Algebraic Number Theory - Science topic

Comparing to elusive truth, how much is an irrational falsity worth?

A sound, archetypical, immemorial, cross-species, well-known, and logical rule is that it is better to reject 100 truths than to accept one falsity. A rejected truth can be accepted tomorrow, but an accepted falsity contaminates the thought immediately.

Truth is worth 1:100 a falsity, maybe more. This applies to all discussions at RG.

This question is now closed, reaching a YES conclusion and preventing bering hijacked by 'wolves" in ResearchGate.

It serves rhe purpose of explanation to those interested, as an open group. Enjoy and pursue the new ideas with your contributions in your space.

DH: What is odd is human ego. The same posts on different topics, different threads, attract different readers.

Howdy Folks,

I am satisfied that mathematics and physics (science) have been well defined and described here. A couple movies are running in my mind's eye that I wish to pass along as afterwords - they are observations not insults.

In Ray Bradbury's work "Medicine for Melancholy" he includes a prose movie of Pablo Picasso sketching a mural in the moist sand of a long beach as the tide is coming in - just like the "then current" theories in science presented by academics as truth, the foaming edges of the waves wash the mural away - new paradigms replace old and the human creation of science is adjusted. It is not "nature" even now.

M. C. Escher's "Metamorphosis" is a great contribution to defined elements fitted together perfectly into a closed, consistent, unnatural whole. Rigorous, however independent of nature, EXCEPT FOR THE FACT that humans and their imagination are natural. I disagree with the separation of "artificial" from natural, except as a verbal convenience.

These are creations of human minds, not figments of human imagination. And I carefully avoided an observation that fresh ideas are not fertilizer and to bury them in a field will not benefit flowers there.

Great exchange, Thanks again, Happy Trails, Len

Stop Hallucination.

Stop the fake fan team.

Stop disturbing and spreading wrong math education.

Show your idea in your thread and stay there.

I have hundreds of students and university colleagues worldwide, and I feel shame asking them to grant a fake recommendation that votes for wrong, bad answers.

I dare any one of your team to join a serious debate in mathematics

about FLT and its consequences.

see

Dear Keyvan Ghaffari

Your efforts are appreciated. No mathematician dares to claim that he proved the Reimann Hypothesis unless they successfully published the results in a reputable Journal. Hundreds of proofs are available online, but all are not trusted before appearing in the right place.

I wish you good luck.

Dear Amirali Fatehizadeh

It would help if you studied advanced abstract algebra, topology, mathematical analysis besides the introductory courses in general number theory.

Regards

A new paper:

Regards,

Joachim

As far as I know, in the entire history of mankind, only two philosophers have seriously dealt with logic, this is Aristotle and Hegel. Of these, only Hegel did mathematics. Nobody else dealt with this problem.

Sincerely, Alexander

If anyone proposes a theory and no one can make it wrong, at least it is not wrong, but it remains a mystery, and that is why it is called a mystery .. This does not mean that the theory is wrong.

Dear Majid K.Neamah ,

I agree with you. Although year after year we are getting closer. A few year ago, it was proven that at least 40% of zeros have to be on the critical line. So a lot of progress has been done. Many conjecture have also been presented, which if proven, they would imply the Riemann Hypothesis. So have gotten some important result, however, we have not quite solved it yet.

Stam Nicolis , you said that

" However it's not necessary for the Hilbert-Polya conjecture to be true, in order for the Riemann Hypothesis to be true. "

Why is that?

Dear Peter Kepp

Why are you asking?

Please respond when you read this.

Let K={|z|<=1} be the closed unit circle (a connected compact) in C, R=O(K) be the ring of holomorphic functions f on K with d(f)=#{zeroes of f in K}, R is a ED.

Let a=a(z)=z^2+2z+1, b=b(z)=z (hence d(a)=2, d(b)=1). Let h=h(z)=sin(z)/z, h(0)=1.

Then

1) z^2+2z+1=(z+2)z+1, q=z+2, r=1 (hence d(r)=0)

2) z^2+2z+1=(z+2+h)z+1-zh, q'=z+2+h, r'=1-zh=1-sin(z) (hence d(r')=0,

since sin(z)=1 iff z=\pi/2 +2\pi*k, k\in Z).

Jean-Yves Boulay

Dear Jean-Yves

You talk about `set´. The naturals (1, 2, 3, ... ) should be elements of that set.

Is that correct?

Before I will give a longer answer, I have this questions:

"What is your definition of a set?” [Maybe that of Cantor?]

"What is your set of?” [Maybe elements?]

"What is your definition of an element?” [Maybe such things which are equal among themselves?]

Please think at first about the difference between `set´ and `classification´.

Would the naturals get classified or defined by the expression `set´?

Are the naturals elements or subsets?

Regards, Peter

Dear Romeo Meštrović

Thank you for your recommendations.

Dear Omran Kouba

Fermat numbers are odd numbers.

I think you are confused with Fibonacci numbers where F_3n even numbers.

Best wishes

Dear Amirali

You can try this also.

Hi, perhaps another solution is to solve two linear systems of the form A.x=b where b is set to (1,0,0...0) and (0,1,0,0....0) respectively, just to obtain the first two columns of the inverse of A.

There is a paper by Bushnell and Reiner on zeta functions of orders which you may find of interest.  You can find a free copy at:

If a<0 then the period is apparently 12.

class number  " h" :is the order of Ideal class group.

Ideal class group :In Z[sqrt(n)] the fractional ideal F is a group ,and the principal ideal P is a subgroup in it so we can take the quotient F/P.

real quadratic field: an extension of the rationals Q by the root of an irreducible quadratic equation over the rationals Q(sqrt((an)2+4a)).

the equation is :   x2- ((an)2+4a)y2 = 4m [or -4m]  with  :  a: odd integer ,n : odd integer.

m: integer .

the relation of "class number" and "real quadratic field" with the integer problem eich i put in question is: " h=2" then the equation represent a norm of ideal so we have an integer solution . now if I know when the equation have an integer solution i can determine the real quadratic fields

the solution wich i need "or wich I need the condition for exist it or not exist "should be (x,y) with " y" don't equal zero.

If you look for "applications in real life" of diophantine equations, the first striking example coming to mind is indeed cryptography with « public key » using elliptic curves, as pointed out by Rogier Brussee. An elliptic curve is a plane irreducible projective cubic. It is miraculously endowed with a natural abelian group structure : given 2 points P,Q on the curve, a simple geometric construction allows to associate to them a 3rd point, denoted  S = P + Q. Given a point P, the sum of d copies of P is written dP. The so called « discrete log » problem on an elliptic curve consists in finding d when knowing P and dP (see any textbook on the subject). It can be quite difficult depending on the choice of the elliptic curve. The front cover of any recent « biometrical » passport in Europe and in the USA is equipped with a chip which contains detailed information on the owner, encrypted via an elliptic curve. If this is not « real life »…

If you accept to include in « real life » the real world of physics, then there is another striking –  perhaps less well known - application based on diophantine approximation. Instead of defining this subject in a general way, let us consider a few particular cases.Typically, diophantine results come in the form of certain finiteness statements, e.g. that certain equations have only a finite number of rational or integral solutions. Consider for example the diophantine equation x^3 – 2y^3 = n (a given integer). By factorizing the LHS inside the decomposition field of the polynomial X^3 – 2,  one can easily show that any possible solution (x, y) must satisfy C /⎹ y^3 ⎹ >=⎹  (x/y – cubic root 2)⎹  , where C is a constant independent of x and y. The problem is thus reduced to approximating irrational numbers by rationals in a certain way. The celebrated theorem of Thue-Siegel-Roth asserts that for any  algebraic number a of degree >= 2, for any e > 0 , the inequality ⎹  p/q - a⎹  >= 1/ q^{2+e}  must hold, with the exception of finitely many rationals p/q where p, q are integers and q > 0.

This shows readily that our diophantine example above admits only a finite number of solutions. Actually, using diophantine approximation (before Roth’s result), Siegel had shown that any affine curve of positive genus defined over a number field contains only a finite number of integral points. In fact, most of the important known results in diophantine equations are consequences of the TSR theorem (including Faltings’ second proof of the Mordell conjecture). Now, where does the physical world come in ? An unexpected close relationship between diophantine approximation and dynamic systems has been revealed by Siegel (1942). When studying the stability of the solar system, one comes across a problem of « small divisors (denominators) » associated with the near resonance of planetary frequencies, which could result in the divergence of the solution expressed as a series. A little bit more mathematically, the dynamical properties of a rotation of angle alpha on a circle of length L are radically different according as the parameter alpha / L is rational or not : in the first case the orbits are all periodic, in the second, they are all dense. Siegel’s idea was to impose a suitable diophantine inequality on the frequencies to ensure that they are not too close to resonance. For more mathematical details, see e.g. Rodrigo A. Pérez, « A brief but historic article of Siegel », Notices of AMS, 58 (4), 2011, 558-567.  

 Applications to other math. domains have been evoked by Rogier Brussee, especially the « Langlands program », which aims to build bridges between 3 islands : 1) Arithmetic varieties (i.e. diophantine equations) and their L-functions   2) Representations of the absolute  Galois group of Q and their L-functions  3) Representations associated to automorphic forms and their L-functions. Will these fundamental (as opposed to applied) developments one day find their way into the real world ? Without discussing the distinction which you seem to make between the real world and the world of ideas (this would bring us back at least to Plato !), a reasonable attitude could be : fundamental science, especially in math., is primarily concerned with knowledge ; the applicable part of it is small, but  this part plays a prominent role in our control of the physical world ; no searcher or scholar can predict what fundamental discovery could be applied one day, and for what purpose. In these times when money tends to go only to applications, what do you think of this fable : it was in the 19-th century ; « deciders» in various governments had decided to pump money and positions into a big plan of research : how to improve lighting by improving the candle ; then a neglected somebody invented the electric bulb.

For small characteristics field like F2^n there are better algorithms quasi-polynomial and so on, much better than the otherwise sub exponential ones (and they do not translate to the general case). A good survey on the state of the art, see:

"For the question assumes the consensualist ought to generally agree with the questioner."

Can we be clear: you are not actually asking a question, you are asking for people to agree with you. This is antithetical to the scientific method where models are rigorously tested both by argument and by experiment. If your theory has merit it ought to stand up to a few basic questions. If it can't, it can't ever be anything more than an opinion. If you don't want people to debate your theory with you, you are on the wrong site. Just to be clear.

Communication protocol  with  high  level Scurry  type   supported   distribution  

As you say, the theory of "complex multiplication" (looking at modular invariants of elliptic curves with CM) solves the problem of determining the maximal abelian extension of an imaginary quadratic field. But as yet no analogous theory is known for real quadratic fields, although class field theory affords  a complete theoretical (but not explicit) description of the abelian extensions of a given number field. You can  take the measure of our ignorance by  listening to the short negative answer given by Serre at the end of his talk on abelian Galois theory at the occasion of Galois' bicentenary (the talk can be found on You Tube, I think).

${F_{{5^k}n}}(q) \equiv 0\bmod {[{5^k}]_q}$ is equivalent with ${F_{{5^k}}}(q) \equiv 0\bmod {[{5^k}]_q}$ 

because ${q^{{5^k} + n}} \equiv {q^{{5^k}}}\bmod {[{5^k}]_q}$ and therefore  ${F_{{5^k}(n + 1)}}(q) \equiv {F_{{5^k}n}}(q){F_{{5^k}n + 1}}(q) \equiv 0\bmod {[{5^k}]_q}.$

Therefore it suffices to show that

${F_{{5^k}}}(q) \equiv 0\bmod {[{5^k}]_q}.$

Use the Laplace transform

It is easy to show that n!+n(n+1)/2 is never an even perfect number. It is known that every even perfect number has form 2^p-1(2^p-1), where p and 2^p-1 are primes. If n!+n(n+1)/2=2^p-1(2^p-1), then n(n+1)<2^p(2^p-1), and so n<2^p-1. If n is odd then n|(n!+n(n+1)/2), hence n|(2^p-1). Since

2^p-1 is a prime, we have n=1, but in this case n!+n(n+1)/2 is not a perfect number. If n is even and 2^k ǁ n, then 2^k-1 ǁ (n!+n(n+1)/2), hence

k-1=p-1 and k=p. But 2^k≤n<2^p, contradiction.

Hi,

Thank you for your responses.

Yes I know that DLS will give me a hydrodynamic diameter, however I watch a webinar of the same company (Malvern) explaining how to understand the different distributions (Volume, Number and intensity).

Unfortunately, they explain and they don't go through the calculations on how they do that, the software itself has an algorithm that does the conversion automatically. That's what I want to know, what is the fundamental of the conversion.

Thanks

Thanks Danny

According to what is expressed in the previous answers, I think that the smallest gap does not exist because the gap tends to zero when n tends to infinity. 

There is something more interesting and is related with the primes:  which is the lower boundary of P (k + 1) - P (k) being both primes when k tends to infinity?

This is my answer:

Dirichlet, demonstrated that:

For any two positive coprime integers a  and b  , there are infinite primes of the form  a+bm, where  n is a non-negative integer (n=1,2,... ). In other words, there are infinite primes which are congruent to a mod b . The numbers of the form  a+bn is an arithmetic progression.

Actually, Dirichlet checks a result somewhat more interesting than the previous claim, since he demonstrated that:

∑_(p=a mod b) [lnp/p]→∞

Which implies that there are infinite primes p≡a mod b.

Therefore

H_1:=〖lim⁡inf〗┬(k→∞)⁡(p_(k+1)-p_k )=2

Becasue according with Dirichlet's theorem:

p_(k+1) when k→∞)=[(a+bn when n→∞)] and

p_(k) when k→∞)=[(b+bn when n→∞)]

Then the smallest gap will be a-b=2, since primes.2 are odd.

This solved the conjecture of twin primes. Zhang came to 246 in  2014 by other form.

The answer is NO. Take the a trivial Lie algebra L, that is just a vector space L (over an arbitrary field) with the product [x,y]=0 for all x,y in L. Let a  f:L-->L a non-linear mapping satisfying f(0)=0.

Then [f(x),f(y)]=0=f(0)=f([x,y]) for all x,y in L. No "aritmetic" helps there.

a strong PV number is a PV number such that the second conjugate µ (for example) is positive < 1 and the remaining conjugates are of the absolute value <µ

Dear Hanifa,

The first answer is NO. Take e.g  a = b = 1/2.

The second is NO, e.g. b = 0.9 , a = - 0.8

Dear Rafiq, 

As it said by Wiwat and Peter, for n=3 or n ≥5, the alternating group A_n is simple ( there are no normal subgroups) . For n≥5, A_n the only normal subgroup of S_n. You can see the link of wiwat or see the book: Introduction à l'algèbre (K. Kostrikin), you will find all details. For n<5, just use Sylow theorems and with information of S_n and A_n, it is so easy to find them, for example for S_4, you have A_4 and if there is an other normal subgroup, then it will be of order 2 or 4 or 8, because by Sylow theorems, there are 4 subgroups of order 3, in fact you can find them: <(123)>, <(124)>, <(134)>, <(234)>. So  there is no normal subgroup of order 3 in S_4. Now, you have to compute for 2, 4 or 8. Bon courage.

I would like to also recommend chapter 2 ("Global Fields") of the classical book 3Algebraic Number Theory", Cassels & Fhröhlich ed., Acad. Press 1969, which gives a compact and complete account of the subject. Concerning your specific question, see especially §7, p.50

Let us write the X-tuples as 01000..., 0110100... (those from your Examples) etc. Consider the real numbers w from the interval [0,1] written in the form w= 0.[X-tuple].

Since the number of 1 in an X-tuple is finite, the X-tuple can be considered as a diadic form of a rational number. Therefore, each number w obtained in the described manner is rational.

Conclusion: The considered set of X-tuples is countable.

Look at

1]:Stark,H.M. A complete determination of the complex quadratic ...elds of

class number one.M ich:M ath:J:14(1967); 1 27:

[2]:Birch,J.B.Weber’s Class Invariants.M athematika 16(1969); 283 294:

[3]:Schertz,R."Complex Multiplication",2010.

[4]:Zagier,

also, Pilar Bayer work is very interesting

Dear Mokhfi,

In the following paper, the authors give an explicit algorithm to calculate what you desire. I have only found references, such as the following article, in french. I hope you find it helpful.

Cheers,

Ana Paula 

Thanks a lot.

With best regards

Yes, there is a relation between the order of the group of points, the size of the base field (check Hasse's condition) and the curve coefficients. You can use a point counting algorithm like Schoof-Elkies-Atkin (SEA) to determine the order. This algorithm is usually already implemented in software like Sage or MAGMA.

Yes. Group theory is very useful in understanding vibrational spectra, as follows:

- Group theory can also help analyze and predict vibrational spectra, both IR and Raman.

Group theory can predict which vibration is IR or Raman active and which is not.

Group theory can predict maximum numbers of active vibrations

Group theory can predict if IR and Raman vibrations will coincide

Group theory can help assign the the measured vibrational spectra, and explain all shoulders and double bands existing in gaseous sample spectra.

Unlike Quantum Theory, Group Theory can not predict positions of the bands. You need to make spectral measurements and then assign them with group theory.  This is a limitation indeed.

Best wishes

Recently  Kevin Ford, Ben Green, Sergei Konyagin, James Maynard, and Terence Tao have produced a paper that proves a lower bound for the maximal gap

G(X) := \sup_{p_n, p_{n+1} \leq X} p_{n+1}-p_n

between primes bounded by X of the form

G(X) \geq c \log X \frac{\log \log X \log\log\log\log X}{\log\log\log X}

for some small constant c.

see: http://arxiv.org/abs/1412.5029 and the as always excellent blog post by Terrence Tao

Dear Fajri Pratama

You look before the world mathematical community as the module on the complex plane with impotentny argument (forgive me for sad humour). Dear Fajri Pratama find to yourself the mistress or present to your girl or to the wife a box of chocolate truffles. Only then before you the world of natural and imaginary numbers, the main thing, in this question strong argument will open in all beauty.

Alexander

1.As for the post of colleague R.C.Mittal, I guess that the topic starter needs not the basic facts about $p$-adic numbers the colleague addresses him to. I even guess that the topic starter is aware of advanced monographs on $p$-adic analysis like the classical Mahler's monograph "$p$-adic numbers and their functions" ore more recent books by Schikhof "Ultrametric calculus",

by Gouvea "Arithmetic of $p$-adic modular forms", by Gouvea "$p$-adic numbers", by Vladimirov et. al. "$p$-adic mathematical physics", several books of Khrennikov on applications of the $p$-adic theory to various disciplines, etc., etc. There are numerous books and numerous papers on the subject, but I guess that the topic starter needs something more definite. My two cents follows.

2. The most known continuous mapping of $p$-adic numbers into reals is the Monna map which is defined as follows: Represent a $p$-adic number $z$ in a canonical form (this representation is unique) $z=\sum_{i=-k}^\infty\alpha_ip^i$ where $k$ is a non-negative (real) integer, $\alpha_i\in\{0,1,\ldots,p-1\}$. The Monna map puts into the correspondence to $z$ the real number $mon(z)=\sum_{i=-k}^\infty\alpha_ip^{-i-1}$. It is clear that the most interesting part is the action of the Monna map on the $p$-adic integers which are represented as $\sum_{i=0}^\infty\alpha_ip^i$: Contrasting to real numbers, the "fractional part" of a $p$-adic number always has a finite canonical representation while the "integral part" not. So further i will speak only of mappings of the space of $p$-adic integers $\mathbb Z_p$ into reals. The space $\mathbb Z_p$ is a unit ball w.r.t. $p$-adic metric and is a sort of analog of a real unit interval. The Monna mapping $mon$ is a continuos mapping of $\mathbb Z_p$ into $[0,1]$ w.r.t the $p$-adic metric on $\mathbb Z_p$ and standard real metric on $[0,1]$. Moreover, $mon$ is a measure-preserving mapping w.r.t the probability Haar measure on $\mathbb Z_p$ and Lebesgue measure on $[0,1]$.

3. The  idea of the construction the Monna map is based on can be extended as follows: Take a real number $\beta>1$ such that the integral part of $\beta$ is $p-1$ and to every $p$-adic integer $z=\sum_{i=0}^\infty\alpha_ip^i$ put into the correspondence the real number $\sum_{i=0}^\infty\alpha_i\alpha_i\beta^{-i-1}$. In this case we deal with the so-called $\beta$-representations which were introduced by Parry and Renyi and now are rather popular research area, see. e.g. numerous papers of Frougny, Sakarovitch, Nikita Sidorov, and many others (too long to list all of them, sorry).

4. There are numerous generalizations for the case when $\beta$ is negative or even a complex number, and they are also related to real (or complex) representations of infinite words that correspond to $p$-adic numbers and have tight relations to automata theory, cf. e.g. the monograph by Lothaire "Algebraic combinatorics on words". The basic idea is that to a $p$-adic integer $z=\sum_{i=0}^\infty\alpha_ip^i$ there corresponds a (left-) infinite word $\ldots\alpha_2\alpha_1\alpha_0$ over a $p$-symbol alphabet; the word may be treated as a representation of a real number in some radix system (e.g., as a $\beta$-representation) and then one can study what reals (or sets of reals) obtained this way can be recognized by finite automata, or transformed by automata, etc, c.f., e.g., the monograph by Allouche and Shallitt "Automatic sequences". From this point of view, real representations of $p$-adic numbers give some information of behavior of automata and of the corresponding discrete dynamical system.

5. As automata map infinite words to infinite words, the automata mappings can be considered as mappings of $p$-adic integers to $p$-adic integers: Actually every automaton mapping is a continuous (w.r.t. the $p$-adic metric) function defined on the space of $p$-adic integers $\mathbb Z_p$ and valuated in $\mathbb Z_p$. Moreover, this mapping is 1-Lipschitz (=satisfies the Lipschtz condition with a constant 1) w.r.t. the $p$-adic metric and vice versa, every 1-Lipschitz mapping $\mathbb Z_p\to\mathbb Z_p$ can be performed by an automaton. From this view, real representations of $p$-adic numbers can visualize the automaton mapping (or the evolution of a discrete dynamical system). But being used for such a visualization, both the Monna map and $\beta$-representations can only give an information about short-time behavior of a system since they put the earliest inputted/outputted symbols to most significant positions in real representations. To study long-term behavior we need maps that act somehow in inverse direction. These "maps" are not maps in a strict sense of the word since put into the correspondence to a single $p$-adic integer some set (maybe, even infinite) of real numbers rather than a single number but these "mappings" can also be judged as 'continuous representations' in some meaning.

6. Namely, given a $p$-adic integer $z=\sum_{i=0}^\infty\alpha_ip^i$ we put into the correspondence to $z$ the set of all accumulation points of the sequence $p^{-k}\sum_{i=0}^{k-1}\alpha_ip^i$, $i=0,1,ldots$. From this view, the dynamics of an automaton map (or of a discrete system) can be represented as a real dynamics on the unit real square or, which is more natural, on the 2-dimensional torus. The representations of this can discover import ant peculiarities in discrete dynamics and are now under research, see e.g.

the monograph by Anashin (me) and Khrennikov "Applied algebraic dynamics" (deGryuter, 2009) and my papers on the theme, e.g. "The non-Archimedean theory of discrete systems" Math. Comp. Sci. 2012 (the paper can be found here on ReserchGate or in arxiv.org).

I am not sure that all these things are really that ones the topic starter is interested in but at least they are related to the question under the discussion. Sorry for the lengthy text, but I tried to be as concrete as possible.

I am not sure whether a closed-form expression for your sum can be found, as this sum appears to be a special case of q-hypergeometric function, see e.g. the link below.

Dear Pieter,

I think you can find your answer in the book of Daniel A. Marcus, "Number Fields", Springer Universitex, 19977, chapter 4, exercise 12, p.118 (see also ex. 14)

Best, T. NQD

Here is an old paper of mine that studies Gauss sums over finite rings rather than fields

Thanks Jeremy

There are 3 main solutions to this problem.

1. Simple to implement is binary search (as already mentioned). Given a pair of values (L,U) such that L^2 < n < U^2 (where n is the number to be tested), we compute ((L+U)/2)^2 and deduce that the solution is contained in an interval of half size. This is polynomial-time (unlike some of the other solutions already stated).

2. Optimal complexity is to use Newton interation. See Chapter 9 of the book by von zur Gathen and Gerhard. In particular, Section 9.5 talks of a 3-adic Newton iteration to compute integer square roots. This leads to a running time of O( M(n)) = O( \log(n) \log\log(n) \log\log\log(n)) the same as integer multiplication. But this is harder to implement.

3. A Monte-Carlo method is to reduce n modulo small primes p and compute the Legendre symbol (n/p) (which will always be 0 or 1 if n is an integer square). One would expect to test around log(n) primes to have good confidence that n is a square. I do not know a reference for a theoretical analysis of this algorithm.

Thank you Andrej.

I have seen all these books, but they focused on the elliptic curves. I am looking for a reference which deals with Abelian varieties in general. It doesn't seem as there is any book on that.

Humm the best person to answer this question is Pieter Moree : he is on this site.

There is a notion of (divisor) class group 'Cl(X) ' associated to algebraic varieties X--affine and projective. (I will define this notion below for elliptic curves which are projective.). Now, if A is a Dedekind domain, (e.g. the ring of integers in a number field F (where F could be a quadratic field you are interested in) Cl(Spec(A)) coincides with the ideal class group of A as defined in Algebraic Number Theory. So, this gives you the connection between affine varieties and ideal class group. (Spec A is affine.)

I will now define Cl(X) for an elliptic curve X. . A divisor D of X is a linear combination of points x_i with coefficients k_i in Z(the integers).. The degree of D (deg D) is the sum of the k_i's. The divisors of X form a group Div(X)--the free Abelian group (Z-module) generated by the x_i's . A principal divisor is a divisor D such that the degree of D = 0.The principal divisors (P(X)) form a subgroup of Div (X) and the quotient group Div(X)/P(X) is denoted by Cl(X). Now, there is homomorphism deg: Cl(X) --> Z whose kernel is denoted by Cl^o(X). Cl^o(X) has the following interesting property:

There is a one-one correspondence between the points of x and the elements of the group Cl^o(X). Moreover by using this one-one correspondence, one can transfer the group law from Cl^o(X) to X itself.

Elliptic curves is a very active area of research probably because the curves have various deep ramifications. For example, an elliptic curve is an Abelian variety of dimension 1 i.e an irreducible projective algebraic group; it is also a curve of genus 1. etc.

for further information you may look at the following books:

1) Basic Algebraic Geometry--Varieties in Projective spaces by I. R. Shavarevich.

Springer-Verlag.

2) Algebraic Geometry-- R. Hartshorne. Springer-Verlag.

Yup Patrick Sole......absolutely....I just shared this paper as it seems to be interesting....

If it is correct, Inventiones or Annals would have been better choices...