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Die folgenden Informationen sind noch nicht freigegeben und deshalb unverbindlich: Riemann-Roch theory for number fields SemesterSoSe 2026 Lecturer Moritz Kerz, Carolyn Echter Type of course (Veranstaltungsart) Seminar Contents The classical Riemann-Roch problem is the problem of counting meromorphic functions on a compact Riemann surface with prescribed bounds on the pole orders. Formally, the quantity in question is the dimension of the space of global sections of a holomorphic line bundle associated to a divisor on the Riemann surface. The classical Riemann-Roch theorem relates this quantity to the degree of the divisor and the genus of the surface. Its algebraic counterpart is the Riemann-Roch theorem for smooth projective curves over the complex numbers, which can be adapted to other base fields. In some respects analogous to the study of curves in geometry is the study of number rings over the integers in number theory, which leads to the following Riemann-Roch-type counting problem: How many elements of a number field satisfy given bounds on their valuations at primes? In the seminar, we will address this question by establishing an arithmetic version of the Riemann-Roch theorem for number fields, following chapter III of Neukirch's book on algebraic number theory. Without going too deep into the analogy with the geometric Riemann-Roch, we will discover the importance of including the infinite primes, which will become even more prominent when we turn to the more general Grothendieck-Riemann-Roch theorem. Literature J. Neukirch, Algebraische Zahlentheorie. Graduate Texts in Mathematics, Springer, 1992. English version: J. Neukirch, Algebraic number theory, vol. 322 of Grundlehren der mathematischen Wissenschaften. Springer, 1999. Recommended previous knowledge Basics of algebraic number theory, for example as was covered by the course taught in the winter term (chapters I and II of Neukirch's book). Depending on the participants' interests, we may have outlook talks on the geometric Riemann-Roch, for which some familiarity with algebraic geometry will be helpful. Time/Date Tue 16-18 Location M 101 Course homepage https://elearning.uni-regensburg.de/course/view.php?id=73938 (Disclaimer: Dieser Link wurde automatisch erzeugt und ist evtl. extern) Registration
BSem, MV, MSem, BSem.2 ECTS 4,5 |