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L-functions and the Langlands program (RH Saga S1E2)

This is the second episode of the RH Saga. We continue the journey into the world of L-functions, by focussing on two specific examples of motivic L-functions. These examples illustrate Langlands reciprocity, i.e. the conjecture that every motivic L-function is also an automorphic L-function. The overall aim of RH Saga Season 1 is to map the landscape of L-functions, as a foundation for future in-depth exploration of some of the most immortal math problems of all time. This video is part of a PeakMath course. Join the journey at https://www.peakmath.org/ --- Chapters: 00:00 - Intro 01:40 - Two examples of L-functions 07:08 - The LMFDB 11:50 - The mystery of K 23:31 - The mystery of E 44:33 - Final remarks on Langlands --- Links: 1. SageMathCell https://sagecell.sagemath.org/ 2. LMFDB https://www.lmfdb.org/ 3. The number defined by the polynomial x^2+1, which we refer to as "K" https://www.lmfdb.org/NumberField/2.0... 4. The elliptic curve "14.a5" which we refer to as "E" https://www.lmfdb.org/EllipticCurve/Q... 5. "Black box" for sequence "K" code in SageMathCell https://sagecell.sagemath.org/?z=eJzz... 6. "Black box" for sequence "E" code in SageMathCell https://sagecell.sagemath.org/?z=eJxz... 7. The point counting code in SageMathCell https://sagecell.sagemath.org/?z=eJxt... 8. Matthew Emerton's homepage https://math.uchicago.edu/~emerton/ 9. Langlands reciprocity survey by Emerton: https://math.uchicago.edu/~emerton/pd... --- Errata: Around 42:30: In the formula for the generating series (modular form), there is traditionally a “q” in front of the product sign. This should be included IF you want the sequence indexed with a1 as the label for the initial term rather than a0. --- Social: https://www.peakmath.org/ #RiemannHypothesis #F1Geometry #Mathematics #PeakMath #RHSaga #Langlands