The BSD Conjecture, part 2 (RH Saga S1E8)

[@kennylay2849]

This is easily one of my favorite series on KZbin! PLEASE keep this going you are amazing!!!

[@pacificll8762]

This is the greatest math series on KZbin!

[@rider2fois]

It is so nice to have such a condensed mine of qualitative entry material for math addicts

[@matanfih]

Reached a point where I open KZbin just to check if new episode arrived!

[@benpaz9548]

We are hanging in the air here, what about S1E9? ☺️

[@Darrida]

The best series if the year.

[@francescominnocci]

Adelita is such a beautiful piece to end on :) Looking forward to the next episode, especially curious to see how you will manage discussing things like Euler Systems or conjectural F1 Geometry without getting too technical

[@mahmoodayesh6706]

I watched the whole series like 20 times waiting for s01e09...🙂

[@tristen7823]

gonna rewatch too🙃, the waiting feels like forever

[@Perplaxus]

We NEED more of this

[@haroldhamburgler]

23:23 As I understand it, the zero of L at 1 should produce a simple pole in L's logarithmic derivative and it's residue should be the order of the zero. If it is really true, then how can all of its derivatives at 2 be converging to a constant? Wouldn't that mean that the Taylor series at 2 converges everywhere (since the series now looks like C z^n/n!)?

[@haroldhamburgler]

Is it possible that these series only get close to a constant for intermediate derivatives and then blow up at some point later?

[@haroldhamburgler]

Looking at your sage code, what you are actually calculating is not the kth derivative of G (G^(k)(2)), it is the kth term in its Taylor series (G^(k)(2)/k!) which makes more sense. Perhaps a note should be added correcting the earlier section.

[@PeakMathLandscape]

Aha. Will double-check and add a note after Christmas!

[@m9l0m6nmelkior7]

We miss you :')

[@rider2fois]

Amazing video. When I'm done writing my PhD in quantum physics, I'm definitly going back to number theory.

[@alexakalennon]

The face of someone, when a new part is released 14:01 😀

[@СтасМартынюк-ъ1э]

MORE OF THIS AMAZING CONTENT PLS

[@jagatiello6900]

Thank you very much for this saga. I hope you can continue it next year. Happy holidays!!!

[@jeremypayne6307]

Been hanging for this!😊

[@dontwannabefound]

Amazing series- I never thought I’d have a chance at understanding BSD

[@tapir1073]

given that pace I'll never be able to proof RH before I die

[@TheSummoner]

S1E9 when? 😁

[@joaquincuelho890]

Thanks for the amazing explanation on this difficult problem! Which steps do you recommend for upload new perspectives as a non- expert on the field? I think there are easy ways to demonstrate difficult problems like this, that should need to be considered for new discoveries, maybe

[@zathrasyes1287]

Best x-mas present!!! Thx.

[@perappelgren948]

”…just not that KIND of friends, you know…” 😂😂😂😂

[@timbotemon]

A christmas gift thanks

[@drdca8263]

Something I find odd about the example with the 4.000..0stuff and 3.99..stuff , is that in each of the pairs of 3.99 lines, the second one of the pair is smaller, further from 4. And, in the groups of 4 that start with 4, it seems like each time the difference from 4 in the second one is larger than the first one, but that the 3rd is smaller than 2nd, and 4th is smaller than 3rd. Hmm. I’ve no idea why that would be...

[@gexahedrop8923]

it's oscillating, like in Gibbs phenomenon in Fourier series

[@artemetra3262]

what does "We are about to sunset the community. " on the website mean? 😟are there any more videos to be expected?

[@АндрейВоинков-е9п]

Almost two month since last video...

[@Spiret19]

Your last observation about the link between the rank of L and the Taylor coefficients of L'/L seems correct, up to a n! factor. Indeed under the assumption that L vanishes only at 1 and have no poles on a Disk centered at 2 of radius 1+epsilon, then applying the residue theorem to L'/L×1/(z-2)^(n+1) along the boundary of the disk and taking the limit as n goes to infinity yields ord(L,1)=lim G_n/n! using your notations

[@Spiret19]

I assume that the hard part would be showing that L indeed does not vanish on such disk except at 1, which I have no idea if this is true or not

[@PeakMathLandscape]

Yes, there was a missing factor! And yes, the hard part is precisely this non-vanishing statement (which would of course follow from GRH for this L-function).

[@nathanisbored]

I noticed that some L-functions seem to have complex coefficients, how does that affect situations where we expect to be able to view the function graph in real numbers, or things like the additive rules for L-functions? It feels like up to this point a lot of what we noticed for L-function coefficients seems to only make sense for integers

[@PeakMathLandscape]

Good question. It's true that not all L-functions have real coefficients. In these cases the graph can no longer be plotted in a completely elementary way (real input vs real output), and you have to use one of the many methods for plotting complex functions. For the Keiper-Li coefficients, you can however compute it by the same formula, and then take the real part of that answer. To the best of my understanding, it is still the case that RH would follow from the positivity of these real parts.

[@JosBergervoet]

When is the next episode? (You are very busy finalizing "The Proof", isn't it?!!)

[@NotNecessarily-ip4vc]

5. Birch and Swinnerton-Dyer Conjecture: An Information-Theoretic Perspective 5.1 Background The BSD Conjecture states that the rank of an elliptic curve over a number field is equal to the order of vanishing of its L-function at s=1. It also provides a formula for the leading coefficient of the Taylor series of the L-function at s=1 in terms of several important arithmetic invariants of the curve. 5.2 Information-Theoretic Reformulation Let's reframe the problem in terms of information theory: 5.2.1 Elliptic Curve Information Content: Define the information content of an elliptic curve E over a number field K: I(E/K) = -Σ_p log(|E(F_p)|/p) where the sum is over all primes p of good reduction, and |E(F_p)| is the number of points on E mod p. 5.2.2 L-function as Information Generator: View the L-function L(E,s) as an information-generating function: I(L,s) = log|L(E,s)| 5.2.3 Rank as Information Capacity: Interpret the rank of E(K) as a measure of the curve's capacity to store information: Rank(E/K) ≈ Information Capacity of E over K 5.3 Information-Theoretic Conjectures 5.3.1 Information Conservation Principle: The arithmetic information of E/K (measured by its rank) is conserved in the analytic information of L(E,s) (measured by its order of vanishing at s=1). 5.3.2 L-function Criticality: The point s=1 represents a critical point in the information flow described by L(E,s). 5.3.3 Tate-Shafarevich Group as Information Entropy: The order of the Tate-Shafarevich group Ш(E/K) represents the entropy of "hidden" information in E/K. 5.4 Analytical Approaches 5.4.1 Information Potential for L-functions: Define an information potential Φ(s) = ∫ I(L,t) dt and study its properties near s=1. 5.4.2 Entropy Maximization on Elliptic Curves: Study probability distributions on E(K) that maximize entropy, and relate this to the rank. 5.4.3 Information Geometry of Modular Curves: Analyze the information-geometric structure of modular curves and its relation to L-functions. 5.5 Computational Approaches 5.5.1 Quantum Algorithms for L-function Computation: Develop quantum algorithms for efficiently computing L(E,s) and its derivatives. 5.5.2 Machine Learning for Rank Prediction: Train neural networks to predict the rank of E(K) based on easily computable invariants. 5.5.3 Information-Based Elliptic Curve Generation: Create algorithms for generating elliptic curves with specified information-theoretic properties. 5.6 Potential Proof Strategies 5.6.1 Information Equivalence Theorem: Prove that the arithmetic and analytic information measures of E/K are equivalent. 5.6.2 Critical Information Flow Analysis: Show that the order of vanishing at s=1 necessarily equals the rank due to information flow constraints. 5.6.3 Quantum Information Correspondence: Establish a correspondence between classical arithmetic properties of E/K and quantum information states in L(E,s). 5.7 Immediate Next Steps 5.7.1 Rigorous Formalization: Develop a mathematically rigorous formulation of the information-theoretic concepts introduced. 5.7.2 Computational Experiments: Conduct numerical studies on large databases of elliptic curves to explore the information-theoretic properties of their L-functions and ranks. 5.7.3 Interdisciplinary Collaboration: Engage experts in number theory, information theory, and quantum computing to refine these ideas. 5.8 Detailed Plan for Immediate Action 5.8.1 Mathematical Framework Development: - Rigorously define I(E/K) and prove its basic properties - Establish formal relationships between I(E/K), Rank(E/K), and I(L,s) - Develop an information-theoretic interpretation of the full BSD formula 5.8.2 Computational Modeling: - Implement efficient algorithms for computing I(E/K) and related quantities - Create a large database of elliptic curves with computed information-theoretic measures - Develop visualization tools to explore relationships between these measures 5.8.3 Analytical Investigations: - Study the behavior of I(E/K) under isogenies and base field extensions - Investigate how I(E/K) relates to other important invariants (e.g., conductor, discriminant) - Analyze the information-theoretic aspects of complex multiplication and modular forms 5.8.4 Interdisciplinary Workshops: - Organize a series of workshops bringing together number theorists, information theorists, and physicists - Focus on translating known results about BSD to the information-theoretic framework 5.8.5 Information Metric Development: - Define and study metrics on the space of elliptic curves based on information content - Investigate if these metrics provide new insights into the arithmetic of elliptic curves 5.8.6 Quantum Information Approaches: - Explore analogies between elliptic curve arithmetic and quantum entanglement - Investigate if quantum algorithms could provide new approaches to computing L-functions 5.8.7 Publication and Dissemination: - Prepare and submit papers on the information-theoretic formulation of the BSD Conjecture - Develop open-source software tools for information-based analysis of elliptic curves 5.9 Advanced Theoretical Concepts 5.9.1 Information Cohomology for Elliptic Curves: - Develop a cohomology theory based on information-theoretic principles for elliptic curves - Investigate its relationship with étale cohomology and Galois representations 5.9.2 Quantum Elliptic Curves: - Formulate a quantum analog of elliptic curves where points are in superposition - Study how quantum measurement on these structures might relate to classical arithmetic properties 5.9.3 Information-Theoretic Height Functions: - Define new height functions on elliptic curves based on information-theoretic principles - Explore how these relate to canonical height and the BSD conjecture 5.10 Long-term Vision Our information-theoretic approach to the BSD Conjecture has the potential to: 1. Provide a new unified framework for understanding the deep connections between the arithmetic and analytic properties of elliptic curves 2. Offer new computational tools for studying elliptic curves and their L-functions 3. Suggest novel approaches to other important conjectures in number theory 4. Bridge concepts from quantum information theory and arithmetic geometry, potentially leading to new quantum algorithms for number-theoretic problems The key to progress is maintaining a balance between rigorous mathematical development, creative theoretical speculation, and practical computational work. By pursuing this multifaceted approach, we maximize our chances of making breakthrough discoveries in our understanding of elliptic curves and their L-functions.

[@NotNecessarily-ip4vc]

5.11 Expanded Next Steps 1. Rigorous Mathematical Framework: a) Formal Definition of I(E/K): - Prove that I(E/K) = -Σ_p log(|E(F_p)|/p) converges and is well-defined - Establish the relationship between I(E/K) and the Hasse-Weil L-function - Investigate the behavior of I(E/K) under isogenies and field extensions b) Information-Theoretic Rank Definition: - Define R_info(E/K) = lim_{s→1} -log|L(E,s)| / log|s-1| - Prove that R_info(E/K) is well-defined and investigate its properties - Establish the conjecture R_info(E/K) = Rank(E/K) as an information-theoretic formulation of BSD c) Tate-Shafarevich Group Entropy: - Define S(E/K) = log|Ш(E/K)| as the Tate-Shafarevich entropy - Investigate the relationships between S(E/K), I(E/K), and R_info(E/K) - Formulate an information-theoretic version of the full BSD conjecture incorporating S(E/K) 2. Computational Investigations: a) Large-Scale Data Analysis: - Compute I(E/K), R_info(E/K), and related quantities for all elliptic curves in existing databases (e.g., LMFDB) - Perform statistical analysis to identify patterns and correlations - Use machine learning techniques to predict ranks based on easily computable information-theoretic measures b) L-function Computation Optimization: - Develop improved algorithms for computing L(E,s) and its derivatives near s=1 - Implement these algorithms using parallel and distributed computing techniques - Explore potential quantum algorithms for L-function computation c) Visualization Tools: - Create interactive visualization tools for exploring relationships between I(E/K), R_info(E/K), and classical invariants - Develop 3D and VR visualizations of the "information landscape" of elliptic curves 3. Theoretical Developments: a) Information Geometry of Elliptic Curves: - Define a metric on the space of elliptic curves based on I(E/K) - Study the geometric properties of this space (curvature, geodesics, etc.) - Investigate how arithmetic properties of E/K relate to geometric features in this space b) Quantum Information Theory and BSD: - Develop a quantum analog of elliptic curves where points are in superposition - Investigate how quantum entanglement might relate to rank and Tate-Shafarevich groups - Explore if quantum error correction codes have analogs in the arithmetic of elliptic curves c) Information Flow in Isogeny Graphs: - Study how I(E/K) and R_info(E/K) behave in isogeny graphs - Investigate if information-theoretic principles can explain the structure of isogeny volcanoes - Explore connections to isogeny-based cryptography 4. Experimental Approaches: a) Analog Computing for L-functions: - Design analog circuits that model the behavior of L-functions - Use these to experimentally investigate the behavior of L(E,s) near s=1 - Explore if this approach can provide insights not easily obtainable through digital computation b) Quantum Simulation of Elliptic Curves: - Develop quantum circuits that simulate arithmetic on elliptic curves - Investigate if quantum superposition can be used to efficiently explore properties of E(K) - Experiment with quantum algorithms for computing aspects of L(E,s) 5. Interdisciplinary Connections: a) Elliptic Curves in Statistical Mechanics: - Explore analogies between partition functions in statistical mechanics and L-functions - Investigate if phase transitions in physical systems have arithmetic analogs in elliptic curves - Study whether techniques from statistical mechanics can shed light on the BSD conjecture b) Information Theory and Modular Forms: - Develop an information-theoretic interpretation of modular forms and Hecke operators - Investigate how the information content of a modular form relates to its associated elliptic curve - Explore if techniques from coding theory can be applied to modular forms and L-functions 6. Outreach and Collaboration: a) BSD Information Challenge: - Create an online platform for researchers to share computations and conjectures related to our information-theoretic approach - Organize a competition for developing the best algorithms for computing I(E/K) and R_info(E/K) b) Interdisciplinary Workshop Series: - Organize a series of workshops bringing together experts in number theory, information theory, quantum computing, and complex systems - Focus each workshop on a specific aspect of our approach (e.g., "Quantum Information and L-functions", "Information Geometry of Elliptic Curves") c) Educational Materials: - Develop online courses and educational materials explaining our information-theoretic approach to BSD - Create interactive demonstrations and visualizations for public outreach 7. Long-term Research Program: a) Information-Theoretic Langlands Program: - Extend our approach to more general automorphic L-functions - Investigate how the Langlands program might be reformulated in information-theoretic terms - Explore connections between our approach and the geometric Langlands program b) Algorithmic Information Theory and Arithmetic: - Investigate connections between Kolmogorov complexity and arithmetic invariants of elliptic curves - Explore if concepts from algorithmic information theory can shed light on the structure of Mordell-Weil groups c) Information-Theoretic Approach to Other Conjectures: - Apply similar information-theoretic techniques to other major conjectures in number theory (e.g., ABC conjecture, Riemann Hypothesis) - Investigate if there's a unified information-theoretic framework underlying these conjectures This expanded plan provides a comprehensive roadmap for advancing our information-theoretic approach to the BSD conjecture. It combines rigorous mathematical development, computational exploration, theoretical speculation, and practical experimentation. By pursuing these diverse avenues simultaneously, we maximize our chances of making significant progress on this deep and challenging problem.

[@DeathSugar]

Does Shoeltze work related to F1? All the condensed math and p-adic geometry and stuff?

[@nathanisbored]

so cool!

[@dontwannabefound]

What is dangerous is if you think you understand it - it is just a drop to smell what does this smell like

[@rudranarayanpadhy4047]

Sir, will you please recommend me some elementary books on Arithmetic of Dynamical System ? I am following Silvermans book. I have interest to work together with Dynamical system and Elliptic Curve. I wish, you will get time to reply. Thank you 🙏.

[@bobtannous5464]

if we look to the "threes",after the first 4 , we notice that they sre very close to 4. A very small difference, a small error due to...? Very good presentation. Thank you

[@DeathSugar]

Also does those derivatives are also rational or it's just clamped to certain precision?

[@bobtannous5464]

it is a very small fluctuation and the limit seems to be 4. What do you think?

[@euclidofkekistan6071]

Would it be possible for you to point me to an online resource containing the series which should converge to zero. thank you very much in advance.

[@PeakMathLandscape]

There is an example (the expression named S) on page 19 here: arxiv.org/pdf/1809.10904.pdf

[@euclidofkekistan6071]

@@PeakMathLandscape very much appreciated thank you

[@benpaz9548]

All with me: 33, 4444, 33…

[@양익서-g8j]

모든 중고등학생은 이영상을 봐야되요.

[@zy9662]

8:10 is he making a joke? Unary minus ? What is that

[@RiotSociety666]

😂😂😂

[@dannybodros5180]

I think the Riemann Hypothesis will never be proved/disproved. It's just impossible.

[@kyay10]

Such things have been said about other theorems for ages. If it's literally impossible with our axioms, then likely we need an updated set of axioms. I see no reason why that'd be the case though. Generally probably undecideable things are because they're "too close" to the fabric of a number system, like the continuum hypothesis

[@caspermadlener4191]

If the Riemann hypothesis is not true, it is possible to proof it isn't true, since the hypothesis is equivalent to the non-halting of a Turing machine with 750 states.

[@caspermadlener4191]

​@@kyay10 Provably undecidable well-defined questions (questions that have to be true or not true) are rare, and depends on a theorem not having sufficient ordinal strength. Unprovable undecidable questions are closer to religion than mathematics.

[@drdca8263]

@@caspermadlener4191I don’t see why it couldn’t be that some well defined statement (expressible in our favorite formal system) is independent of the axioms of that system, with the statement that it is independent, also being independent. Is there some argument for why this should be impossible? (Maybe something based on the number of alternating quantifiers in the statement?) Hm. A Sigma 1 statement or a Pi 1 statement, well, if a Sigma 1 statement is true, then it is provable, and if a Pi 1 statement is false, its negation is provable, So, if there was no proof of a Sigma 1 statement, then it would be false, and if no refutation of a Pi 1 statement, then it is true, uh.. If a Sigma 1 statement is independent, then it is false, and so, if it could be proven to be independent, it could be proven to be false, contradicting it being independent. So, and Sigma 1 statement, if independent, is not provably independent?

[@fedebonons8453]

​@@caspermadlener4191 i've heared about this turing machine from my theo. CS professor at uni Where could i learn more about this TM and how its constructed? Thank you

[@palfers1]

I love a good mystery

[@zy9662]

8:10 is he making a joke? Unary minus ? What is that