All Questions
Tagged with complex-multiplication modular-forms
12 questions
5
votes
0
answers
174
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Effective Hecke Equidistribution
In 1918 and 1920 Hecke introduced his L-functions attached to his Grössencharakteren (Hecke characters) and proved they are equidistributed in a sense to made precise momentarily. One can identify ...
- 177
4
votes
2
answers
197
views
Eisenstein $E_2$ at imaginary quadratic arguments
In the paper On Epstein's zeta-function, Chowla and Selberg give a formula for evaluating the Dedekind eta function
$$\eta (\tau)=e^{\pi i\tau/12}\prod_{n=1}^\infty (1-e^{2\pi i n\tau}),\quad \Im\tau\...
- 317
14
votes
2
answers
1k
views
Complex Multiplication and algebraic integers
Let $q=e^{2\pi i\tau}$ and
$$E_2(\tau) = 1 - 24 \sum_{n=1}^\infty\frac{nq^n}{1-q^n}$$ be the Eisenstein Series of weight $2$
and let $E_2^*(\tau) = E_2(\tau) - \frac{3}{\pi\cdot Im(\tau)}$ be the ...
- 598
11
votes
0
answers
380
views
What are the possible bad reductions for an abelian variety of dimension $g$ and a maximal endomorphism ring?
Perhaps the most basic fact about abelian varieties with CM is they have an everywhere potential good reduction (Serre-Tate). On the face of it it might appear that there isn't much more to be added ...
- 13.8k
23
votes
3
answers
2k
views
Why are values of Eisenstein $E_2^*$ algebraic integers?
I'm looking for a proof that the following term is an algebraic integer whenever $\tau_N=\frac{N+\sqrt{-N}}{2}$ is a quadratic irrationality with class number $1$:
$$A_N:=\sqrt{-N}\cdot\frac{E_2(\...
- 598
5
votes
2
answers
581
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 ...
- 874
15
votes
0
answers
289
views
What's the dimension of the space of CM cusp forms?
I would guess that the following is very well known, but I don't know the answer and I couldn't find anything with some googling.
Let $\Gamma \subset \mathrm{SL}(2,\mathbf Z)$ be a congruence ...
- 40.2k
9
votes
1
answer
318
views
Existence of CM Newforms in Level p
If $p$ is a prime and $k \geq 2$ is an even integer, what can we say about the existence of CM forms in the space $S_k^\text{new}(\Gamma_0(p))$? If it helps at all, I'm specifically interested in the ...
- 91
11
votes
3
answers
1k
views
Motive of CM elliptic curve and modular forms
I am trying to get some insight into the Deligne/Scholl construction of the motive of a modular form. First of all I would like to understand the case of weight two, especially when there is complex ...
- 111
8
votes
0
answers
429
views
Twists of CM modular forms
Let $K$ be an imaginary quadratic field of class number 1 and $E$ an elliptic curve over $\mathbf{Q}$ with CM by $\mathcal{O}_K$. Let $\psi$ be the Groessencharacter of $K$ attached to $E$, and
$$ g_{...
- 37k
19
votes
3
answers
4k
views
Definition of CM modular form
Dear friends,
I have some trouble finding a precise definition of what a modular form with complex multiplication. Could anyone provide such a definition and references for the study of CM modular ...
- 191
10
votes
1
answer
1k
views
Adelic formulations of complex multiplication and modular curves
In modular curves and modular forms, there is an adelic formulation, in which smaller open subgroups of some adelic group relate to higher level structure. As we know, higher level structure ...
- 15.4k