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It would be difficult to overestimate the influence and importance of modular forms, modular curves, and modular abelian varieties in the development of num- ber theory and arithmetic geometry during the last fifty years. These subjects lie at the heart of many past achievements and future challenges. For example, the theory of complex multiplication, the classification of rational torsion on el- liptic curves, the proof of Fermat's Last Theorem, and many results towards the Birch and Swinnerton-Dyer conjecture all make crucial use of modular forms and modular curves. A conference was held from July 15 to 18, 2002, at the Centre de Recerca Matematica (Bellaterra, Barcelona) under the title "Modular Curves and Abelian Varieties". Our conference presented some of the latest achievements in the theory to a diverse audience that included both specialists and young researchers. We emphasized especially the conjectural generalization of the Shimura-Taniyama conjecture to elliptic curves over number fields other than the field of rational numbers (elliptic Q-curves) and abelian varieties of dimension larger than one (abelian varieties of GL2-type).
It would be difficult to overestimate the influence and importance of modular forms, modular curves, and modular abelian varieties in the development of num- ber theory and arithmetic geometry during the last fifty years. These subjects lie at the heart of many past achievements and future challenges. For example, the theory of complex multiplication, the classification of rational torsion on el- liptic curves, the proof of Fermat's Last Theorem, and many results towards the Birch and Swinnerton-Dyer conjecture all make crucial use of modular forms and modular curves. A conference was held from July 15 to 18, 2002, at the Centre de Recerca Matematica (Bellaterra, Barcelona) under the title "Modular Curves and Abelian Varieties". Our conference presented some of the latest achievements in the theory to a diverse audience that included both specialists and young researchers. We emphasized especially the conjectural generalization of the Shimura-Taniyama conjecture to elliptic curves over number fields other than the field of rational numbers (elliptic Q-curves) and abelian varieties of dimension larger than one (abelian varieties of GL2-type).
Buongiorno Italia! is the highly successful course for beginners learning Italian. Whether learning at home or in a class, you will learn to use and understand basic colloquial Italian in a wide range of everyday situations. You can gain valuable insights into modern Italian culture, and extended reading material in Italian will enhance your learning experience, introducing more advanced language and specialist topics. Focusing on listening, reading, writing and speaking skills, Buongiorno Italia! will help you reach the equivalent of a first qualification, such as GCSE. The Language CD pack contains 1 Book and 3 Audio CDs.
Buongiorno Italia is a beginners course in Italian. Based on authentic conversations and interviews recorded in Italy, it offers a unique introduction to simple, everyday language. It is now fully revised in a new edition, including the euro and up-to-date cultural information about life in Italy. those studying in a class. Divided into 20 short units, the book contains dialogues, language notes, information about Italian society, vocabulary and practice activities. There is also a full reference section, including answers, grammar notes and a glossary. The audio offers the chance to develop listening and speaking skills, with real-life conversations, stage-by-stage pronunciation practice and further activities.
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