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/

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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

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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...

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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.

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Social:

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