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2
votes
1answer
121 views

isogeny clases of CM abelian varieties

Let $A$ be an abelian variety defined over $\overline{\mathbb{Q}}$ and with complex multiplication by a CM field $K$. Looking at the action of $K$ on $H^0(A, \Omega^1_A)$ one gets a CM type of $K$, ...
3
votes
1answer
122 views

CM abelian varieties over the rationals

Let $K$ be a number field and let $A$ be an abelian variety of dimension $g$ over $K$. Let $L$ be a CM field and suppose that $[L:{\bf Q}]=2g$. Suppose that there exists an embedding ...
5
votes
1answer
240 views

notion of $\mathrm{Gal}(\overline{\mathbb{Q}}_p / \mathbb{Q}_p)$ representation with complex multiplication

In usual Hodge theory, there is the notion of Hodge structure $H$ with complex multiplication, that can be defined in several ways, i.e. asking that there exists a CM number field $E$ such that $\dim ...
0
votes
1answer
58 views

The group of real points on quadratic twist of elliptic curve has one connected component

I am trying to understand the proof of assertion (i) in Proposition 3.10 (page 14) of this paper http://arxiv.org/pdf/1312.3884v3.pdf $M$ stands for a square free integer which is prime to $7$, $A$ ...
2
votes
1answer
174 views

Difference between Frobenii on Tate modules of special and generic fibre

Let $E$ be elliptic curve over $\mathbb Q$ and $p$ a prime of good reduction for $E$. Fix $\ell \neq p$. If $E_p$ is ordinary then we have Frobenius $F_p$ on $E_p$. Assume $F_p$ lifts to ...
16
votes
0answers
343 views

Why is the CM closure of $\mathbb{Q}$ the “ultimate” coefficient field for motives?

In a rough way, a category of motives over a field $k$ with coefficients in a field $K$ gives a universal cohomology theory with coefficients in $K$ for algebraic varieties defined over $k$. I had the ...
4
votes
1answer
182 views

Elliptic curves with maximal order in an imaginary field

Let $K/\mathbb{Q}$ be an imaginary quadratic extension with discriminant $-D$. Then there is an elliptic curve $E$ over $\overline{\mathbb{Q}}$ such that End$(E)^{0}: =$ End$(E) \otimes Q = K$. Now ...
12
votes
0answers
200 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 ...
1
vote
1answer
252 views

The special point count on Shimura varieties

This, in view of the analogies between CM points on Shimura curves and torsion points on elliptic curves, is a sequel to an earlier question I had asked: The torsion point count in higher dimension . ...
0
votes
0answers
94 views

Galois groups of CM fields which are a degree two extension of a cyclotomic field

Let $E$ be a CM number field. Assume that $E$ is a degree two extension of the cyclotomic field $\mathbb{Q}(\mu_n)$, so $E=\mathbb{Q}(\mu_n)(\sqrt{\kappa})$ for some $\kappa \in \mathbb{Q}(\mu_n)$. ...
0
votes
0answers
79 views

the CM type of a CM abelian variety

Let $(A, F, i)$ be a CM abelian variety, by which I mean an abelian variety $A$ defined over $\overline{\mathbb{Q}}$, say of dimension $n$, a CM number field $K$ of degree $2n$ and an embedding $i: F ...
4
votes
1answer
276 views

motives of elliptic curves, modular forms, Hecke characters

Let $E$ be an elliptic curve over $\mathbb{Q}$. By the modularity theorem, $L(E, s)$ is the $L$-function of some modular form $f$. Now one has the following motives: (a) The Chow motive $h^1(E)$ ...
2
votes
1answer
210 views

Canonical lifts from $\mathbb F_q$ and CM-theory

One knows that (ordinary) Jacobians of hyperelliptic curves over a finite field $\mathbb F_q$ (mostly of genus 1 (elliptic curves) and 2) are extensively studied by cryptographers, as a platform for ...
2
votes
0answers
93 views

Genus 2 hyperelliptic cryptography : typical discriminant and class number

As far as I know, there is no standard yet for cryptography based on the DLP over Jacobians of genus 2 curves. Yet, what can we say about the class number, and the discriminant of the complex ...
3
votes
2answers
195 views

abelian varieties with the same CM type are isogenous

Does anybody have a reference for the following fact? All abelian varieties with complex multiplication and same CM type are isogenous over $\overline{\mathbb{Q}}$? Here abelian variety with ...
1
vote
1answer
127 views

Hasse-Weil L-Functions of CM Abelian Varieties

In Shimura's paper "On the Zeta Function of an Abelian Variety With Complex Multiplication", in his terminology, the `one-dimensional part' of the zeta function is identified with a Hecke $L$-function ...
0
votes
1answer
177 views

how to see CM types as functions on the Galois group?

Let $K$ be a CM field, that is, an imaginary quadratic extension of a totally real number field. Its degree $[K: \mathbb{Q}]$ is a en even number $2n$. (1) For me a CM type is a subset $\Phi \subset ...
3
votes
2answers
466 views

Example of elliptic curve with CM (complex multiplication) by \sqrt{-7}

Can someone give me an example of elliptic curve with CM by sqrt(-7) with the action. I've found a list of examples in the following link but not the action. ...
3
votes
0answers
133 views

modular curves with complex multiplication

Is the complete list of values of $N$ for which the modular curve $X_0(N)$ has complex multiplication? I guess the answer is no... Are there at least some examples known? The only one which comes ...
6
votes
2answers
208 views

Rational points and torsion points of CM elliptic curve

Let $E$ be a CM elliptic curve defined over a quadratic imaginary field $K$ with maximal order i.e., $\mathrm{End}_K(E)\cong \mathcal{O}=\mathcal{O}_K$. Let $\mathfrak{p}$ be a prime of $K$ such that ...
5
votes
2answers
176 views

can all CM types be realized by Jacobians?

The question is kind of self contained, but let me develop a bit further. Assume K is a CM field of degree $2g$, that is, a quadratic imaginary extension of a totally real field. A CM type of $K$ is ...
2
votes
0answers
69 views

CM Hodge structures

There are several definitions of Hodge structures with complex multiplication. One of them is the following: Definition. A Hodge structure $H$ is CM if it is polarizable and its Mumford-Tate group ...
2
votes
3answers
479 views

transcendence of periods of CM elliptic curves

Let $E$ be an elliptic curve over $\overline{\mathbb{Q}}$ defined by a Weierstrass equation $$ y^2=4x^3+g_2x+g_3. $$ Then $H^1_{dR}(E/\overline{\mathbb{Q}})$ is spanned by the classes of the ...
9
votes
1answer
218 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 ...
1
vote
1answer
125 views

Hecke Character vs Grossencharakter

I would like to know if there is any difference between (1) an algebraic Hecke character (2) a Hecke character (3) a Grössencharakter All of the above in the setting of ellitpic curves with complex ...
1
vote
1answer
394 views

Hecke Characters

My question today concerns Hecke characters or Größencharaktere. I'm doing a project on complex multiplication of elliptic curves and need some help understanding Hecke characters. My main ...
0
votes
0answers
86 views

field of definition of abelian varieties with extra endormorphism

Let $A$ be a complex abelian variety such that $\mathrm{End}(A)$ is strictly bigger than $\mathbb{Z}$. Question: Is is true that $A$ is defined over $\overline{\mathbb{Q}}$? This is of course what ...
7
votes
1answer
301 views

motive of CM elliptic curve and modular forms

I am trying to get some insight into Deligne/Scholl construction of the motive of a modular form. First of all I would like to understand the case of weight two, especialy when there is complex ...
7
votes
0answers
251 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 $$ ...
9
votes
1answer
605 views

How did Takagi prove Kronecker's Jugendtraum for Q(i)?

In Noah Snyder's historical undergraduate thesis on Artin L-Functions, it mentions that Takagi proved Kronecker's Jugendtraum in the case of Q(i) in his doctoral thesis. Since I don't know how to get ...
8
votes
3answers
881 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 ...
1
vote
1answer
279 views

complex multiplication

For an abelian variety $A$, it is said to be have $complex \ multiplication$ if $\mathrm{End}(A) \otimes_{\mathbb{Z}} \mathbb{Q}$ contains a number filed $F$ of degree $2 \cdot \mathrm{dim} (A)$. ...
1
vote
1answer
433 views

David Hilbert on Complex Multiplication [closed]

I have tried vainly to understand the significance of the following statement attributed to David Hilbert: The theory of complex multiplication is not only the most beautiful part of mathematics ...
10
votes
0answers
666 views

What numbers are integrally represented by $4 x^2 + 2 x y + 7 y^2 - z^3$

This is related to my first MO question and Kevin Buzzard's conjecture at Integers not represented by $ 2 x^2 + x y + 3 y^2 + z^3 - z $ In December 2010 my question appeared in the M.A.A. Monthly, ...
20
votes
1answer
983 views

Can one prove complex multiplication without assuming CFT?

The Kronecker-Weber Theorem, stating that any abelian extension of $\mathbb Q$ is contained in a cyclotomic extension, is a fairly easy consequence of Artin reciprocity in class field theory (one just ...
3
votes
0answers
228 views

Formal non-CM in local fields

An elliptic curve $E$ with complex multiplication by an imaginary quadratic field $F$ has $\ell$-adic Galois representations that essentially encode the class field theory of $F$ - in other words, the ...
2
votes
1answer
839 views

books (or notes) on complex multiplication

This would be a vague question, but I still want to ask here. Do you have any recommended book on complex multiplicaton. I know only 2 books: Shimura's book ...
5
votes
1answer
543 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 ...
6
votes
1answer
811 views

CM rational points on modular curves

Dear MO Community, I am trying to understand Mazur's 1976 notes "Rational points on Modular Curves" (which can be found in Springer Lecture Notes in Mathematics 601). Let N be a prime number, and ...
6
votes
4answers
2k views

Class Field Theory for Imaginary Quadratic Fields

Let $K$ be a quadratic imaginary field, and E an elliptic curve whose endomorphism ring is isomorphic to the full ring of integers of K. Let j be its j-invariant, and c an integral ideal of K. ...
10
votes
0answers
940 views

Are there analogues of Beilinson's conjectures for motives with coefficients?

There's a body of wisdom (following Beilinson, Bloch, Deligne, ...) relating mixed Tate motives, motivic cohomology, algebraic K-theory, special values of L-functions, and polylogarithms. My ...
24
votes
5answers
2k views

A problem of Shimura and its relation to class field theory

In Chapter II.10 of The Map of My Life, Goro Shimura mentions a certain problem: The second topic concerns a polynomial $F(x)$ with integer coefficients. Take $$ F(x) = x^3 + x^2 - 2x - 1, $$ ...
10
votes
2answers
796 views

Legitimacy of reducing mod p a complex multiplication action of an elliptic curve?

I scoured Silverman's two books on arithmetic of elliptic curves to find an answer to the following question, and did not find an answer: Given an elliptic curve E defined over H, a number field, ...