBp+_#B7pRUFn_CNNPih*Ce3FYH@P`A z<3h!UduhPR&E_#64j>4SZ%}J}gzLbrNy28A5mJo`9pJmU@r@lW^<}6gqC{)+mlc|M zgHy-^++j{9pplxn!x4+lsM?W?B`t82WvNxD<{PsV8v~5a_VS42JRLj7mCmR|fJDN4WoW ze4Cdoek}4$H`@4}t6eZ6C}GLqdG98a6L3;Wpf>2|wtrPf9@tmwxj4}=7-;f|sV@P+ zE909tguA2mt9(l_>gTw#sTQiOal zfs@V(&VQUhdKPN~`A6EBGQ}U)_Ciad)m*8lh-8jvI8nB1He6f<2#L8qjs@Zpa!N?p 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b/number-theory/langlands-functoriality/ref.bib @@ -423,4 +423,34 @@ @inproceedings{deligne1971formes author = {Deligne, Pierre}, booktitle = {S{\'e}minaire Bourbaki. Vol. 1968/69: Expos{\'e}s 347--363}, year = {1971} +} + +@book{serre2012course, + title = {A course in arithmetic}, + author = {Serre, Jean-Pierre}, + volume = {7}, + year = {2012}, + publisher = {Springer Science \& Business Media} +} + +@article{eichler1955darstellbarkeit, + title = {{\"U}ber die Darstellbarkeit von Modulformen durch Thetareihen.}, + author = {Eichler, Martin}, + year = {1955}, + publisher = {Walter de Gruyter, Berlin/New York Berlin, New York} +} + +@incollection{eichler1973basis, + title = {The basis problem for modular forms and the traces of the Hecke operators}, + author = {Eichler, Martin}, + booktitle = {Modular functions of one variable I}, + pages = {75--152}, + year = {1973}, + publisher = {Springer} +} + +@inproceedings{Hecke1940AnalytischeAD, + title = {Analytische Arithmetik der positiven quadratischen Formen}, + author = {Erich Hecke}, + year = {1940} } \ No newline at end of file From ae47f403ecaf55f8a963d4d369869f0cd5b2e237 Mon Sep 17 00:00:00 2001 From: Seewoo Lee Date: Sun, 2 Apr 2023 19:49:11 -0700 Subject: [PATCH 4/4] reformat --- .gitignore | 10 +- .../bibliography-mimosis.tex | 177 ++++++++++++++ .../langlands-functoriality/main.pdf | Bin 374719 -> 646008 bytes .../langlands-functoriality/main.tex | 193 +++++++++++++-- .../langlands-functoriality/mimosis.cls | 220 ++++++++++++++++++ .../langlands-functoriality/src/abstract.tex | 7 + .../{ => src}/autoind.tex | 9 +- .../{ => src}/basechange.tex | 5 +- .../{ => src}/intro.tex | 2 +- .../{ => src}/jacquetlanglands.tex | 9 +- .../{ => src}/rankinselberg.tex | 9 +- .../{ => src}/symmetricpower.tex | 15 +- .../{ => src}/thetacorrespondence.tex | 14 +- .../langlands-functoriality/src/title.tex | 25 ++ 14 files changed, 644 insertions(+), 51 deletions(-) create mode 100644 number-theory/langlands-functoriality/bibliography-mimosis.tex create mode 100644 number-theory/langlands-functoriality/mimosis.cls create mode 100644 number-theory/langlands-functoriality/src/abstract.tex rename number-theory/langlands-functoriality/{ => src}/autoind.tex (95%) rename number-theory/langlands-functoriality/{ => src}/basechange.tex (95%) rename number-theory/langlands-functoriality/{ => src}/intro.tex (96%) rename number-theory/langlands-functoriality/{ => src}/jacquetlanglands.tex (95%) rename number-theory/langlands-functoriality/{ => src}/rankinselberg.tex (97%) rename number-theory/langlands-functoriality/{ => src}/symmetricpower.tex (99%) rename number-theory/langlands-functoriality/{ => src}/thetacorrespondence.tex (96%) create mode 100644 number-theory/langlands-functoriality/src/title.tex diff --git a/.gitignore b/.gitignore index 3d87dfb..42db3c2 100644 --- a/.gitignore +++ b/.gitignore @@ -7,4 +7,12 @@ *.synctex.gz *.out *.toc -.DS_Store \ No newline at end of file +.DS_Store +*.acn +*.bcf +*.glo +*.idx +*.ilg +*.ind +*.ist +*.run.xml \ No newline at end of file diff --git a/number-theory/langlands-functoriality/bibliography-mimosis.tex b/number-theory/langlands-functoriality/bibliography-mimosis.tex new file mode 100644 index 0000000..b78df4f --- /dev/null +++ b/number-theory/langlands-functoriality/bibliography-mimosis.tex @@ -0,0 +1,177 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Some adjustments to make the bibliography more clean +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% +% The subsequent commands do the following: +% - Removing the month field from the bibliography +% - Fixing the Oxford commma +% - Suppress the "in" for journal articles +% - Remove the parentheses of the year in an article +% - Delimit volume and issue of an article by a colon ":" instead of +% a dot "" +% - Use commas to separate the location of publishers from their name +% - Remove the abbreviation for technical reports +% - Display the label of bibliographic entries without brackets in the +% bibliography +% - Ensure that DOIs are followed by a non-breakable space +% - Use hair spaces between initials of authors +% - Make the font size of citations smaller +% - Fixing ordinal numbers (1st, 2nd, 3rd, and so) on by using +% superscripts + +% Remove the month field from the bibliography. It does not serve a good +% purpose, I guess. And often, it cannot be used because the journals +% have some crazy issue policies. +\AtEveryBibitem{\clearfield{month}} +\AtEveryCitekey{\clearfield{month}} + +% Fixing the Oxford comma. Not sure whether this is the proper solution. +% More information is available under [1] and [2]. +% +% [1] http://tex.stackexchange.com/questions/97712/biblatex-apa-style-is-missing-a-comma-in-the-references-why +% [2] http://tex.stackexchange.com/questions/44048/use-et-al-in-biblatex-custom-style +% +\AtBeginBibliography{% + \renewcommand*{\finalnamedelim}{% + \ifthenelse{\value{listcount} > 2}{% + \addcomma + \addspace + \bibstring{and}% + }{% + \addspace + \bibstring{and}% + } + } +} + +% Suppress "in" for journal articles. This is unnecessary in my opinion +% because the journal title is typeset in italics anyway. +\renewbibmacro{in:}{% + \ifentrytype{article} + {% + }% + % else + {% + \printtext{\bibstring{in}\intitlepunct}% + }% +} + +% Remove the parentheses for the year in an article. This removes a lot +% of undesired parentheses in the bibliography, thereby improving the +% readability. Moreover, it makes the look of the bibliography more +% consistent. +\renewbibmacro*{issue+date}{% + \setunit{\addcomma\space} + \iffieldundef{issue} + {\usebibmacro{date}} + {\printfield{issue}% + \setunit*{\addspace}% + \usebibmacro{date}}% + \newunit} + +% Delimit the volume and the number of an article by a colon instead of +% by a dot, which I consider to be more readable. +\renewbibmacro*{volume+number+eid}{% + \printfield{volume}% + \setunit*{\addcolon}% + \printfield{number}% + \setunit{\addcomma\space}% + \printfield{eid}% +} + +% Do not use a colon for the publisher location. Instead, connect +% publisher, location, and date via commas. +\renewbibmacro*{publisher+location+date}{% + \printlist{publisher}% + \setunit*{\addcomma\space}% + \printlist{location}% + \setunit*{\addcomma\space}% + \usebibmacro{date}% + \newunit% +} + +% Ditto for other entry types. +\renewbibmacro*{organization+location+date}{% + \printlist{location}% + \setunit*{\addcomma\space}% + \printlist{organization}% + \setunit*{\addcomma\space}% + \usebibmacro{date}% + \newunit% +} + +% Display the label of a bibliographic entry in bare style, without any +% brackets. I like this more than the default. +% +% Note that this is *really* the proper and official way of doing this. +\DeclareFieldFormat{labelnumberwidth}{#1\adddot} + +% Ensure that DOIs are followed by a non-breakable space. +\DeclareFieldFormat{doi}{% + \mkbibacro{DOI}\addcolon\addnbspace + \ifhyperref + {\href{http://dx.doi.org/#1}{\nolinkurl{#1}}} + % + {\nolinkurl{#1}} +} + +% Use proper hair spaces between initials as suggested by Bringhurst and +% others. +\renewcommand*\bibinitdelim {\addnbthinspace} +\renewcommand*\bibnamedelima{\addnbthinspace} +\renewcommand*\bibnamedelimb{\addnbthinspace} +\renewcommand*\bibnamedelimi{\addnbthinspace} + +% Make the font size of citations smaller. Depending on your selected +% font, you might not need this. +\usepackage{relsize} +\renewcommand*{\citesetup}{% + \biburlsetup + \relsize{-.5}% +} + +\DeclareLanguageMapping{english}{english-mimosis} + +% Make hyperlinks extend to the author name if `\textcite` is being used +% instead of another cite command. + +\DeclareFieldFormat{citehyperref}{% + % Need this to avoid nested links + \DeclareFieldAlias{bibhyperref}{noformat}% + \bibhyperref{#1}% +} + +\DeclareFieldFormat{textcitehyperref}{% + % Need this to avoid nested links + \DeclareFieldAlias{bibhyperref}{noformat}% + \bibhyperref{% + #1% + \ifbool{cbx:parens} + {\bibcloseparen\global\boolfalse{cbx:parens}} + {}% + }% +} + +\savebibmacro{cite} +\savebibmacro{textcite} + +\renewbibmacro*{cite}{% + \printtext[citehyperref]{% + \restorebibmacro{cite}% + \usebibmacro{cite}}% +} + +\renewbibmacro*{textcite}{% + \ifboolexpr{ + ( not test {\iffieldundef{prenote}} and + test {\ifnumequal{\value{citecount}}{1}} ) + or + ( not test {\iffieldundef{postnote}} and + test {\ifnumequal{\value{citecount}}{\value{citetotal}}} ) + }% + {\DeclareFieldAlias{textcitehyperref}{noformat}} + {}% + \printtext[textcitehyperref]{% + \restorebibmacro{textcite}% + \usebibmacro{textcite}}% +} diff --git a/number-theory/langlands-functoriality/main.pdf b/number-theory/langlands-functoriality/main.pdf index 4129281755c303b335865c7410c4933a9406380f..555dd6a71bf0bff2616566693b8f341917aabb6e 100644 GIT binary patch delta 616083 zcmbrlV~lUXw*}a?ZQHhO+qUgr+qP}qwrzLcwr$)tZ+Fjq^M8|>P|9bYM1x6LwLgb*h1PFHcVRr6mi^?HLW4kC>m80fdspH(gt6s~QjM#h9;w z&d8k9^FmE3Ap@@3m(qUf0IaUy9h;7K$XEc@zqgw0u9=okvh75w9}?!-UElADs=}p~ z2B6`yTFvd9rt#w}D@&nU+711`)XKLDRajET`d$!@v{+GTF1mzv!*bi5S{2N;DSpc# z%OC@PR`vTxw}MHpjUCLXKXtS=GqIo=y!H)EIbW8z)Q`J%=-W+`ag$Hq`2T%hjWBrx zFWm<$!whP_E}{i?g*cf-epo2p=mlht*Ivg8x2}l!rMI2#41bakp;=ge{31De`DWQUOl3%*^lHwpfW?fQ#Gnkfmh z+Zhg6_KVuDDl$Wue{@h^;2a2SyZ-+0z=&o4R)dN}F-x5s;cQy{E(N%l2181NsFnWD 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zM@eTt>j<8htJ=bCf_ww5$XDTXeDVR>a%G0yER3CFJP$cIbr4~Eiv#M)u!*Ky9ho*@ zR_YHlen7TwQ*M$s7oJgsoYP9Tg*ZJE&Q+1KQT@7@3rhHcIJP*jg4BlCzyl;M+2~mH zIq~RtRg*WrlaDHY}U!a%r`WQfk Wb!fbQqC-|LPF5Ilaxn#QnEwLi3FIFD diff --git a/number-theory/langlands-functoriality/main.tex b/number-theory/langlands-functoriality/main.tex index 0205315..c7b85d8 100644 --- a/number-theory/langlands-functoriality/main.tex +++ b/number-theory/langlands-functoriality/main.tex @@ -1,11 +1,124 @@ -\documentclass{amsart} -\usepackage[utf8]{inputenc} +\documentclass{mimosis} \usepackage{amsmath, amsthm, amssymb} -\usepackage{url} +\usepackage{metalogo} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Some of my favourite personal adjustments +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% +% These are the adjustments that I consider necessary for typesetting +% a nice thesis. However, they are *not* included in the template, as +% I do not want to force you to use them. + +% This ensures that I am able to typeset bold font in table while still aligning the numbers +% correctly. +\usepackage{etoolbox} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Hyperlinks & bookmarks +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\usepackage[% + colorlinks = true, + citecolor = RoyalBlue, + linkcolor = RoyalBlue, + urlcolor = RoyalBlue, + unicode, + ]{hyperref} + +\usepackage{bookmark} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Bibliography +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% +% I like the bibliography to be extremely plain, showing only a numeric +% identifier and citing everything in simple brackets. The first names, +% if present, will be initialized. DOIs and URLs will be preserved. + +\usepackage[% + autocite = plain, + backend = biber, + doi = true, + url = true, + giveninits = true, + hyperref = true, + maxbibnames = 99, + maxcitenames = 99, + sortcites = true, + style = numeric, + ]{biblatex} + +\input{bibliography-mimosis} +\addbibresource{ref.bib} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Fonts +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\ifxetexorluatex + \usepackage{unicode-math} + \setmainfont{EB Garamond} + \setmathfont{Garamond Math} + + % Load some missing symbols from another font. + \setmathfont{STIX Two Math}[% + range = { + \sharp, + \natural, + \flat, + \clubsuit, + \spadesuit, + \checkmark + } + ] + \setmonofont[Scale=MatchLowercase]{Source Code Pro} +\else + \usepackage[lf]{ebgaramond} + \usepackage[oldstyle,scale=0.7]{sourcecodepro} + \singlespacing +\fi + +\newacronym[description={Principal component analysis}]{PCA}{PCA}{principal component analysis} +\newacronym {SNF}{SNF}{Smith normal form} +\newacronym[description={Topological data analysis}] {TDA}{TDA}{topological data analysis} + +\newglossaryentry{LaTeX}{% + name = {\LaTeX}, + description = {A document preparation system}, + sort = {LaTeX}, +} + +\newglossaryentry{Real numbers}{% + name = {$\real$}, + description = {The set of real numbers}, + sort = {Real numbers}, +} + +\makeindex +\makeglossaries + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Ordinals +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\makeatletter +\@ifundefined{st}{% + \newcommand{\st}{\textsuperscript{\textup{st}}\xspace} +}{} +\@ifundefined{rd}{% + \newcommand{\rd}{\textsuperscript{\textup{rd}}\xspace} +}{} +\@ifundefined{nd}{% + \newcommand{\nd}{\textsuperscript{\textup{nd}}\xspace} +}{} +\makeatother + +\renewcommand{\th}{\textsuperscript{\textup{th}}\xspace} + +%%% \newtheorem{theorem}{Theorem}[section] \newtheorem{conjecture}{Conjecture}[section] -% \newtheorem*{conjecture*}{Conjecture}[section] \newtheorem{lemma}{Lemma} \newtheorem{proposition}{Proposition}[section] \newtheorem{corollary}{Corollary}[section] @@ -19,25 +132,71 @@ \DeclareMathOperator{\Oo}{\mathrm{O}} \DeclareMathOperator{\Gal}{\mathrm{Gal}} \DeclareMathOperator{\AI}{\mathrm{AI}} +%%% + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Incipit +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\title{Langlands Functoriality Conjecture - A Survey} +\title{Langlands functoriality} +\subtitle{view towards classical examples} \author{Seewoo Lee} -% \date{July 2021} \begin{document} -\maketitle +\frontmatter + \include{src/title.tex} + \include{src/abstract.tex} + + \tableofcontents + +\mainmatter + + % \part[A good part]{% + % A good part\\ + % % + % \vspace{1cm} + % % + % \begin{minipage}[l]{\textwidth} + % % + % \textnormal{% + % \normalsize + % % + % \begin{singlespace*} + % \onehalfspacing + % % + % You can also use parts in order to partition your great work + % into larger `chunks'. This involves some manual adjustments in + % terms of the layout, though. + % \end{singlespace*} + % } + % \end{minipage} + % } + + \include{src/intro.tex} + \include{src/basechange.tex} + \include{src/rankinselberg.tex} + \include{src/symmetricpower.tex} + \include{src/jacquetlanglands.tex} + \include{src/thetacorrespondence.tex} + + + +% This ensures that the subsequent sections are being included as root +% items in the bookmark structure of your PDF reader. +\bookmarksetup{startatroot} +\backmatter -\input{intro} +% \begingroup +% \let\clearpage\relax +% \glsaddall +% \printglossary[type=\acronymtype] +% \newpage +% \printglossary +% \endgroup -\input{autoind} -\input{basechange} -\input{rankinselberg} -\input{symmetricpower} -\input{jacquetlanglands} -\input{thetacorrespondence} -\newpage + \printindex + \printbibliography -\bibliographystyle{acm} -\bibliography{ref} \end{document} diff --git a/number-theory/langlands-functoriality/mimosis.cls b/number-theory/langlands-functoriality/mimosis.cls new file mode 100644 index 0000000..0741a4d --- /dev/null +++ b/number-theory/langlands-functoriality/mimosis.cls @@ -0,0 +1,220 @@ +\NeedsTeXFormat{LaTeX2e} +\ProvidesClass{mimosis}[2017/08/01 Minimal modern thesis class] + +\PassOptionsToPackage{ + DIV = 10, % TODO: Make configurable + BCOR = 10mm, % TODO: Make configurable +}{typearea} + +\LoadClass[a4paper, + twoside, + 11pt, + cleardoublepage=empty, + numbers=noenddot, + titlepage, + toc=bibliography, + toc=index,]{scrbook} + +\RequirePackage{iftex} +\newif\ifxetexorluatex\xetexorluatexfalse +\ifxetex + \xetexorluatextrue +\fi +\ifluatex + \xetexorluatextrue +\fi + +\ifxetexorluatex + \RequirePackage{fontspec} +\else + \RequirePackage[T1]{fontenc} + \RequirePackage[utf8]{inputenc} +\fi + +% Makes it possible to switch between different languages in the text +% while keeping hyphenation rules correct. Should you add another one +% in the list, please ensure that `english` is the last one. The last +% language is used to control standard hyphenation. +\usepackage[ngerman,french,english]{babel} + +\RequirePackage[autostyle=true]{csquotes} % Context-sensitive quotation marks +\RequirePackage{makeidx} % For creating indices +\RequirePackage{xspace} % For automatically "eating" spaces + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Multi-line comments +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\newcommand{\comment}[1]{} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Fonts & colours +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\RequirePackage[usenames,dvipsnames]{xcolor} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Graphics +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\RequirePackage{graphicx} +\graphicspath{% + {Figures/} + {./} +} + +% Suppress warnings about page groups in PDFs. This is not justified +% in most of the cases. I am pretty sure I am including my images in +% the right manner. +\begingroup\expandafter\expandafter\expandafter\endgroup +\expandafter\ifx\csname pdfsuppresswarningpagegroup\endcsname\relax +\else + \pdfsuppresswarningpagegroup=1\relax +\fi + +\RequirePackage{subcaption} + +% Make sub-references using \subref being typeset with parentheses. +% Otherwise, only the counter will be printed. +\captionsetup{subrefformat=parens} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Glossaries +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\RequirePackage[% + acronym, + nogroupskip, + nopostdot, + nonumberlist, + toc, + ]{glossaries} + +% New style that prevents line-breaks between the full description and +% the acronym. Furthermore, it ensures that the acronym is always +% printed in an upright font. +\newacronymstyle{long-short-mimosis} +{% + \GlsUseAcrEntryDispStyle{long-short}% +}% +{% + \GlsUseAcrStyleDefs{long-short}% + \renewcommand*{\genacrfullformat}[2]{% + \glsentrylong{##1}##2~\textup{(\firstacronymfont{\glsentryshort{##1}})}% + }% + \renewcommand*{\Genacrfullformat}[2]{% + \Glsentrylong{##1}##2~\textup{(\firstacronymfont{\glsentryshort{##1}})}% + }% + \renewcommand*{\genplacrfullformat}[2]{% + \glsentrylongpl{##1}##2~\textup{(\firstacronymfont{\glsentryshortpl{##1}})}% + }% + \renewcommand*{\Genplacrfullformat}[2]{% + \Glsentrylongpl{##1}##2~\textup{(\firstacronymfont{\Glsentryshortpl{##1}})}% + }% +} + +% A new glossary style that based on the long style of the glossaries +% package. It ensures that descriptions and entries are aligned. +\newglossarystyle{long-mimosis}{% + \setglossarystyle{long} + + \renewcommand{\glossentry}[2]{% + \glsentryitem{##1}\glstarget{##1}{\glossentryname{##1}} & + \glossentrydesc{##1}\glspostdescription\space ##2\tabularnewline + }% +} + +% Constrain the description width to enforce breaks. +\setlength{\glsdescwidth}{10cm} + +\setacronymstyle{long-short-mimosis} +\setglossarystyle{long-mimosis} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Paragraph lists & compact enumerations +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\RequirePackage[% + olditem, % Do not modify itemize environments by default + oldenum % Do not modify enumerate environments by default + ]{paralist} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Spacing +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\RequirePackage{setspace} +\onehalfspacing + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Tables +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\RequirePackage{booktabs} +\RequirePackage{multirow} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Proper typesetting of units +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\RequirePackage{siunitx} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Mathematics +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\RequirePackage{amsmath} +\RequirePackage{amsthm} +\RequirePackage{dsfont} + +% Fix the spacing of \left and \right. Use these with the proper bracket +% in order to ensure that they scale automatically. +\let\originalleft\left +\let\originalright\right +\renewcommand{\left}{\mathopen{}\mathclose\bgroup\originalleft} +\renewcommand{\right}{\aftergroup\egroup\originalright} + +\DeclareMathOperator*{\argmin} {arg\,min} +\DeclareMathOperator {\dist} {dist} +\DeclareMathOperator {\im} {im} + +\newcommand{\domain}{\ensuremath{\mathds{D}}} +\newcommand{\real} {\ensuremath{\mathds{R}}} + +% Proper differential operators +\newcommand{\diff}[1]{\ensuremath{\operatorname{d}\!{#1}}} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Penalties +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\clubpenalty = 10000 +\widowpenalty = 10000 +\displaywidowpenalty = 10000 + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Headers +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\RequirePackage{scrlayer-scrpage} +\pagestyle{scrheadings} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Typefaces for parts, chapters, and sections +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\renewcommand{\partformat}{\huge\partname~\thepart\autodot} +\renewcommand{\raggedpart}{\flushleft} + +\setkomafont{part}{\normalfont\huge\scshape} + +\setkomafont{sectioning}{\normalfont\scshape} +\setkomafont{descriptionlabel}{\normalfont\bfseries} + +\setkomafont{caption}{\small} +\setkomafont{captionlabel}{\usekomafont{caption}} + +% Large number for chapter +\renewcommand*{\chapterformat}{% + \fontsize{50}{55}\selectfont\thechapter\autodot\enskip +} diff --git a/number-theory/langlands-functoriality/src/abstract.tex b/number-theory/langlands-functoriality/src/abstract.tex new file mode 100644 index 0000000..360c493 --- /dev/null +++ b/number-theory/langlands-functoriality/src/abstract.tex @@ -0,0 +1,7 @@ +\begin{center} + \textsc{Abstract} +\end{center} +% +\noindent +% +Survey on the Langlands functoriality conjecture. \ No newline at end of file diff --git a/number-theory/langlands-functoriality/autoind.tex b/number-theory/langlands-functoriality/src/autoind.tex similarity index 95% rename from number-theory/langlands-functoriality/autoind.tex rename to number-theory/langlands-functoriality/src/autoind.tex index 279e389..33622b8 100644 --- a/number-theory/langlands-functoriality/autoind.tex +++ b/number-theory/langlands-functoriality/src/autoind.tex @@ -1,7 +1,6 @@ -\newpage -\section{Automorphic induction} +\chapter{Automorphic induction} -\subsection{From $\GL_{1}/K$ to $\GL_{2}/\mathbb{Q}$} +\section{From $\GL_{1}/K$ to $\GL_{2}/\mathbb{Q}$} Let $K$ be a quadratic field (over $\mathbb{Q}$) and $\xi$ be a Hecke character for $K$. By Hecke and Maass, it was proven that one can associate $\GL_{2}$ automorphic forms. Hecke attached modular forms to Hecke characters for imaginary quadratic fields, and Maass attached Maass forms to those for real quadratic fields. @@ -41,7 +40,7 @@ \subsection{From $\GL_{1}/K$ to $\GL_{2}/\mathbb{Q}$} Both theorem can be proved using converse theorems for $L$-functions. By showing that the $L$-function attached to Hecke character $\xi$ satisfies suitable functional equations, converse theorem shows that the $L$ function should coincides with one comes from modular forms or Maass forms. -\subsection{From $\GL_{n}/K$ to $\GL_{rn}/F$} +\section{From $\GL_{n}/K$ to $\GL_{rn}/F$} In view of Langlands functoriality conjecture, Hecke and Maass' results can be considered as a special case when $G = \mathrm{GL}_{1} / K$ and $G' = \GL_{2}/\mathbb{Q}$. Automorphic induction, which is a vast generalization of this, is a functoriality from $\GL_{n}/K$ to $\GL_{rn}/F$, where $K/F$ is a degree $r$ extension. % It is more natural to see Galois counterpart for the functoriality. @@ -78,7 +77,7 @@ \subsection{From $\GL_{n}/K$ to $\GL_{rn}/F$} % Here are brief reviews of proofs for known cases. We give a sketch of proofs for known cases. -\subsubsection{Local automorphic induction (Henniart-Herb)} +\subsection{Local automorphic induction (Henniart-Herb)} In \cite{henniart1995automorphic}, Henniart and Herb proved automorphic induction over local fields of characteristic zero, i.e. for $p$-adic fields. diff --git a/number-theory/langlands-functoriality/basechange.tex b/number-theory/langlands-functoriality/src/basechange.tex similarity index 95% rename from number-theory/langlands-functoriality/basechange.tex rename to number-theory/langlands-functoriality/src/basechange.tex index 28a3e61..9999eaf 100644 --- a/number-theory/langlands-functoriality/basechange.tex +++ b/number-theory/langlands-functoriality/src/basechange.tex @@ -1,7 +1,6 @@ -\newpage -\section{Base change} +\chapter{Base change} -\subsection{Doi-Naganuma lifting} +\section{Doi-Naganuma lifting} In \cite{doi1969functional}, Doi and Naganuma constructed a lifting from the space of elliptic modular forms to the space of Hilbert modular forms of the same (parallel) weight. The proved the following theorem: \begin{theorem}[Doi-Naganuma \cite{doi1969functional}] diff --git a/number-theory/langlands-functoriality/intro.tex b/number-theory/langlands-functoriality/src/intro.tex similarity index 96% rename from number-theory/langlands-functoriality/intro.tex rename to number-theory/langlands-functoriality/src/intro.tex index 7e11fc5..bdd5c99 100644 --- a/number-theory/langlands-functoriality/intro.tex +++ b/number-theory/langlands-functoriality/src/intro.tex @@ -1,4 +1,4 @@ -\section{Introduction} +\chapter{Introduction} \begin{conjecture}[Langlands functoriality conjecture] Let $G$ and $G'$ be reductive groups over a global field $F$. \end{conjecture} diff --git a/number-theory/langlands-functoriality/jacquetlanglands.tex b/number-theory/langlands-functoriality/src/jacquetlanglands.tex similarity index 95% rename from number-theory/langlands-functoriality/jacquetlanglands.tex rename to number-theory/langlands-functoriality/src/jacquetlanglands.tex index 91e7529..f9dc2ec 100644 --- a/number-theory/langlands-functoriality/jacquetlanglands.tex +++ b/number-theory/langlands-functoriality/src/jacquetlanglands.tex @@ -1,7 +1,6 @@ -\newpage -\section{Jacquet-Langlands correspondence} +\chapter{Jacquet-Langlands correspondence} -\subsection{Basis problem and quaternionic modular forms} +\section{Basis problem and quaternionic modular forms} Here's a fundamental question that we can ask about modular forms. @@ -26,9 +25,9 @@ \subsection{Basis problem and quaternionic modular forms} \cite{eichler1973basis} -\subsection{Jacquet-Langlands correspondence} +\section{Jacquet-Langlands correspondence} -\subsection{Basis problem by means of Jacquet-Langlands correspondence} +\section{Basis problem by means of Jacquet-Langlands correspondence} In this section, we will give a solution to the basis problem by using Jacquet-Langlands correspondence. Our goal is to prove the following theorem. \begin{theorem}[Kimball, \cite{martin2020basis}] diff --git a/number-theory/langlands-functoriality/rankinselberg.tex b/number-theory/langlands-functoriality/src/rankinselberg.tex similarity index 97% rename from number-theory/langlands-functoriality/rankinselberg.tex rename to number-theory/langlands-functoriality/src/rankinselberg.tex index c78ca25..a869ef3 100644 --- a/number-theory/langlands-functoriality/rankinselberg.tex +++ b/number-theory/langlands-functoriality/src/rankinselberg.tex @@ -1,7 +1,6 @@ -\newpage -\section{Rankin-Selberg product} +\chapter{Rankin-Selberg product} -\subsection{Rankin-Selberg convolution of modular forms} +\section{Rankin-Selberg convolution of modular forms} Let $f, g$ be two holomorphic cusp forms of weight $k$ and level 1. Assume that two forms have Fourier expansions @@ -77,7 +76,7 @@ \subsection{Rankin-Selberg convolution of modular forms} \end{align*} \end{proof} -\subsection{Modularity of $\GL(2) \times \GL(2)$} +\section{Modularity of $\GL(2) \times \GL(2)$} It is also possible to construct Rankin-Selberg $L$-function attached to two Maass cusp forms with similar properties. In general, for given automorphic representations $\pi_{1}, \pi_{2}$ on $\GL(2)$, one can define Rankin-Selberg $L$-function $L(s, \pi_{1}\times \pi_{2})$. @@ -131,5 +130,5 @@ \subsection{Modularity of $\GL(2) \times \GL(2)$} which shows that $\pi' \simeq \pi \otimes \chi$ for some $\chi$ by cuspidality criterion that is also proved by him in loc. cit. -\subsection{$\GL(n)\times \GL(n)$} +\section{$\GL(n)\times \GL(n)$} diff --git a/number-theory/langlands-functoriality/symmetricpower.tex b/number-theory/langlands-functoriality/src/symmetricpower.tex similarity index 99% rename from number-theory/langlands-functoriality/symmetricpower.tex rename to number-theory/langlands-functoriality/src/symmetricpower.tex index e020c03..d70e186 100644 --- a/number-theory/langlands-functoriality/symmetricpower.tex +++ b/number-theory/langlands-functoriality/src/symmetricpower.tex @@ -1,4 +1,3 @@ -\newpage \newcommand\px[1]{\partial x_{#1}} \newcommand\py[1]{\partial y_{#1}} @@ -13,13 +12,13 @@ \newcommand\Sym{\mathrm{Sym}} -\section{Symmetric power lifting} +\chapter{Symmetric power lifting} Automorphic forms on $\mathrm{GL}(2)$ are often \emph{classified} into two kinds of objects: \emph{modular forms} and \emph{Maass forms}\footnote{and constant functions.}. These functions oftenly considered as a starting point for studying automorphic forms and representations for $\mathrm{GL}(n)$ and other groups. However, there are not many references for $\mathrm{GL}(3)$. -\subsection{Automorphic forms on $\mathrm{GL}(3, \mathbb{R})$} +\section{Automorphic forms on $\mathrm{GL}(3, \mathbb{R})$} We first introduce the theory of automorphic forms on $\mathrm{GL}(3)$ (We follow the Bump's book \cite{bump2006automorphic}). We only consider the level 1 automorphic forms. @@ -263,7 +262,7 @@ \subsection{Automorphic forms on $\mathrm{GL}(3, \mathbb{R})$} -\subsection{Symmetric square lifting by Gelbart-Jacquet} +\section{Symmetric square lifting by Gelbart-Jacquet} Let $f$ be a level 1 Maass cusp form (of weight 0) on $\GL(2,\mathbb{R})$ with eigenvalue $\lambda = \nu(1-\nu)$ which is also a normalized eigenform. Let $\{a_{n}\}_{n\geq 1}$ be Fourier coefficients of $f$. @@ -336,7 +335,9 @@ \subsection{Symmetric square lifting by Gelbart-Jacquet} $$ where $\Ad(\pi) = \Sym^{2}(\pi) \otimes \omega_{\pi}^{-1}$. \end{theorem} -\subsection{Higher symmetric power} + + +\section{Higher symmetric power} Since symmetric power map $\Sym^{r}:\GL(2) \to \GL(r+1)$ is defined for arbitrary power $r$, we expect the presence of lifting from $\GL(2)$ automorphic representations to $\GL(r+1)$ automorphic representations. Until now, this is proved for $r = 3, 4$ cases. @@ -374,7 +375,7 @@ \subsection{Higher symmetric power} \end{theorem} -\subsection{Ramanujan's conjecture and Selberg's 1/4 conjecture} +\section{Ramanujan's conjecture and Selberg's 1/4 conjecture} The importance of symmetric power lifting is due to it's application on Ramanujan's conjecture and Selberg's eigenvalue conjecture. @@ -532,7 +533,7 @@ \subsection{Ramanujan's conjecture and Selberg's 1/4 conjecture} We refer to \cite{sarnak2005notes,blomer2013role,winnie2020ramanujan} for further discussion about Ramanujan's conjecture. -\subsection{Sato-Tate conjecture} +\section{Sato-Tate conjecture} Another consequence of symmetric power lifting is Sato-Tate conjecture. It is a conjecture about equidistribution of \emph{Frobenius angle}, which we are going to explain briefly. diff --git a/number-theory/langlands-functoriality/thetacorrespondence.tex b/number-theory/langlands-functoriality/src/thetacorrespondence.tex similarity index 96% rename from number-theory/langlands-functoriality/thetacorrespondence.tex rename to number-theory/langlands-functoriality/src/thetacorrespondence.tex index cd15026..be86d0a 100644 --- a/number-theory/langlands-functoriality/thetacorrespondence.tex +++ b/number-theory/langlands-functoriality/src/thetacorrespondence.tex @@ -1,7 +1,7 @@ -\newpage -\section{Theta correspondence and Howe duality} +\chapter{Theta correspondence and Howe duality} + -\subsection{Half-integral weight modular forms} +\section{Half-integral weight modular forms} The theta series $$ \theta(z) = \sum_{n\in\mathbb{Z}} q^{n^{2}}, \qquad q = e^{2\pi i z} @@ -84,7 +84,7 @@ \subsection{Half-integral weight modular forms} -\subsection{Shimura correspondence and Shintani lift} +\section{Shimura correspondence and Shintani lift} Let $f(z) = \sum_{n \geq 1} a_{n}q^{n}$ be a Hecke eigenform of half-integral weight $k/2$ and level $N$ with character $\chi$, i.e. $T_{\chi, p^{2}}f = \lambda_{p}f$ for all $p$ with $p\nmid N$. Then for every square-free integer $t$, we have @@ -112,8 +112,8 @@ \subsection{Shimura correspondence and Shintani lift} \end{theorem} The proof is based on Weil's converse theorem. -\subsection{Waldspurger's work} +\section{Waldspurger's work} -\subsection{Theta correspondence and Howe duality} +\section{Theta correspondence and Howe duality} -\subsection{Gan-Gross-Prasad conjecture} \ No newline at end of file +\section{Gan-Gross-Prasad conjecture} \ No newline at end of file diff --git a/number-theory/langlands-functoriality/src/title.tex b/number-theory/langlands-functoriality/src/title.tex new file mode 100644 index 0000000..70cac8d --- /dev/null +++ b/number-theory/langlands-functoriality/src/title.tex @@ -0,0 +1,25 @@ +\begin{titlepage} + \vspace*{5cm} + \makeatletter + \begin{center} + \begin{Huge} + \@title + \end{Huge}\\[0.1cm] + % + \begin{Large} + \@subtitle + \end{Large}\\ + % + \emph{by}\\ + \@author + % + \vfill + Survey on the Langlands functoriality conjecture with many examples + \end{center} + \makeatother +\end{titlepage} + +\newpage +\null +\thispagestyle{empty} +\newpage