中科院数学与系统科学研究院
数学研究所
学术报告
数论研讨班
报告人：Prof. Preda Mihailescu (University of Göttingen)
题 目：Semilocal galois approximation for some popular Diophantine Equations
时 间：2023.08.16（星期三），13:30-14:30
地 点：数学院南楼N913室
摘 要：We consider the equation (x^p*+y^p)/(x+y) = p^e z^q, where x,y,z are non zero and mutually coprime integers, p and q are primes, not necessarily different and p > 3, while e = 0 is p does not divide z and 1 otherwise. We present an approach combining class field theory and approximation in semilocal products of completions at primes dividing x, which completely proves that the case p = q has no solutions, while providing good upper bounds for the case p neq q. We also indicate how to prove that the homogenous case of distinct exponents (Fermat-Catalan) has no solutions when either q > 2p or q does not divide the relative class number of the o-th cyclotomic field and that field contains no q-primary roots of unity which are not q-powers. We discuss further applications.
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报告人：Prof. Preda Mihailescu (University of Göttingen)
题 目：The Kummer - Vandiver Conjecture
时 间：2023.08.16（星期三），15:00-17:00
地 点：数学院南楼N913室
摘 要：We provide a complete proof of the Kummer - Vandiver Conjecture, stating that p does not divide the class number of the maximal real subfield of the p-th cyclotomic extension of Q. If K is the p-th cyclotomic field and K_infty/K its cyclotomic Z_p-extension, the negation of the conjecture implies the existence of some unramified field L/K which is abelian of degree p and unramified over K, while being galois over Q. It comes together with its cyclotomic Z_p-extension L_infty. The proof is based on an investigation of the local behavior of the real units in the fields L_n, connected to the capitulation that happens in K_n/L_n. Universal norms and a proof of the Leopoldt Conjecture for L_n are important prerequisites used.
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