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Elliptic Functions and Arithmetic

In: The Shaping of Arithmetic after C. F. Gauss’s Disquisitiones Arithmeticae

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  • Christian Houzel

Abstract

8. Conclusion We have tried to showthe beginnings of the intervention of the theory of elliptic functions in arithmetic: complex multiplication, q-calculus, and the functional equation for the theta functions, Kronecker’s limit formula, addition theorem and Diophantine analysis. The germs of most of these ideas were already present in Gauss’s work. Gauss clearly saw the necessity to use methods not restricted to the consideration of natural numbers, in order to prove properties of natural numbers. After Abel and Jacobi, the mathematicians who developed these theories in the first two thirds of the XIXth century were Eisenstein, Kronecker, and Hermite. We saw the particular importance of Kronecker’s work which vindicates this author’s insistence on formulae as the essence of mathematics. At the end of the XIXth century, and throughout the XXth century, the theories mentioned have continually developed and this growth continues to this day. This is a testimony to the continuing presence of Gauss’s work.

Suggested Citation

  • Christian Houzel, 2007. "Elliptic Functions and Arithmetic," Springer Books, in: Catherine Goldstein & Norbert Schappacher & Joachim Schwermer (ed.), The Shaping of Arithmetic after C. F. Gauss’s Disquisitiones Arithmeticae, chapter 0, pages 291-312, Springer.
    • Handle: RePEc:spr:sprchp:978-3-540-34720-0_10
    DOI: 10.1007/978-3-540-34720-0_10
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