Elliptic curves lie at the intersection of several areas of mathematics which have been approached in different ways. So the problem in learning the subject is deciding which approach to adopt. Lawrence Washington's book is very approachable and doable, written by a serious mathematician with the novice in mind, and it is realistic to expect that it can be worked out. Knapp's notes are very readable, and after the outlines of the subject are familiar furnish enrichment and orientation-they are compulsively readable-and can mislead one into thinking that his notes can be easily mastered! Silverman's two volumes are the definitive treatment of the arithmetic aspects of the subject for the average researcher. As to the quibbles about scheme theory being omitted in his treatment, one has to ask oneself what one is learning the subject for. As Silverman remarks, it is easy to write down a particular elliptic curve and pursue interesting arithmetic research questions without using scheme theory. One could view Mordell's finite generation theorem as either being "elementary" or "interesting". Both points of view have a measure of truth in them. This is a foundational theorem in the subject, and its proof requires no more than high school mathematics! This is of course a testament to the fact that elliptic curves are amazing objects. Not sure Andrew Wiles used scheme theory in his proof of FLT. He may well have relied on deep results obtained using scheme theory, but that does not make his proof less valid. Another example is Mazur's theorem on the torsion subgroup for elliptic curves over Q. The result is very useful, even if we do not understand its proof, it is easy to apply. Very few of us have the time to learn algebraic geometry in its rigorous modern formulation. Silverman remarks on this matter in his introduction. One has to view the study of this subject as a prerequisite to engaging in research which really is the point of any serious study. I have been lucky enough to write and publish papers on the subject with very limited knowledge. Elliptic curves are deep mathematical objects especially when viewed from an arithmetic perspective, but interesting problems can be pursued with modest equipment. So unless one wants to be an algebraic geometer, it is possible to pursue arithmetic questions as topics for research without knowing about schemes. I have found myself referring to Silverman's books and to Knapp's notes when the "chips were down" and I was tackling research problems. My problems may have been superficial, but the results were published-so there must be a community that finds them interesting. Maybe I am slow, but I am leary of advice that gives a sequence of books to be read for learning something. I suspect most of us will read very few books thoroughly-and skimming a mathematics book doesn't seem like a useful pursuit to me. As Goethe (?) said. "Art is long. Life is short. Decisions are difficult, and opportunities are fleeting." My two cents for whatever it is worth.

Elliptic curves lie at the intersection of several areas of mathematics which have been approached in different ways. So the problem in learning the subject is deciding which approach to adopt. 

Lawrence Washington's book is very approachable and doable, written by a serious mathematician with the novice in mind, and it is realistic to expect that it can be worked out. 

Knapp's notes are very readable, and after the outlines of the subject are familiar furnish enrichment and orientation-they are compulsively readable-and can mislead one into thinking that his notes can be easily mastered! 

Silverman's two volumes are the definitive treatment of the arithmetic aspects of the subject for the average researcher. As to the quibbles about scheme theory being omitted in his treatment, one has to ask oneself what one is learning the subject for. As Silverman remarks, it is easy to write down a particular elliptic curve and pursue interesting arithmetic research questions without using scheme theory. 

One could view Mordell's finite generation theorem as either being "elementary" or "interesting". Both points of view have a measure of truth in them. This is a foundational theorem in the subject, and its proof requires no more than high school mathematics! This is of course a testament to the fact that elliptic curves are amazing objects. Not sure Andrew Wiles used scheme theory in his proof of FLT. He may well have relied on deep results obtained using scheme theory, but that does not make his proof less valid. 

Another example is Mazur's theorem on the torsion subgroup for elliptic curves over Q. The result is very useful, even if we do not understand its proof, it is easy to apply. Very few of us have the time to learn algebraic geometry in its rigorous modern formulation. Silverman remarks on this matter in his introduction. One has to view the study of this subject as a prerequisite to engaging in research which really is the point of any serious study. 

I have been lucky enough to write and publish papers on the subject with very limited knowledge. Elliptic curves are deep mathematical objects especially when viewed from an arithmetic perspective, but interesting problems can be pursued with modest equipment. So unless one wants to be an algebraic geometer, it is possible to pursue arithmetic questions as topics for research without knowing about schemes. 

I have found myself referring to Silverman's books and to Knapp's notes when the "chips were down" and I was tackling research problems. My problems may have been superficial, but the results were published  - so there must be a community that finds them interesting. Maybe I am slow, but I am leary of advice that gives a sequence of books to be read for learning something. I suspect most of us will read very few books thoroughly  - and skimming a mathematics book doesn't seem like a useful pursuit to me. 

As Goethe (?) said, "Art is long. Life is short. Decisions are difficult, and opportunities are fleeting." My two cents for whatever it is worth.

Elliptic curves lie at the intersection of several areas of mathematics which have been approached in different ways. So the problem in learning the subject is deciding which approach to adopt. Lawrence Washington's book is very approachable and doable, written by serious mathematician with the novice in mind, and it is realistic to expect that it can be worked out. Knapp's notes are very readable, and after the outlines of the subject are familiar furnish enrichment and orientation-they are compulsively readable-and can mislead one into thinking that his notes can be easily mastered! Silverman's two volumes are the definitive treatment of the arithmetic aspects of the subject for the average researcher. As to the quibbles about scheme theory being omitted in his treatment, one has to ask oneself what one is learning the subject for. As Silverman remarks, it is easy to write down a particular elliptic curve and pursue interesting arithmetic research questions without using scheme theory. One could view Mordell's finite generation theorem as either being "elementary" or "interesting". Both points of view have a measure of truth in them. This is a foundational theorem in the subject, and its proof requires no more than high school mathematics! This is of course a testament to the fact that elliptic curves are amazing objects. Not sure Andrew Wiles used scheme theory in his proof of FLT. He may well have relied on deep results obtained using scheme theory, but that does not make his proof less valid. Another example is Mazur's theorem on the torsion subgroup for elliptic curves over Q. The result is very useful, even if we do not understand its proof, it is easy to apply. Very few of us have the time to learn algebraic geometry in its rigorous modern formulation. Silverman remarks on this matter in his introduction. One has to view the study of this subject as a prerequisite to engaging in research which really is the point of any serious study. I have been lucky enough to write and publish papers on the subject with very limited knowledge. Elliptic curves are deep mathematical objects especially when viewed from an arithmetic perspective, but interesting problems can be pursued with modest equipment. So unless one wants to be an algebraic geometer, it is possible to pursue arithmetic questions as topics for research without knowing about schemes. I have found myself referring to Silverman's books and to Knapp's notes when the "chips were down" and I was tackling research problems. My problems may have been superficial, but the results were published-so there must be a community that finds them interesting. Maybe I am slow, but I am leary of advice that gives a sequence of books to be read for learning something. I suspect most of us will read very few books thoroughly-and skimming a mathematics book doesn't seem like a useful pursuit to me. My two cents for whatever it is worth. As Goethe (?) said. "Art is long. Life is short. Decisions are difficult, and opportunities are fleeting." My two cents for whatever it is worth.

Elliptic curves lie at the intersection of several areas of mathematics which have been approached in different ways. So the problem in learning the subject is deciding which approach to adopt. Lawrence Washington's book is very approachable and doable, written by a serious mathematician with the novice in mind, and it is realistic to expect that it can be worked out. Knapp's notes are very readable, and after the outlines of the subject are familiar furnish enrichment and orientation-they are compulsively readable-and can mislead one into thinking that his notes can be easily mastered! Silverman's two volumes are the definitive treatment of the arithmetic aspects of the subject for the average researcher. As to the quibbles about scheme theory being omitted in his treatment, one has to ask oneself what one is learning the subject for. As Silverman remarks, it is easy to write down a particular elliptic curve and pursue interesting arithmetic research questions without using scheme theory. One could view Mordell's finite generation theorem as either being "elementary" or "interesting". Both points of view have a measure of truth in them. This is a foundational theorem in the subject, and its proof requires no more than high school mathematics! This is of course a testament to the fact that elliptic curves are amazing objects. Not sure Andrew Wiles used scheme theory in his proof of FLT. He may well have relied on deep results obtained using scheme theory, but that does not make his proof less valid. Another example is Mazur's theorem on the torsion subgroup for elliptic curves over Q. The result is very useful, even if we do not understand its proof, it is easy to apply. Very few of us have the time to learn algebraic geometry in its rigorous modern formulation. Silverman remarks on this matter in his introduction. One has to view the study of this subject as a prerequisite to engaging in research which really is the point of any serious study. I have been lucky enough to write and publish papers on the subject with very limited knowledge. Elliptic curves are deep mathematical objects especially when viewed from an arithmetic perspective, but interesting problems can be pursued with modest equipment. So unless one wants to be an algebraic geometer, it is possible to pursue arithmetic questions as topics for research without knowing about schemes. I have found myself referring to Silverman's books and to Knapp's notes when the "chips were down" and I was tackling research problems. My problems may have been superficial, but the results were published-so there must be a community that finds them interesting. Maybe I am slow, but I am leary of advice that gives a sequence of books to be read for learning something. I suspect most of us will read very few books thoroughly-and skimming a mathematics book doesn't seem like a useful pursuit to me. As Goethe (?) said. "Art is long. Life is short. Decisions are difficult, and opportunities are fleeting." My two cents for whatever it is worth.