All Questions
86 questions
10
votes
2
answers
286
views
Finding a model for $X_G$ with $G\subseteq \mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})$
I'm studying modular curves as part of my doctorate work and would like to understand how one gets from a subgroup $G$ of $\operatorname{GL}_2(\mathbb{Z}/N\mathbb{Z})$ to an equation for $X_G$. By $...
- 195
3
votes
0
answers
345
views
Modern integral $p$-adic Hodge theory and modularity lifting and Fontaine-Mazur
As a follow-up to a comment on this answer, I'm wondering if there are expected to be applications of the new point of view on integral $p$-adic Hodge theory, à la Bhatt-Morrow-Scholze and others, to ...
- 15.4k
17
votes
3
answers
2k
views
Are some congruence subgroups better than others?
When I first started studying modular forms, I was told that we can consider any congruence subgroup $\Gamma\subset\operatorname{SL}_2(\mathbb{Z})$ as a level, but very soon the book/lecturer begins ...
- 821
10
votes
1
answer
550
views
Igusa's $\chi_{10}$ and Borcherds products
Igusa defined a genus 2 Siegel modular form $\chi_{10}$, which vanishes on the Humbert surface $G_{1}$ (the image of a "degenerate" Hilbert modular surface, the product of modular curves, ...
- 3,309
3
votes
1
answer
384
views
Overconvergent modular forms and the level at $p$
I am a little bit confused about the basic theory of overconvergent modular forms, so here is a question that I think will be straightforward for those who know the theory but would help me a lot.
The ...
- 241
4
votes
0
answers
204
views
$\DeclareMathOperator\sym{sym}$Does $L(s, \sym^2 f \times \sym^2 g)$ have a pole at $s=1$?
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\sym{sym}$I encountered a question on the poles of $\GL_3\times \GL_3$ $L$-function, which needs the knowledge of the experts ...
- 563
6
votes
1
answer
308
views
An explicit equation of the canonical morphism $X_1(N) \to X_0(N)$
I know there are some research about explicit equations for affine models in $\mathbb{A}^2$ of many modular curves over $\mathbb{Q}$, for example of $X_i(N), X(N)$ (where $i = 0, 1, 2$) for small $N$.
...
- 1,364
3
votes
0
answers
186
views
Maximum value of newform from Galois representation
One can attach $\ell$-adic Galois representations to holomorphic cuspidal newforms of weight $2$ on the upper half-plane.
If a newform is $L^2$-normalized, can one extract its maximum value from the ...
- 39
2
votes
1
answer
227
views
Global section of vertical differential 1 forms on universal elliptic curve
Let $B$ be a modular curve (of some level) over a number field $K$ (here, we implicitly assume that $K$ is large enough to make sense the phrase "$B$ is a $K$-variety"). Let $E\to B$ the ...
- 1,428
16
votes
0
answers
274
views
Why should an abelian variety with few places of bad reduction and a lot of endomorphisms not have many points?
In the paper "Points of Order 13 on Elliptic Curves" by Mazur-Tate, they say in the introduction:
It seemed ... that if such an abelian variety $J$, which has bad reduction at only one ...
- 7,746
5
votes
0
answers
147
views
Is the cohomology of Hilbert modular surfaces spanned by special cycles?
We consider the Hilbert modular surface $X$ that parametrize abelian surface with real multiplication by $\mathcal{O}_F$ and $\mathfrak{a}$-polarization, where $F$ is a real quadratic field with ...
- 409
4
votes
1
answer
282
views
Mean square estimate for the Kloosterman sums
For $m,n\in \mathbb{N}$, denote the Kloosterman sum
$$S(m,n;c)=\sum_{a\bmod c}e\left( \frac{ma+n \overline{a}}{c}\right),$$where $\overline{a}$
denotes the multiplicative inverse of $a\bmod c$.
Does ...
- 353
4
votes
1
answer
211
views
anti-holomorphic Hilbert modular forms as global sections
The classical definition of Hilbert modular cuspforms as given in say, Hida's "On $p$-adic Hecke algebras for $\mathrm{GL}_2$ over totally real fields", defines them as holomorphic functions ...
- 302
0
votes
0
answers
219
views
How canonical are integral lifts of Hasse invariants and other mod-$p$ modular forms?
Let $p$ be an odd prime. Recall that the mod $p$ Hasse invariant $A$ of an elliptic curve is an $\mathrm{SL}(2,\mathbb Z)$-modular form of weight $p-1$ defined over $\mathbb{F}_p$. Writing $\overline{\...
- 54.6k
4
votes
1
answer
347
views
Weight 3 modular form associated to singular abelian surfaces?
Given an extremal K3 surface $S$ over $\mathbb{Q}$ (i.e. a K3 surface with maximal Picard rank) there is a 2-dimensional Galois representation on the transcendental lattice $T(S)$, and an associated ...
- 1,701
2
votes
0
answers
105
views
Possible Context for this "Siegel-like" Modular Form Construction?
The following construction of something very nearly a Siegel modular form of degree 2 arose in my research. I'm outside the worlds of automorphic forms and number theory, so I'm wondering if it ...
- 1,701
3
votes
1
answer
335
views
Comparison of two definitions of the modular sheaf $\omega$
I have seen two equivalent definitions of the modular sheaf $\omega$. Let $S$ be some base scheme. If $p \colon \mathcal{E} \to X$ is the universal generalized elliptic curve over the modular curve $X$...
- 949
4
votes
0
answers
230
views
Extra line bundles from torsors
Another math.stackexchange question (here: $\mathbb{G}_m$-torsors and line bundles) goes over a way to construct a line bundle $L$ from a $\mathbb{G}_m$-torsor $T \to B$, by using a decomposition $\...
- 949
6
votes
2
answers
1k
views
About different cohomology theories used to study Shimura varieties
The classical theory of Eichler-Shimura realizes the space of cusp forms of certain weights and levels in the parabolic cohomology of modular curves, which is the image of the cohomology with compact ...
- 159
2
votes
0
answers
214
views
Cohomology of modular curves: vanishing and decomposition
Let $\pi:E\to Y$ be a universal elliptic curve over an open modular curve $Y$. Take a prime $\ell$ and take $\mathcal{H}=(R^1\pi_*\mathbb{Q}_\ell)^\vee$ where the dual, $(-)^\vee$, means the internal ...
- 1,428
3
votes
0
answers
166
views
Automorphy Factor from Vector Bundles on Compact Dual
So I'm coming from an algebraic geometry perspective and I'm trying to carefully piece together the story of interpreting automorphic forms as sections of vector bundles on Shimura varieties. I think ...
- 1,701
10
votes
1
answer
562
views
Are there infinitely many real multiplication fields of abelian surfaces over $\mathbb Q$?
Do there exist infinitely many real quadratic fields $F$ such that there is an abelian surface $A$ over $\mathbb Q$ whose ring of endomorphisms, tensored with $\mathbb Q$, is $F$?
Do there exist ...
- 148k
2
votes
1
answer
290
views
Dimension formulae for Jacobi forms
I'm interested in locating dimension formulae for (more general) Jacobi forms associated with a lattice $L$ (where the Jacobi forms of Eichler-Zagier correspond to $L=A_1$).
Unfortunately, the ...
- 61
5
votes
0
answers
201
views
The Geometry of Jacobi Forms and their Asymptotic Expansions
A Jacobi form of weight $k$ and index $m$ is a meromorphic function $\varphi: \mathbb{H} \times \mathbb{C} \to \mathbb{C}$ satisfying
$$\varphi\bigg(\frac{a \tau + b}{c \tau + d}, \frac{z}{c \tau + d}...
- 1,701
5
votes
0
answers
158
views
Maass-Saito-Kurokawa Lift of Weak Jacobi Forms
Given a holomorphic Jacobi form $\varphi_{k,1} \in \mathbb{J}_{k,1}$ of weight $k$ and index 1, we know we can use the Hecke operators $V_{m}$ to lift $\varphi_{k,1}$ to a Siegel modular form
$$\...
- 1,701
3
votes
1
answer
480
views
Index one weak Jacobi forms and weakly holomorphic modular forms?
At the beginning of Eichler and Zagier's book on Jacobi Forms, there is the following diagram, summarizing part of the special role played by Jacobi forms of index 1.
One of the things confusing me ...
- 1,701
0
votes
1
answer
278
views
Proof of the coherence of ${\Bbb F}_q[[X_1,\dots,X_{\infty}]]$
We shall define the infinitely-many-variable formal power series ring $A = {\Bbb F}_q[[X_1,\ldots,X_{\infty}]]$ over a finite field ${\Bbb F}_q$ as the following$\colon$
$A \colon= \underset{n \geq ...
- 563
5
votes
0
answers
323
views
Existence of infinitely many Heegner points that are divisible by $p^{n}$ in $K_{\lambda}$
Let $E$ be an elliptic curve over $\mathbb{Q}$, $p>2$ a prime and $n,s\in\mathbb{N}$.
For $j\in\{1,...,s\}$ let $n_{j}\in\mathbb{N}$ be a natural number which may or may not be coprime to $p$.
Let $...
- 1,805
4
votes
2
answers
570
views
What are positive divisors of degree 2 on Elliptic Curve $y^2=x^3-x-1$ over $\mathbb{F}_3$?
Let (E) be an elliptic curve $y^2=x^3-x-1$ over $\mathbb{F}_3$, $a_0=1$, $a_n$ is the number of positive divisor of degree $n\geq 1$. $a_1$ in this case is the number of points of E, i.e., $a_1=1$ ...
- 41
8
votes
2
answers
403
views
Mazur's Question on Mod $N$ Galois representations
In Rational Isogenies of Prime Degree, Mazur poses:
"the problem of determining all elliptic curves $E'/\mathbb{Q}$ with symplectic $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ isomorphisms $E'[N]\...
- 901
4
votes
0
answers
413
views
quasi-finite group schemes
The following is what Mazur wrote on page 91 of his paper, Modular curves and the Eisenstein ideal, published in Publ. IHES in 1977, DOI: 10.1007/BF02684339 (freely available at eudml):
Let $m$ be an ...
- 335
13
votes
1
answer
444
views
Finite generation of module of modular forms
Given a commutative $\mathbb{Z}[\frac1n]$-algebra $R$, we can consider the ring of modular forms $M_*(\Gamma_1(n), R)$. If $R$ is a subring of $\mathbb{C}$, these can be defined as those (holomorphic) ...
- 12.1k
7
votes
1
answer
482
views
Geometry of Hecke Operators on Jacobi Forms?
In something I've been thinking about recently, the following object appears:
$$\mathcal{F}_{g} = \sum_{n=0}^{\infty} Q^{n} T_{n}\big( \phi_{2g-2}(\tau, z) \big)$$
where $T_{n}$ is the $n$-th Hecke ...
- 1,701
4
votes
1
answer
311
views
Are there "primitive" modular functions for $\Gamma(n)$ with Fourier coefficients in $\mathbb{Q}$?
The modular curve $Y(n)$ which over $\mathbb{C}$ is $\mathcal{H}/\Gamma(n)$ is often viewed as a curve defined over $\mathbb{Q}(\zeta_n)$. However, if one twists the moduli problem to be given by ...
- 10.7k
4
votes
0
answers
279
views
On De Shalit's Lemma in Wiles' proof of R=T
In Wiles' proof of ``$R = {\Bbb T}$", if we associate the extra primes the set of which is denoted $D$, there is a famous result in the following$\colon$
Theorem(De Shalit's Lemma). Let ${\Bbb T}_N$ ...
- 781
15
votes
2
answers
1k
views
Modular forms from counting points on algebraic varieties over a finite field
Suppose we are given some polynomial with integer coefficients, which we regard as carving out an affine variety $E$, for example:
$$ 3x^2y - 12 x^3y^5 + 27y^9 - 2 = 0 \tag{$*$} $$
(We might ...
- 1,821
7
votes
1
answer
794
views
Special fiber of the Néron model of the generalized Jacobian of a singular curve
Let $C$ be a curve over $\mathbf{Q}_p$ (or a finite extension) whose minimal regular model $\mathcal{C}$ over $\mathbf{Z}_p$ has a "nice" special fiber (maybe singular, with at most ordinary double ...
- 1,109
6
votes
1
answer
408
views
Is the ring of meromorphic modular forms on a fine modular curve generated in degree 1?
Consider a torsion-free congruence subgroup $\Gamma\le SL_2(\mathbb{Z})$ (for example, $\Gamma(N)$ for $N\ge 3$, or $\Gamma_1(N)$ for $N\ge 4$).
By a meromorphic modular form for $\Gamma$ of weight $...
- 7,527
17
votes
1
answer
852
views
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 (...
- 734
3
votes
0
answers
311
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 ...
- 845
21
votes
3
answers
1k
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 $X(p)$,...
- 148k
5
votes
0
answers
224
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 $\...
- 845
7
votes
1
answer
1k
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 $E$ ...
- 509
3
votes
1
answer
182
views
Does a modular function primitive for $\Gamma$ generate the function field of $\mathcal{H}/\Gamma$?
Let $f$ be a modular function (that is, a meromorphic modular form of weight 0) holomorphic on $\mathcal{H}$ which is invariant under $\Gamma\le SL_2(\mathbb{Z})$ (not necessarily congruence!), and ...
- 10.7k
8
votes
0
answers
403
views
rings of modular functions on the upper half plane
Let $\Gamma_1\le SL_2(\mathbb{Z})$ be a noncongruence subgroup of finite index.
Let $\Gamma_2\le SL_2(\mathbb{Z})$ be another subgroup of finite index.
Let $M_0(\Gamma_i)$ denote the ring of modular ...
- 10.7k
2
votes
0
answers
246
views
Morphism of Shimura varieties and differential equations
Is there a way of constructing a morphism between Shimura varieties using differential equations? Maybe, this looks like a completely ridiculous question, so I think that I should explain the context ...
- 21
9
votes
1
answer
864
views
Definition of p-adic modular forms
I have been reading Hida's book "p-Adic automorphism forms on Shimura varieties" and I don't understand a point.
He first describes p-adic modular forms of tame level N as functions on the Igusa ...
- 845
17
votes
1
answer
745
views
Special fiber of $X(p)$ in characteristic $p$
Let $p \geq 5$ be a prime. Let $Y(p)$ be the fine moduli space representing elliptic curves + basis of the $p$-torsion over $\mathbb{Q}_p$ and let $Y_0(p)$ be the fine moduli space representing ...
- 1,109
5
votes
2
answers
674
views
Why is the supersingular locus the zero locus of a modular form?
This question is related to my other question here: Examples of subspaces singled out by modular forms.
Here I am wondering if there is a philosophical explanation about why the supersingular locus ...
- 231
3
votes
0
answers
154
views
Examples of subspaces singled out by modular forms
I am wondering what subspaces of modular varieties defined as the zero locus of modular forms have been studied in the literature.
To be more clear let me explain the example I have in mind.
Let $N\...
- 231