# Tagged Questions

49 views

### Identities for L-series and Euler product

It is a mabe a stupid question for many experts here. There is something wrong in the following reasoning, and now I could not find it. Could someone help me out? Any advice will be highly ...
22 views

38 views

Let $f(z)$ be a modular form of some integral weight $k \geq 0$ and level $\Gamma_1(N)$ (I insist I want $\Gamma_1(N)$, not $\Gamma_0(N)$ or $\Gamma(N)$). Thus for any $d \in (\mathbb Z/N\mathbb ... 0answers 12 views ### Jacobi Form and its Fourier expansion Let k,m be non negative integers. A Jacobi form of weight k and index m is a holomorphic function f on$\mathbb{H} x \mathbb{C}$(where$\mathbb{H}$denotes the upper half plane) satisfying the ... 1answer 145 views ### Serre's Modularity Conjecture — Weight I was reading Serre's paper "Sur les Représentations Modulaires de Degré$2$de Gal($\bar{\mathbb{Q}}/\mathbb{Q}$)" where he states his modularity conjecture (which is now a theorem). Following his ... 1answer 32 views ### Identities of Hecke operators While studying, I recently came across the following interesting problem. Let's say that the (level one) weight$k$modular forms$M_k(\Gamma(1))$have dimension$d$. We know by the ring structure ... 1answer 60 views ### Undergraduate Introduction to Modular Forms What are the best introductory texts (or lecture notes) on modular forms aimed at an advanced undergraduate audience (for a student with a course in complex analysis and two courses in algebra and ... 1answer 73 views ### Proving the Uniformization Theorem for Elliptic Curves (An Exercise from Silverman's Advanced Topics on Elliptic Curves ) In Silverman's Advanced Topics in the Arithmetic of Elliptic Curves there are two demonstrations of the Uniformization Theorem for the Elliptic Curves (It says that, given an Elliptic Curve$E$, ... 1answer 55 views ### Rewriting$\tau(p)\Delta(\tau)$when$p$is prime$p$is a prime, and$\tauis Ramanujan's tau function: $$p^{11}\Delta(p\tau)+\frac{1}{p}\sum_{k=0}^{p-1}\Delta\bigg(\frac{\tau + ... 1answer 71 views ### Modular Form, but Cusp Form in Disguise Suppose I have a holomorphic modular form f \in M_k(\Gamma_0(N), \chi) with k \in \mathbb Z^+ and f = \sum_{n=1}^\infty a(n)q^n. Considering the q-expansion, one may suspicious that f is ... 1answer 47 views ### Why a modular form is a highest weight vector of a discrete series summand of L_2(SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R}))? It is said that a modular form is a highest weight vector of a discrete series summand of L_2(SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R})). Why a modular form is a highest weight vector of a ... 1answer 24 views ### Diamond operator on a meromorphic function Is there a standard way in which the diamond operator for modular forms is defined for an arbitrary meromorphic function?. I know the definition for a weight k modular form, but since I usually wont ... 1answer 67 views ### Equidistribution of lattice points on spheres in dimensions d \ge 4 (Pommerenke's theorem) I am looking for either an English translation of: Pommerenke, C.: Über die Gleichverteilung von Gitterpunkten auf m-dimensionalen Ellipsoiden. Acta Arith. 5, 227-257 (1959) Or a reference in ... 0answers 13 views ### Is it true that \theta_{1,1}^{4N} \in J_{2N,2N}(2N)? I need examples of Jacobi forms for full congruence subgroups \Gamma(N) of SL(2,Z). As a particular case, take the theta function \theta_{1,1}(t,z) := \sum_{n\in\mathbb{Z}} exp(\pi it(n + ... 0answers 130 views ### The conjecture A159634(n) = \psi(2n)/3 where \psi is the Dedekind psi function. Steven Finch is the author of the OEIS-sequence A159634 which is defined as "Coefficient for dimensions of spaces of modular & cusp forms of weight k/2, level 4n >and trivial character, ... 0answers 103 views ### Proving that \sigma_7(n) = \sigma_3(n) + 120 \sum_{m=1}^{n-1} \sigma_3(m)\sigma_3(n-m) without using modular forms? This problem appears as a (starred!) exercise in D. Zagier's notes on modular forms. I have to admit that I have no idea how to do it. Here, \sigma_k(n) =\sum_{d\mid n} d^k, as usual. This ... 0answers 48 views ### Learning roadmap for p-adic modular forms and eigenvarieties. What are some good sources for breaking into the field of p-adic modular forms? It was suggested to me to read Katz' paper "P-adic Properties of Modular Schemes and Modular Forms". I have found this ... 1answer 62 views ### Definition of Automorphic form I am looking at how one maps the space of cusp forms S_k(N,\psi) into the space of automorphic forms on GL_2 over the Adeles. And I have a question about the definition of K-finiteness. I am ... 0answers 50 views ### Are the Fourier coefficients of a new form real? Let f\in S^\text{new}_k(Γ_0(N)) be a \text{newform} . Are all its Fourier coefficients real? Of course the Hecke operators T_n are selfadjoint for (n,N)=1, but is it also true for all n? 5answers 438 views ### Other interesting consequences of d=163? Question: Any other interesting consequences of d=163 having class number h(-d)=1 aside from the list below? Let \tau = \tfrac{1+\sqrt{-163}}{2}. We have (see notes at end of list), ... 3answers 98 views ### How to show that \Delta\left(\frac{az+b}{cz+d}\right)=(cz+d)^{12}\Delta(z)? Let (a, b; c, d) \in SL_2(\mathbb{Z}) and \Delta(z)=q\prod_{n=1}^{\infty} (1-q^n)^{24}, q=e^{2\pi i z}. How to show that \Delta(\frac{az+b}{cz+d})=(cz+d)^{12}\Delta(z)? Thank you very much. I ... 0answers 23 views ### The filtration of mod l modular form I'm reading a paper of Swinnerton-Dyer, "On l-adic representations for coefficients of modular forms." He defines the notion of "filtration" for mod l modular forms: If \tilde{f} \in \tilde{M} ... 1answer 106 views ### Real life applications of Maass wave forms Explaining my work on Maass wave forms to friends and family (all non-mathematician) typically earns me blank faces. So I wonder whether there is some good example to explain their meaning to laymen. ... 4answers 362 views ### What are applications of number theory in physics? I was reading Goro Shimura's The Map of My Life. He wrote the following quote in the book. It made me come up with the title question. In particular, is there any application of the theory of modular ... 0answers 103 views ### Modular Functions with Rational Fourier Expansions I have been reading the paper of Cox, McKay and Stevenhagen "Principal Moduli and Class Fields", http://arxiv.org/pdf/math/0311202v1.pdf, and I have a question regarding the nature of the function ... 1answer 38 views ### A quick fact check about filtrations of modular forms and E_{p+1} Suppose f is a mod p modular form of level N with p>3 and p not dividing N. Is it true that w(f E_{p+1})=w(f)+p+1, where w(f) is the filtration of f? (The filtration of f is the ... 1answer 116 views ### modular form -Petersson inner product my question is about Petersson inner product. i need to prove that (E_k,f) =0 \forall f \in S_k(SL_2(\mathbb{Z})) the only thing that i think that should help me is that the space of cusp form ... 2answers 69 views ### Other names for E_{p+1} \pmod{p}? If I want to know properties of E_{p+1} modulo p, do you know a name for this modular form, so that it is easier to search via the internet? So far, what I know is that E_{p-1} is the Hasse ... 3answers 209 views ### Inverting the modular J function Klein's modular function J(z) is defined and studied in e.g. Apostol's book Modular functions and Dirichlet series in number theory. Certain specific evaluations are available, for example, ... 1answer 120 views ### Recognizing if a power series is a q-expansion of a modular form Given a power series in q, is it possible to tell if it is the q-expansion of a modular form (of level N say)? I don't need to show results of this sort, but it has come up enough that I'm ... 1answer 69 views ### Expressing Eisenstein series E_k in terms of E_4 and E_6 Given an Eisenstein series E_k (of level 1), it is a polynomial P_k(E_4,E_6) in E_4 and E_6, and http://en.wikipedia.org/wiki/Eisenstein_series#Recurrence_relation should give a finite ... 0answers 263 views ### Can 262537412640768743.99999999999925 be beaten with simple expressions? [closed] We know:$$\begin{align}e^{\pi \sqrt{163}} &= 262537412640768743.9999999999992500726\dots\\ x^{24} - 24&=262537412640768743.9999999999992511239\dots\end{align}$$where x is the real solution ... 1answer 150 views ### Elementary tools for proving congruences of modular forms My impression is that the specialists in the field use geometric modular forms when proving congruences of modular forms. While this is probably the right way, I don't think I will be able to get a ... 2answers 213 views ### Definition of Modular Forms over finite Fields I'm still severely lacking in background at the moment, but I'm interested in doing something with congruence properties of modular forms (relations between coefficients of the q-expansions that hold ... 2answers 74 views ### Routine question about derivatives of automorphic forms being L^2 I consider Automorphic forms on G = SL_2 (\mathbb{R}), which are \Gamma-invariant, K-finite, Z(g) finite, and of moderate growth. If I have such an automorphic form, which happens to be in ... 1answer 135 views ### p-adic modular form example In Serre's paper on p-adic modular forms, he gives the example (in the case of p = 2,3,5) of \frac{1}{Q} and \frac{1}{j} as p-adic modular forms, where Q = E_4 = 1 + 540\sum ... 1answer 89 views ### Coefficients of powers of the theta function Let q=\exp(2 \pi i z) and$$\theta(z)=\sum_{n=-\infty}^\infty q^{n^2}.$$Now, I shall show that the powers of \theta are given by$$\theta(z)^r = \sum_{n=0}^\infty S_r(n) q^n$$where S_r(n) ... 1answer 118 views ### Stark's formula for the j-invariant In his paper On the "gap" in Heegner's proof (which you can find : here) Stark gives the following formula for the j-invariant (for some \tau \in \mathcal{H} and q=e^{2i\pi\tau})$$ j(\tau) = ... 2answers 222 views ### How to calculate|\operatorname {SL}_2(\mathbb Z/N\mathbb Z)|$? the answer should be $$|\operatorname {SL}_2(\mathbb Z/N\mathbb Z)|=N^{3}\prod_{p|N}(1-{1 /p^2})$$ But first how to prove $$|\operatorname {SL}_2(\mathbb Z/p^e\mathbb Z)|=p^{3e}(1-{1 /p^2})$$ 0answers 157 views ### When does a modular form satisfy a differential equation with rational coefficients? Given a modular form$f$of weight$k$for a congruence subgroup$\Gamma$, and a modular function$t$for$\Gamma$with$t(i\infty)=0$, we can form a function$F$such that$F(t(z))=f(z)$(at least ... 5answers 1k views ### Intuition for the Importance of Modular Forms I am learning about modular forms for the first time this term and am just starting to wrap my head around what might be the big picture of things. I was wondering if the following interpretation of ... 1answer 306 views ### how to calculate$q$-expansion coefficients of a modular form? Can you please explain for me how we can calculate the coefficients of$q$-expansion series of a modular form function$f$? I am really confused. 1answer 84 views ### What is the difference between the dimension of$\mathcal{M}_1(\Gamma)$and$\mathcal{S}_1(\Gamma)$? There is a formula given in the book saying that $$\dim{\mathcal{M}_1(\Gamma)}=\dim\mathcal{S}_1(\Gamma)+\frac{\varepsilon^{\text{reg}}_\infty}{2}$$ Where$\varepsilon^{\text{reg}}_\infty$means ... 1answer 142 views ### In one version of the Modularity Theorem, what does “arise from modular forms” mean? One version of Modularity Theorem says that The elliptic curves with rational$j$-values arise from modular forms. Where $$j(\tau)=1728\frac{g_2^3(\tau)}{\Delta(\tau)}$$ I know every ... 1answer 54 views ### Question about a property of elliptic function Let$E=\mathbb{C}/\Lambda$be a complex elliptic curve where$\Lambda=\omega_1\mathbb{Z}\oplus\omega_2\mathbb{Z}$Let$f$be a nonconstant elliptic function with respect to$\Lambda$. Let ... 1answer 68 views ### A question about complex tori. Let$\Lambda$,$\Lambda'$be two lattice in$\mathbb{C}$and$m\neq 0\in\mathbb{C}\$ satisfying $$m\Lambda\subset\Lambda'$$ The, the book I'm reading says that by the theory of finite Abelian groups ...
I'm new to modular form, reading the book A First Course in Modular Forms We have the weight 2 Eisenstein series $$G_2(\tau)=\sum_{c\in\mathbb{Z}}\sum_{d\in\mathbb{Z}_c'}\frac{1}{(c\tau+d)^2}$$ ...