


Number Theory Seminar
Speaker: Somnath Jha, IIT Kanpur
Time & Date: 2.30 pm, Monday, 26 December 2022
Venue: Ramanujan Hall
Title: Title: nSelmer group of elliptic curves over number fields
Abstract: The recent work of Bhargava et al., MazurRubin and others on the nSelmer group of an elliptic curve has made a significant impact on the arithmetic of the elliptic curve. Let E be an elliptic curve over Q with a rational 3isogeny. In this talk, we will discuss the 3Selmer group of E. We will indicate some applications to a classical Diophantine problem related to rational cube sum. This talk is based on a joint work with D. Majumdar and P. Shingavekar.
Number Theory Seminar
Speaker: Kartik Prasanna, University of Michigan
Time & Date: 4 pm, Monday, 26 December 2022
Venue: Ramanujan Hall
Title: Motivic realizations of functoriality
Abstract: Langlands functoriality predicts relations between automorphic forms on different groups. I will discuss in some examples how functoriality interacts with the theory of motives.
Virtual Commutative Algebra Seminar. Speaker: Mitsuyasu Hashimoto, Metropolitan University, Sumiyoshiku, Osaka, Japan Date/Time: Friday, 30 December 2022, 5:30 pm Gmeet link: [1]meet.google.com/yduyqgusxq [2] Title: Asymptotic behaviors of the Frobenius pushforwards of the ring of invariants Abstract: Let k be an algebraically closed field of characteristic p > 0, n a positive integer, and V = k^d. Let G be a finite subgroup of GL(V) without pseudoreflections. Let S = Sym V be the symmetric algebra of V, and A = S^G be the ring of invariants. The functor (?)^G gives an equivalence between the category {}^*Ref(G,S), the category of Qgraded Sfinite Sreflexive (G,S)modules and the category {}^*Ref(A), the category of Qgraded Afinite Areflexive Amodules. As the ring of invariants of the Frobenius pushforward ({}^e S)^G is the Frobenius pushforward {}^eA, the study of the (G,S)module {}^e S for various e yields good information on {}^eA. Using this principle, we get some results on the properties of A coming from the asymptotic behaviors of {}^eA. In this talk, we will discuss the following: (1) The generalized Fsignature of A (with Y. Nakajima and with P. Symonds). (2) Examples of G and V such that A is Frational, but not Fregular. (3) Examples of G and V such that (the completion of) A is not of finite Frepresentation type (work in progress with A. Singh). Generalizing finite groups to finite group schemes G, we have that s(A)>0 if and only if G is linearly reductive, and if this is the case, s(A)=1/G, where G is the dimension of the coordinate ring k[G] of G, provided the action of G on Spec S is 'small' (with F. Kobayashi). For more information and links to previous seminars, visit the website of VCAS: https://sites.google.com/view/virtualcommalgebraseminar [3] Links:  [1] http://goog_9085540/ [2] http://meet.google.com/yduyqgusxq [3] https://sites.google.com/view/virtualcommalgebraseminar