Questions about modular forms and related areas

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What is the relation between roots of classical and Atkin modular polynomial?

Modular polynomials are a good tools in elliptic curves. I need to find the roots of $l$-th classical modular polynomial $\Phi_l(X,Y)$ over the prime field. Unfortunately the coefficients of these ...
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2answers
294 views

How many integer solutions of $a^2+b^2=c^2+d^2+n$ are there?

Are there any references to study the integer solutions (existence and how many) of Diophantine equations like $a^2+b^2=c^2+d^2+2$, $a^2+b^2=c^2+d^2+3$, $a^2+b^2=c^2+d^2+5$...? Actually, I can prove ...
2
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1answer
112 views

Clarification request on sign changes of Hecke eigenvalues

In their paper 'Sign changes of Hecke eigenvalues', Matomaki and Radziwill established in Lemma 6.2 the following result: There exists absolute positive constants $c$ and $\eta$ such that uniformly in ...
4
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1answer
169 views

Does viewing an Eisenstein series as a sum over cusps explain the antagonism between Eisenstein serieses and cusp forms?

I'm trying to understand the relationship between various aspects of the concept of "Eisenstein series" (as discussed for example in Diamond & Shurman's "A First Course in Modular Forms"), in ...
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99 views

Kitagawa's p-adic modular symbols for different weights: a confusing observation

References are to K. Kitagawa, "On standard $p$-adic $L$-functions of families of elliptic cusp forms", Contemp. Math. 165, 1994. Let $\mathcal O$ be the ring of integers in a finite extension of ...
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1answer
227 views

On $\eta(6z)\eta(18z)$ and the splitting / modularity of $x^3 - 2$

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} + ...
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1answer
185 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 ...
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1answer
447 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 ...
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73 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 ...
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144 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 ...
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171 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 ...
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121 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 ...
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1answer
133 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 ...
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1answer
184 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 ...
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1answer
114 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 ...
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1answer
153 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 ...
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1answer
721 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 ...
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3answers
740 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 ...
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2answers
206 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 ...
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126 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 ...
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1answer
166 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 ...
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64 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 ...
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332 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 ...
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400 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) ...
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1answer
222 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 ...
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1answer
194 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 ...
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143 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 ...
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1answer
164 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 ...
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164 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 ...
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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 ...
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1answer
235 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 ...
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310 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 ...
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282 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 $$X_{0}(N\cdot {p}^{2})\rightarrow E$$ ...
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1answer
222 views

Adelic and classical modular forms on quaternion algebras

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 ...
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118 views

Riemann-Roch for curves over Dedekind domains and base-change for modular forms

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 ...
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403 views

What do we know about the structure of $J_{0}(N)$ over $\mathbb{Q}[{\mu}_{{p}^{\infty}},{{k}}^{\frac{{1}}{{p}^{n}}}])$?

What is known about the structure of $J_{0}(N)$ over $\mathbb{Q}[\mu_{p^{\infty}}]$? More generally, what do we know about $J_{0}(N)$ over $\mathbb{Q}[\mu_{p^{\infty}},k^{1/p^{n}}]$, where ...
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1answer
169 views

Polygonal Mersenne numbers [closed]

I posted the same question on Math SE since this one got put on hold. Link to Math SE question:Polygonal Mersenne numbers Polygonal numbers are of the form $\cfrac {n^2(s-2)-n(s-4)}{2}$, where $s$ ...
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1answer
155 views

How to determine whether a power of eta function is a eigenform? [closed]

I find that it is complicated to do this from the definition. In fact, I know that $\eta^k(m z)$ is a eigenform for $k=1,2,3,4,6,8,12,24$ and $m=\frac{24}{gcd(k,24)}$. What I want to know is the cases ...
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466 views

Analytic continuation for $L$-functions of elliptic curves

Let $E$ be an elliptic curve over a number field. When $E$ has no CM and is a $\mathbb Q$-curve (i.e. it is $\overline{\mathbb Q}$-isogenous to all of its conjugates), it is nowadays known that $E$ ...
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1answer
147 views

Upper bound on level of a congruence subgroup of the modular group

Let $\Gamma = PSL(2,\mathbb{Z}) = \langle S,T \ | \ S^2=(ST)^3=1 \rangle$. Let $G$ be some mystery normal subgroup of $\Gamma$ that we happen to think may be congruence. Recall that a subgroup of ...
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1answer
145 views

How do the Dim($K_f / \mathbf{Q}$) vary for all f in a given $S_k(\Gamma(N))$?

For the theory of classical modular forms, the space of new forms $S_k^{new}(\Gamma(N))$ has a basis of Hecke eigenforms $\{ f_i = \sum a_n q^n : a_1=1, a_n \in \bar{\mathbb{Q}}\}$ Given $k$ and $N$ ...
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1answer
276 views

Jacquet's approach to Rankin--Selberg L-functions

In his book "Automorphic Forms on GL(2), II", Springer Lecture Notes vol. 278, Jacquet defines the Rankin--Selberg L-function of $\pi_1 \times \pi_2$, where $\pi_i$ are automorphic representations of ...
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78 views

Tables of eigenvalues for Hilbert newforms of level $\mathfrak{p}$

Bit of a naïve question but are there tables of Hecke eigenvalues for Hilbert newforms over say real quadratic fields (of parallel weight not necessarily equal to 2 and level ...
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1answer
494 views

How much can an Eisenstein series be truncated?

For ease of exposition, I will stick to the simplest case: consider the Eisenstein series for $SL_2(\bf R)$ $$E(z,s)=\sum_{\gamma\in P_{\bf Z}\backslash SL_2(\bf Z)}\text{Im}(\gamma ...
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4answers
339 views

$p$-th Fourier coefficients of newforms of level $\Gamma_1(N)$ with $p|N$

Let $f$ be a newform of level $\Gamma_1(N)$ and character $\chi$ which is not induced by a character mod $N/p$. I learned from these notes by Ribet and Stein that $|a_p|=p^{(k-1)/2}$ where $k$ is the ...
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2answers
500 views

Computing millions of coefficients of non self-dual modular forms

To test some conjectures made by some colleagues, I need to compute millions of coefficients of non self-dual modular forms, preferably in low weight (say 2 or 3). A form such as this. For elliptic ...
8
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1answer
499 views

Plot of Ramanujan tau function

There is a picture on wikipedia of Ramanujan tau function. At first I noticed that there are exceptional red point (where the red points are sparse in the lower part), this should be due to Sato-Tate ...
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1answer
132 views

How are holomorphic and real-analytic Eisenstein series related?

This is certainly not a research level question, but I didn't get an answer to my question on MSE, so here goes: The holomorphic Eisenstein series can be given as $$G_{2k}(z)=\sum_{(c,d)\in{\bf ...
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1answer
124 views

degree of Hecke field (number field of an eigenform)

Let $f\in S_k(\Gamma_1(N))$ be an eigenform, and $K_f$ be its number field, which is of finite degree over $\mathbb{Q}$. Consider the following statements. 1, $[K_f:\mathbb{Q}]=\#\{$Galois conjugates ...
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2answers
291 views

Do “most” modular forms have no extra twists?

Let $f$ be a modular form -- more specifically, a normalized new eigenform which is not of CM type. We say $f$ has extra twists if there exists some $\sigma \in \operatorname{Aut}( \mathbf{C})$ such ...