# Tagged Questions

Questions about modular forms and related areas

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### Can something finite over $\mathbb{C}(q)$ be a modular form?

If $f\in\mathbf{C}[[q]]$ is non-constant, and algebraic over $\mathbf{C}[q]$ (in the sense that it is a root of a polynomial with coefficients in in $\mathbf{C}[q]$) then can $f$ be the $q$-expansion ...
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### Why is there a factor $p$ in the definition of $T_p$ via Hecke correspondences on modular curves?

Fix $N\ge4$. Let $Y_1(N)$ and $X_1(N)$ be the usual modular curves. I want to view them as schemes over $\mathbb Z$ representing the moduli functors of (usual or generalized) elliptic curves with (...
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### Restriction to the diagonal of Hilbert eigenforms

Do you know of any reference that discusses whether the restriction to the diagonal of a Hilbert eigenform is an (elliptic) eigenform?
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### Definition of Hecke operators on orthogonal modular forms

In his paper Automorphic forms with singularities on Grassmannians, Borcherds poses Problem 16.5: "Describe how the correspondence in this paper behaves under the action of Hecke operators." Since ...
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### Calculate Ramanujan's class invariant by using modular equation of degree $5$

Let $$K(k):=\int_{0}^{\frac{\pi}{2}}\frac{d\phi}{\sqrt{1-k^2\sin^2\phi}}=\frac{\pi}{2}{ _2F_1\bigg(\frac{1}{2},\frac{1}{2},1;k^2 \bigg)}$$ where $0<k<1$ Let $K, K′, L$ and $L′$ denote the ...
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### Elementary proof of Ramanujan's “most beautiful identity”

Ramanujan presented many identities, Hardy chose one which for him represented the best of Ramanujan. There are many proofs for this identity. (for example, H. H. Chan’s proof, M. Hirschhorn's proof....
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Consider one of the simplest non-abelian examples of modularity. Let $$\eta(6z)\eta(18z) = q\prod_{n=1}^\infty (1 - q^{6n})(1 - q^{18n}) = q - q^7 - q^{13} -q^{19} + q^{25} + 2q^{31} - q^{37} + 2q^{43}... 1answer 202 views ### Non-vanishing of L-function of modular form There is a theorem by Langlands and Shalika (link) that the L-function of a cuspidal automorphic representation does not vanish on the line \mathrm{Re}( s)=1 (in their normalization which might be ... 1answer 462 views ### What is the motivation behind Ramanujan's conjecture? One motivation I have seen given for Ramanujan's conjecture for the estimate$$ |a_p|< C p^{k - \frac{1}{2}} $$for the Fourier coefficients of a cusp form of weight 2k is that it allows one to ... 0answers 75 views ### Split multiplicative galois representation and specialization My questions stems from my attempt to understand the paper of Greenberg and Stevens about the Mazur-tate-Teitelbaum conjecture (you can find the paper here). To understand this question you probably ... 0answers 148 views ### Comparison of algebraic and analytic q-expansion I would like to check that algebraic and analytic q-expansion of a modular form coincide. I'm thinking about modular forms as global sections of some sheaf on modular curves. If X is a modular ... 1answer 178 views ### Special fibre of the modular curve X(N) Let N be an integer \geq 3 and X(N)\rightarrow \mathrm{Spec } \mathbb{Z}[1/N] is the projective smooth modular curve defined in Deligne-Rappoport. Is there an exemple of N for which the ... 0answers 171 views ### Equivalent definitions of the Hasse invariant As probably many others before me, I got stuck in verifying all the nice properties of the Hasse invariant. Let me start by recalling one definition: Let E\to S be an elliptic curve in ... 1answer 137 views ### Congruence Primes and Modular Degrees Let \mathcal{S}=S_2(\Gamma_0(N) \cap \mathbf{Z} [[ q ]] be the set of cusp forms of weight 2 on \Gamma_0(N) with integral coefficients. Let f \in \mathcal{S} be a normalized newform, so it ... 1answer 276 views ### An electronic copy of Vishik's work on p-adic L-functions for modular forms This question is very simple. Would someone be so nice as to send me an electronic copy of M. M. Vishik, Non-Archimedean measures connected with Dirichlet series, Mat. Sb. (N.S.), 1976, Volume 99(... 1answer 124 views ### Maass form properties and their fourier coefficients Some Maass form can be written (K_{iR} is the K-Bessel function):$$f(x+iy)=\sum_{n \ne 0}^{\infty} a_n \sqrt{y} \;K_{iR}(2\pi |n| y) \; e^{2 i\pi nx}$$with the a_n multiplicative, but inversly ... 1answer 159 views ### Applications of Level Lowering What are some applications/consequences of level lowering of Galois representations? I understand the application of Ribet's theorem in the proof of Fermat's last theorem but I am wondering what other ... 1answer 754 views ### Sphere packings : what next after the recent breakthrough of Viazovska (et al.)? Given the march 2016 breakthrough concerning sphere packings by Viazovska for the case of dimension 8, and by Cohn, Kumar, Miller, Radchenko and Viazovska for the case of dimension 24, it follows that ... 3answers 760 views ### Does X(13) have potentially good reduction at 13? The complete level modular curve X(p) does not have potentially good reduction at p for any p \neq 2,3,5,7,13 because then there are cusp forms on X_0(p) showing up in the cohomology of X(p),... 2answers 216 views ### Field cut out by a CM modular form is imaginary Let f=\sum_{n=1}^\infty a_nq^n be a newform of level N and weight k\ge 2. Suppose that f is a CM modular form in the sense of §3 of Ribet's paper Galois representations attached to eigenforms ... 0answers 139 views ### The Fricke involution and expansions at infinity Let p be prime, and f be a modular form for \Gamma_0 (p) whose expansion at infinity has coefficients in {\mathbb Z}\left[1/p\right]. I'd like a down to earth proof that the same holds true ... 1answer 175 views ### Computing coefficients for the slash operator of a modular form Suppose f is a classical modular form of weight r for a (congruence) group \Gamma. Let \gamma be any matrix in \operatorname{SL}_2(\mathbb{Z}). Then the slash operator |_\gamma is usually ... 0answers 70 views ### the shifted convolution sums and the sub convexity problem for l functions in the paper of gergely harcos, an additive problem in the fourier coefficients of cusp forms, a bound for the shifted convolution sums for hecke eigenvalues was explicited and i thought that his ... 1answer 347 views ### Is Faltings' p-adic Eichler-Shimura isomorphism the p-adic comparison isomorphism? This is a question about Faltings' p-adic Eichler-Shimura isomorphism from his 1987 article "Hodge-Tate structures and Modular Forms". Let N\ge5, k\ge2 be integers. Denote by X(N) the proper ... 0answers 412 views ### What is the best way to learn about Modular Forms? I am a senior Mathematics Major, and I am interesting in learning about Modular Forms. I have a layman's general sense of what they are but I was wondering if there is a lecture(I am willing to pay) ... 1answer 223 views ### Up to 2000, A145722(n-1) \equiv \sigma(4n-3) \pmod{5} A145722 is Expansion of f(q) * f(q^5) / phi(-q^2)^2 in powers of q where f(), phi() are Ramanujan theta functions. Using the pari program and offset 0, up to 2000... 1answer 205 views ### Weight 12 cusp forms for \Gamma_0(p) Let S_k be the space of weight 12 cusp forms of \Gamma_0(p), (p prime), then Sage tells that \dim S_k^{\text{new}}=\dim S_k-2. Thus the old forms spans a 2-dimensional subspace. One of the ... 0answers 147 views ### Comparison of sheaves of modular forms Let \pi:E\to X the universal generalized elliptic curve over the compactified modular curve, with zero section e: X\to E. Now consider the following two sheaves on X: e^*\Omega^1_{E/X} and \... 1answer 180 views ### p-adic modular forms, Hecke algebra, deformation theory and modular curves. Let h^{ord}(N,\mathcal{O}) be the p-ordinary Hecke algebra, and \mathfrak{m} be a maximal ideal of the semi local ring h^{ord}(N,\mathcal{O}) corresponding to a residual representation \bar{\... 0answers 171 views ### Control theory for Kitagawa's \Lambda-adic modular symbols Let p be a prime, \Gamma=1+p\mathbb Z_p and \Lambda=\mathbb Z_p[[\Gamma]] the Iwasawa algebra. Let \kappa\colon\Gamma\rightarrow\mathbb Z_p^\times be the inclusion. For a character \... 1answer 123 views ### How different can characters be for a sum of modular forms to still be in Gamma_0? I have a modular form I am constructing out of sums and products of various dissected divisor-sum series, namely forms of the type$$f_i = \sum_{j=0}^\infty \sigma_1(36j+i) q^{36j+i}.$$Each of these ... 1answer 245 views ### Is special value of Epstein zeta function in 3 variables a period? Kontsevich-Zagier's article "Periods" contains the following question Is \displaystyle \sum_{x,y,z \in \mathbb{Z}}' \frac{1}{(x^2+y^2+z^2)^2} an extended period? (\sum' means we do not sum ... 2answers 313 views ### Lifting the Hasse invariant mod 2 Katz defines in Section 2.0 p-adic properties of modular schemes and modular forms the Hasse invariant as a mod p modular form A of weight p-1. In other words, it is a section of \omega^{\... 0answers 284 views ### Lifting a real quadratic twist of an Elliptic Curve to the modular curve Let E be an elliptic curve of conductor N\cdot p^2 over \mathbb{Q}, defined by the equation$$y^2=x^3+p^2b\cdot x + p^3\cdot c$$and parametrized by a map$$X_{0}(N\cdot {p}^{2})\rightarrow E ...
Let $R$ be an Eichler order of an indefinite quaternion algebra $B/\mathbb{Q}$ (suppose B is not the collection of $2\times 2$ matrices) and $S$ the corresponding Shimura Curve. Modular forms of ...
In p-adic properties of modular schemes and modular forms Katz formulates the following base change theorem as Theorem 1.7.1 Let $n\geq 3$ and $\overline{\mathcal{M}}_n$ be the compactified moduli ...