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## Mathematics Colloquium | Neelam Saikia

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Title:      AGM and jellyfish swarms of elliptic curves over finite fields

Abstract:  The classical $\\AGM$ produces wonderful interdependent infinite sequences of arithmetic and geometric means with a common limit. In this talk, we discuss a finite field analogue $\\AGM_{\\F_q}$ over finite fields $\\F_q,$ with $q\\equiv 3\\pmod 4,$  that spawns directed finite graphs instead of infinite sequences. The compilation of these graphs reminds one of a {\\it jellyfish swarm}, as the 3D renderings of the connected components resemble {\\it jellyfish} (i.e. tentacles connected to a bell head).

These swarms turn out to be more than the stuff of child's play; they are taxonomical devices in number theory. Each jellyfish is an isogeny graph of elliptic curves with isomorphic groups of $\\F_q$-points, which can be used to prove that each swarm has at least $(1/2-\\varepsilon)\\sqrt{q}$ many jellyfish.

Moreover, this interpretation gives a description of the  {\\it class numbers} of Gauss, Hurwitz, and Kronecker which is akin to counting types of spots on jellyfish. This is joint work with Michael J. Griffin, Ken Ono, and Wei-Lun Tsai.

About the Speaker:  Neelam Saikia is a Fulbright-Nehru postdoctoral fellow at the University of Virginia, U.S.A, under the guidance of Prof. Ken Ono. Previously, she was DST Inspire faculty at Indian Institute of Science Bangalore, India, and at Indian Institute of Technology Guwahati, India. Before that, she was a visiting scientist at the Indian Statistical Institute, Delhi center. She received her Ph.D. degree from the Indian Institute of Technology Delhi, India.

Her research interests are in the areas of Number Theory. More specifically, she is interested in hypergeometric series and their relationship to Modular forms, elliptic curves, counting points on varieties over finite fields, Kloosterman sums, and Sato-Tate type distributions for hypergeometric varieties.