---
date: 2023-01-02 17:18
modification date: Monday 2nd January 2023 17:18:20
title: Notes Beilinson Conjectures
aliases:
  - Beilinson Conjectures
annotation-target: https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf
status: ✅ Started
page_current: 0
page_total: 0
flashcard: 
created: 2023-01-02T17:18
updated: 2024-01-29T17:51
---


# Beilinson Conjectures

>%%
>```annotation-json
>{"created":"2023-01-02T22:29:20.708Z","text":"E.g. \n$$\\zeta(2) = {\\pi^2\\over 6}$$\n$$\\zeta(4) = {\\pi^4 \\over 90}$$","updated":"2023-01-02T22:29:20.708Z","document":{"title":"NotesF6.pdf","link":[{"href":"urn:x-pdf:4156188b07698653b3e79866f0d62bdc"},{"href":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf"}],"documentFingerprint":"4156188b07698653b3e79866f0d62bdc"},"uri":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf","target":[{"source":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf","selector":[{"type":"TextPositionSelector","start":1561,"end":1659},{"type":"TextQuoteSelector","exact":"Some classical identities. The following identities are instances of specialvalues of L-functions:","prefix":" ZHU1. Classical motivation1.1. ","suffix":"1 + 122 + 132 + 142 + ···= π26(1"}]}]}
>```
>%%
>*%%PREFIX%%ZHU1. Classical motivation1.1.%%HIGHLIGHT%% ==Some classical identities. The following identities are instances of specialvalues of L-functions:== %%POSTFIX%%1 + 122 + 132 + 142 + ···= π26(1*
>%%LINK%%[[#^r1pcucv83eo|show annotation]]
>%%COMMENT%%
>E.g. 
>$$\zeta(2) = {\pi^2\over 6}$$
>$$\zeta(4) = {\pi^4 \over 90}$$
>%%TAGS%%
>
^r1pcucv83eo



>%%
>```annotation-json
>{"created":"2023-01-02T22:28:15.109Z","text":"What is a **regulator** $R$?","updated":"2023-01-02T22:28:15.109Z","document":{"title":"NotesF6.pdf","link":[{"href":"urn:x-pdf:4156188b07698653b3e79866f0d62bdc"},{"href":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf"}],"documentFingerprint":"4156188b07698653b3e79866f0d62bdc"},"uri":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf","target":[{"source":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf","selector":[{"type":"TextPositionSelector","start":1147,"end":1249},{"type":"TextQuoteSelector","exact":"The central concept is that of a regulator map from a K-theoretical objectto a Hodge-theorical object.","prefix":"ature of these special val-ues. ","suffix":" Please inform me of corrections"}]}]}
>```
>%%
>*%%PREFIX%%ature of these special val-ues.%%HIGHLIGHT%% ==The central concept is that of a regulator map from a K-theoretical objectto a Hodge-theorical object.== %%POSTFIX%%Please inform me of corrections*
>%%LINK%%[[#^5u5re2u068t|show annotation]]
>%%COMMENT%%
>What is a **regulator** $R$?
>%%TAGS%%
>
^5u5re2u068t


>%%
>```annotation-json
>{"created":"2023-01-02T22:31:51.335Z","text":"What is $\\zeta(2m)$, i.e. special values at even integers?\n\n$$\\zeta(2 m)=\\frac{(-1)^{m+1}(2 \\pi)^{2 m} B_{2 m}}{2(2 m) !}, m \\in \\mathbb{Z}_{>0}$$\nwhere the Bernoulli numbers are defined as \n$$t\\left(e^t-1\\right)^{-1}=\\sum_{k \\geq 0} B_k \\frac{t^k}{k !}$$","updated":"2023-01-02T22:31:51.335Z","document":{"title":"NotesF6.pdf","link":[{"href":"urn:x-pdf:4156188b07698653b3e79866f0d62bdc"},{"href":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf"}],"documentFingerprint":"4156188b07698653b3e79866f0d62bdc"},"uri":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf","target":[{"source":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf","selector":[{"type":"TextPositionSelector","start":2512,"end":2547},{"type":"TextQuoteSelector","exact":"Euler shows the following formula a","prefix":"s at integer arguments) of ζ(s).","suffix":"t positive even integers:ζ(2m) ="}]}]}
>```
>%%
>*%%PREFIX%%s at integer arguments) of ζ(s).%%HIGHLIGHT%% ==Euler shows the following formula a== %%POSTFIX%%t positive even integers:ζ(2m) =*
>%%LINK%%[[#^w3gsdqijei|show annotation]]
>%%COMMENT%%
>What is $\zeta(2m)$, i.e. special values at even integers?
>
>$$\zeta(2 m)=\frac{(-1)^{m+1}(2 \pi)^{2 m} B_{2 m}}{2(2 m) !}, m \in \mathbb{Z}_{>0}$$
>where the Bernoulli numbers are defined as 
>$$t\left(e^t-1\right)^{-1}=\sum_{k \geq 0} B_k \frac{t^k}{k !}$$
>%%TAGS%%
>
^w3gsdqijei


>%%
>```annotation-json
>{"created":"2023-01-02T22:33:32.295Z","updated":"2023-01-02T22:33:32.295Z","document":{"title":"NotesF6.pdf","link":[{"href":"urn:x-pdf:4156188b07698653b3e79866f0d62bdc"},{"href":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf"}],"documentFingerprint":"4156188b07698653b3e79866f0d62bdc"},"uri":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf","target":[{"source":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf","selector":[{"type":"TextPositionSelector","start":3328,"end":3517},{"type":"TextQuoteSelector","exact":"The rest cases are the positive odd integers and the negative even integers. Theyare of course again related by the functional equation. The nature of these zetavalues remain very mysteriou","prefix":"dentity using formal operations.","suffix":"s. It is known that infinitely m"}]}]}
>```
>%%
>*%%PREFIX%%dentity using formal operations.%%HIGHLIGHT%% ==The rest cases are the positive odd integers and the negative even integers. Theyare of course again related by the functional equation. The nature of these zetavalues remain very mysteriou== %%POSTFIX%%s. It is known that infinitely m*
>%%LINK%%[[#^3v8arzmgex4|show annotation]]
>%%COMMENT%%
>
>%%TAGS%%
>
^3v8arzmgex4


>%%
>```annotation-json
>{"created":"2023-01-02T22:34:03.490Z","text":"$$\\zeta_F(s)=\\sum_{\\mathfrak{a} \\in \\Id(\\OO_F)}\\left|\\mathcal{O}_F / \\mathfrak{a}\\right|^{-s}=\\prod_{\\mathbf{p} \\in \\spec \\OO_F }\\left(1-\\left|\\mathcal{O}_F / \\mathfrak{p}\\right|^{-s}\\right)^{-1}$$\n\nConverges absolutely for $\\Re(s) > 1$, has a meromorphic extension to $\\CC$ with a simple pole at $s=1$ and a function equation relating $\\zeta_F(s)$ to $\\zeta_F(1-s)$.","updated":"2023-01-02T22:34:03.490Z","document":{"title":"NotesF6.pdf","link":[{"href":"urn:x-pdf:4156188b07698653b3e79866f0d62bdc"},{"href":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf"}],"documentFingerprint":"4156188b07698653b3e79866f0d62bdc"},"uri":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf","target":[{"source":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf","selector":[{"type":"TextPositionSelector","start":4136,"end":4159},{"type":"TextQuoteSelector","exact":"Dedekind zeta functions","prefix":"erms of algebraic K-theory.1.3. ","suffix":". Generalizing Riemann’s zeta fu"}]}]}
>```
>%%
>*%%PREFIX%%erms of algebraic K-theory.1.3.%%HIGHLIGHT%% ==Dedekind zeta functions== %%POSTFIX%%. Generalizing Riemann’s zeta fu*
>%%LINK%%[[#^9ipxpc27er4|show annotation]]
>%%COMMENT%%
>$$\zeta_F(s)=\sum_{\mathfrak{a} \in \Id(\OO_F)}\left|\mathcal{O}_F / \mathfrak{a}\right|^{-s}=\prod_{\mathbf{p} \in \spec \OO_F }\left(1-\left|\mathcal{O}_F / \mathfrak{p}\right|^{-s}\right)^{-1}$$
>
>Converges absolutely for $\Re(s) > 1$, has a meromorphic extension to $\CC$ with a simple pole at $s=1$ and a function equation relating $\zeta_F(s)$ to $\zeta_F(1-s)$.
>%%TAGS%%
>
^9ipxpc27er4


>%%
>```annotation-json
>{"created":"2023-01-02T22:48:31.981Z","text":"$$\\Res_{s=1} \\zeta_F(s) = \\frac{2^{r_1}(2 \\pi)^{r_2}}{\\left|d_K\\right|^{1 / 2} w(K)} h(F) R(F)$$\nwhere $r_1, r_2$ are the number of real/complex places of $F$, $h(F) \\da \\size \\Pic(\\OO_F)$, and $R(F) = \\abs{\\det\\log \\abs{u_i}_j}$ where $\\ts{u_i}$ are a $\\ZZ\\dash$basis for $\\OO_F\\units/\\tors$ and $\\abs{\\wait}_j$ is the absolute value at the $j$th archimedean place.","updated":"2023-01-02T22:48:31.981Z","document":{"title":"NotesF6.pdf","link":[{"href":"urn:x-pdf:4156188b07698653b3e79866f0d62bdc"},{"href":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf"}],"documentFingerprint":"4156188b07698653b3e79866f0d62bdc"},"uri":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf","target":[{"source":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf","selector":[{"type":"TextPositionSelector","start":4957,"end":4979},{"type":"TextQuoteSelector","exact":"(Class number formula)","prefix":"n, we have the followingTheorem ","suffix":". The residue of ζF(s) at s = 1 "}]}]}
>```
>%%
>*%%PREFIX%%n, we have the followingTheorem%%HIGHLIGHT%% ==(Class number formula)== %%POSTFIX%%. The residue of ζF(s) at s = 1*
>%%LINK%%[[#^fgmyv6njhli|show annotation]]
>%%COMMENT%%
>$$\Res_{s=1} \zeta_F(s) = \frac{2^{r_1}(2 \pi)^{r_2}}{\left|d_K\right|^{1 / 2} w(K)} h(F) R(F)$$
>where $r_1, r_2$ are the number of real/complex places of $F$, $h(F) \da \size \Pic(\OO_F)$, and $R(F) = \abs{\det\log \abs{u_i}_j}$ where $\ts{u_i}$ are a $\ZZ\dash$basis for $\OO_F\units/\tors$ and $\abs{\wait}_j$ is the absolute value at the $j$th archimedean place.
>%%TAGS%%
>
^fgmyv6njhli


>%%
>```annotation-json
>{"created":"2023-01-02T22:50:54.941Z","text":"Regulators determine the transcendental part of the zeta value. Assemble the maps $\\log\\abs{\\wait}_j$ to form a regulator map $r: \\OO_F\\units\\to \\RR^{r_1+r_2-1}$, noting that the domain is $\\K_1(\\OO_F)$ which is rank $r_1+r_2-1$ by Dirichlet's unit theorem, and $R(F) = \\covol(r)$.","updated":"2023-01-02T22:50:54.941Z","document":{"title":"NotesF6.pdf","link":[{"href":"urn:x-pdf:4156188b07698653b3e79866f0d62bdc"},{"href":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf"}],"documentFingerprint":"4156188b07698653b3e79866f0d62bdc"},"uri":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf","target":[{"source":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf","selector":[{"type":"TextPositionSelector","start":6127,"end":6156},{"type":"TextQuoteSelector","exact":"We can regard R(F) as follows","prefix":"+ √2), so to speak, as in (1.4).","suffix":". The maps log |·|j form a regul"}]}]}
>```
>%%
>*%%PREFIX%%+ √2), so to speak, as in (1.4).%%HIGHLIGHT%% ==We can regard R(F) as follows== %%POSTFIX%%. The maps log |·|j form a regul*
>%%LINK%%[[#^ebppwyo1hcu|show annotation]]
>%%COMMENT%%
>Regulators determine the transcendental part of the zeta value. Assemble the maps $\log\abs{\wait}_j$ to form a regulator map $r: \OO_F\units\to \RR^{r_1+r_2-1}$, noting that the domain is $\K_1(\OO_F)$ which is rank $r_1+r_2-1$ by Dirichlet's unit theorem, and $R(F) = \covol(r)$.
>%%TAGS%%
>
^ebppwyo1hcu


>%%
>```annotation-json
>{"created":"2023-01-02T22:52:46.130Z","text":"$\\dim_\\QQ \\K_{2m-1}(\\OO_F) \\tensor \\QQ = d_m$ where $d_m = r_2$ for $m$ even and $r_1+r_2$ for $m$ odd. Note that $d_m$ is also the order of vanishing of $\\zeta_F(s)$ at $s=1-m$.","updated":"2023-01-02T22:52:46.130Z","document":{"title":"NotesF6.pdf","link":[{"href":"urn:x-pdf:4156188b07698653b3e79866f0d62bdc"},{"href":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf"}],"documentFingerprint":"4156188b07698653b3e79866f0d62bdc"},"uri":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf","target":[{"source":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf","selector":[{"type":"TextPositionSelector","start":6392,"end":6496},{"type":"TextQuoteSelector","exact":"Higher regulators. One might ask whether the higher K groups Ki(OF) canbe related to other values of ζF.","prefix":" equal to thecovolume of r.1.4. ","suffix":"Theorem (Borel, 1972). Let m ∈Z>"}]}]}
>```
>%%
>*%%PREFIX%%equal to thecovolume of r.1.4.%%HIGHLIGHT%% ==Higher regulators. One might ask whether the higher K groups Ki(OF) canbe related to other values of ζF.== %%POSTFIX%%Theorem (Borel, 1972). Let m ∈Z>*
>%%LINK%%[[#^h8vn3emc31p|show annotation]]
>%%COMMENT%%
>$\dim_\QQ \K_{2m-1}(\OO_F) \tensor \QQ = d_m$ where $d_m = r_2$ for $m$ even and $r_1+r_2$ for $m$ odd. Note that $d_m$ is also the order of vanishing of $\zeta_F(s)$ at $s=1-m$.
>%%TAGS%%
>
^h8vn3emc31p


>%%
>```annotation-json
>{"created":"2023-01-02T22:54:10.519Z","text":"Can one define a regulator $r_m: \\K_{2m-1}(\\OO_F) \\to d_m$ so that $\\covol(r_m)$ is related to $\\zeta_F(m)$?","updated":"2023-01-02T22:54:10.519Z","document":{"title":"NotesF6.pdf","link":[{"href":"urn:x-pdf:4156188b07698653b3e79866f0d62bdc"},{"href":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf"}],"documentFingerprint":"4156188b07698653b3e79866f0d62bdc"},"uri":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf","target":[{"source":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf","selector":[{"type":"TextPositionSelector","start":6824,"end":6863},{"type":"TextQuoteSelector","exact":"Lichtenbaum asked the following questio","prefix":"n, ζ∗F(1 −m) is related to ζ(m).","suffix":"n: For m ∈ Z>1, can we define a "}]}]}
>```
>%%
>*%%PREFIX%%n, ζ∗F(1 −m) is related to ζ(m).%%HIGHLIGHT%% ==Lichtenbaum asked the following questio== %%POSTFIX%%n: For m ∈ Z>1, can we define a*
>%%LINK%%[[#^emkvfiacgdm|show annotation]]
>%%COMMENT%%
>Can one define a regulator $r_m: \K_{2m-1}(\OO_F) \to d_m$ so that $\covol(r_m)$ is related to $\zeta_F(m)$?
>%%TAGS%%
>
^emkvfiacgdm


>%%
>```annotation-json
>{"created":"2023-01-02T22:55:00.405Z","updated":"2023-01-02T22:55:00.405Z","document":{"title":"NotesF6.pdf","link":[{"href":"urn:x-pdf:4156188b07698653b3e79866f0d62bdc"},{"href":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf"}],"documentFingerprint":"4156188b07698653b3e79866f0d62bdc"},"uri":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf","target":[{"source":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf","selector":[{"type":"TextPositionSelector","start":7320,"end":7424},{"type":"TextQuoteSelector","exact":"Conjecturally, one can associate L-functions to motives, thelatter being objects of a conjectural nature","prefix":"ve forward tohigher dimensions. ","suffix":" that should capture cohomologic"}]}]}
>```
>%%
>*%%PREFIX%%ve forward tohigher dimensions.%%HIGHLIGHT%% ==Conjecturally, one can associate L-functions to motives, thelatter being objects of a conjectural nature== %%POSTFIX%%that should capture cohomologic*
>%%LINK%%[[#^fmqz7kwhdio|show annotation]]
>%%COMMENT%%
>
>%%TAGS%%
>
^fmqz7kwhdio


>%%
>```annotation-json
>{"created":"2023-01-02T22:55:16.652Z","updated":"2023-01-02T22:55:16.652Z","document":{"title":"NotesF6.pdf","link":[{"href":"urn:x-pdf:4156188b07698653b3e79866f0d62bdc"},{"href":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf"}],"documentFingerprint":"4156188b07698653b3e79866f0d62bdc"},"uri":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf","target":[{"source":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf","selector":[{"type":"TextPositionSelector","start":7590,"end":7759},{"type":"TextQuoteSelector","exact":"These represent the rough ideaof \"L-functions of a geometric origin\", as opposed to other interesting L-functionsof an analytic origin, namely the automorphic L-function","prefix":"rred to as motivic L-functions. ","suffix":"s. Of course Langlands’program c"}]}]}
>```
>%%
>*%%PREFIX%%rred to as motivic L-functions.%%HIGHLIGHT%% ==These represent the rough ideaof "L-functions of a geometric origin", as opposed to other interesting L-functionsof an analytic origin, namely the automorphic L-function== %%POSTFIX%%s. Of course Langlands’program c*
>%%LINK%%[[#^ascfxuc7l2g|show annotation]]
>%%COMMENT%%
>
>%%TAGS%%
>
^ascfxuc7l2g


>%%
>```annotation-json
>{"created":"2023-01-02T22:55:29.101Z","updated":"2023-01-02T22:55:29.101Z","document":{"title":"NotesF6.pdf","link":[{"href":"urn:x-pdf:4156188b07698653b3e79866f0d62bdc"},{"href":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf"}],"documentFingerprint":"4156188b07698653b3e79866f0d62bdc"},"uri":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf","target":[{"source":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf","selector":[{"type":"TextPositionSelector","start":7772,"end":7893},{"type":"TextQuoteSelector","exact":"Langlands’program conjectures that the motivic L-functions form a well characterized subsetof the automorphic L-functions","prefix":"omorphic L-functions. Of course ","suffix":".2.1. Realizations of motives. W"}]}]}
>```
>%%
>*%%PREFIX%%omorphic L-functions. Of course%%HIGHLIGHT%% ==Langlands’program conjectures that the motivic L-functions form a well characterized subsetof the automorphic L-functions== %%POSTFIX%%.2.1. Realizations of motives. W*
>%%LINK%%[[#^cwparh4vsg|show annotation]]
>%%COMMENT%%
>
>%%TAGS%%
>
^cwparh4vsg


>%%
>```annotation-json
>{"created":"2023-01-02T22:56:45.740Z","updated":"2023-01-02T22:56:45.740Z","document":{"title":"NotesF6.pdf","link":[{"href":"urn:x-pdf:4156188b07698653b3e79866f0d62bdc"},{"href":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf"}],"documentFingerprint":"4156188b07698653b3e79866f0d62bdc"},"uri":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf","target":[{"source":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf","selector":[{"type":"TextPositionSelector","start":7209,"end":7228},{"type":"TextQuoteSelector","exact":"Motivic L-functions","prefix":"l notdiscuss the former here.2. ","suffix":"Number fields are zero dimension"}]}]}
>```
>%%
>*%%PREFIX%%l notdiscuss the former here.2.%%HIGHLIGHT%% ==Motivic L-functions== %%POSTFIX%%Number fields are zero dimension*
>%%LINK%%[[#^c38d8rowet|show annotation]]
>%%COMMENT%%
>
>%%TAGS%%
>
^c38d8rowet


>%%
>```annotation-json
>{"created":"2023-01-02T22:56:54.754Z","updated":"2023-01-02T22:56:54.754Z","document":{"title":"NotesF6.pdf","link":[{"href":"urn:x-pdf:4156188b07698653b3e79866f0d62bdc"},{"href":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf"}],"documentFingerprint":"4156188b07698653b3e79866f0d62bdc"},"uri":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf","target":[{"source":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf","selector":[{"type":"TextPositionSelector","start":8145,"end":8156},{"type":"TextQuoteSelector","exact":"Tate twist,","prefix":"e symbol (n) is called the n-th ","suffix":" c.f. Example 2.1.3 below. There"}]}]}
>```
>%%
>*%%PREFIX%%e symbol (n) is called the n-th%%HIGHLIGHT%% ==Tate twist,== %%POSTFIX%%c.f. Example 2.1.3 below. There*
>%%LINK%%[[#^cw4cooiqnb|show annotation]]
>%%COMMENT%%
>
>%%TAGS%%
>
^cw4cooiqnb


>%%
>```annotation-json
>{"created":"2023-01-02T22:57:01.172Z","updated":"2023-01-02T22:57:01.172Z","document":{"title":"NotesF6.pdf","link":[{"href":"urn:x-pdf:4156188b07698653b3e79866f0d62bdc"},{"href":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf"}],"documentFingerprint":"4156188b07698653b3e79866f0d62bdc"},"uri":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf","target":[{"source":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf","selector":[{"type":"TextPositionSelector","start":8285,"end":8302},{"type":"TextQuoteSelector","exact":"Betti realization","prefix":" the variousrealizations of M.• ","suffix":": Consider MB = Hi(X(C),Q(n)). H"}]}]}
>```
>%%
>*%%PREFIX%%the variousrealizations of M.•%%HIGHLIGHT%% ==Betti realization== %%POSTFIX%%: Consider MB = Hi(X(C),Q(n)). H*
>%%LINK%%[[#^b5rix6b1zwa|show annotation]]
>%%COMMENT%%
>
>%%TAGS%%
>
^b5rix6b1zwa


>%%
>```annotation-json
>{"created":"2023-01-02T22:57:04.307Z","updated":"2023-01-02T22:57:04.307Z","document":{"title":"NotesF6.pdf","link":[{"href":"urn:x-pdf:4156188b07698653b3e79866f0d62bdc"},{"href":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf"}],"documentFingerprint":"4156188b07698653b3e79866f0d62bdc"},"uri":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf","target":[{"source":"https://math.mit.edu/juvitop/old/notes/2015_Fall/NotesF6.pdf","selector":[{"type":"TextPositionSelector","start":8818,"end":8835},{"type":"TextQuoteSelector","exact":"l-adic realizatio","prefix":"compositionMB ⊗C = ⊕p+q=wHp,q.• ","suffix":"n. Let l be a prime number. Cons"}]}]}
>```
>%%
>*%%PREFIX%%compositionMB ⊗C = ⊕p+q=wHp,q.•%%HIGHLIGHT%% ==l-adic realizatio== %%POSTFIX%%n. Let l be a prime number. Cons*
>%%LINK%%[[#^424vezvwibc|show annotation]]
>%%COMMENT%%
>
>%%TAGS%%
>
^424vezvwibc


>%%
>```annotation-json
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>```
>%%
>*%%PREFIX%%omic character of Gal(  ̄Q/Q).•%%HIGHLIGHT%% ==De Rham realizatio== %%POSTFIX%%n: Consider MdR = Hi(X,Ω·)(n). T*
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>%%
>```annotation-json
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>```
>%%
>*%%PREFIX%%independent of the choice of ιp.%%HIGHLIGHT%% ==Conjecture 2.2.1. Plp(M,T) is a polynomial in T with coefficients in Z, andindependent of l.This conjecture is known when X has good reduction at p, which means thatthere is a smooth projective model Xof X over Zp== %%POSTFIX%%. In this case Ip acts trivially*
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>%%
>```annotation-json
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>```
>%%
>*%%PREFIX%%1),s) = ζ(s + 1).Example 2.2.3.%%HIGHLIGHT%% ==L-functions of zero dimensional motives recover the classical no-tion of Artin L-functions (including Dedekind zeta functions discussed above andDirichlet L-functions)== %%POSTFIX%%. For instance for F a number fi*
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>%%
>```annotation-json
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>```
>%%
>*%%PREFIX%%, whichhas little to do with C2.%%HIGHLIGHT%% ==Conjecture 3.3.5 (Beilinson).== %%POSTFIX%%Assume w = −1. Then(1) There is*
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>%%
>```annotation-json
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>```
>%%
>*%%PREFIX%%ust up to Q×. c.f. [Kin03].3.4.%%HIGHLIGHT%% ==Known cases. The conjectures for Dedekind zeta functions and Dirichlet L-functions are proved by Borel and Beilinson. In higher dimensions only partialresults are known, all related to elliptic curves or Shimura varieties== %%POSTFIX%%. c.f. [Nek94]§8.10 YIHANG ZHU4.*
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>%%
>```annotation-json
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>```
>%%
>*%%PREFIX%%ies:(1) Motives. (Grothendieck.)%%HIGHLIGHT%% ==Let V be a suitable subcategory of the categoryof algebraic varieties over a field k. We want to construct a universalcohomology theory on V.== %%POSTFIX%%In particular we want to constr*
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>%%
>```annotation-json
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>```
>%%
>*%%PREFIX%%Motivic complexes. (Beilinson.)%%HIGHLIGHT%% ==We want to do better than Philosophy(1). For X ∈V, we would like to produce an object RΓ(X,n) of the derivedcategory D(M) of M== %%POSTFIX%%, supposing the latter makes sen*
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>%%
>```annotation-json
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>```
>%%
>*%%PREFIX%%CONJECTURES (TALK NOTES) 11N.B.%%HIGHLIGHT%% ==Beilinson knows that for the above speculations about the motivic com-plexes and motivic cohmology to work, the category M has to be some categoryof mixed motives (i.e. allowing non-smooth varieties), whose associated categoryof semi-simple objects would be the category of pure motives== %%POSTFIX%%. This is in sharpcontrast to Gr*
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>%%
>```annotation-json
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>```
>%%
>*%%PREFIX%%es. This is in sharpcontrast to%%HIGHLIGHT%% ==Grothendieck’s philosophy about motives, where one is totally allowedto restrict attention to smooth projective varieties and consider only pure motives== %%POSTFIX%%.Unfortunately, unlike the situa*
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>%%
>```annotation-json
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>```
>%%
>*%%PREFIX%%nly pure motives.Unfortunately,%%HIGHLIGHT%% ==unlike the situation with pure motives, where a proof of the stan-dard conjectures about algebraic cycles would automatically lead to a good theory,as for now people have not found a good framework to formulate the theory ofmixed motives== %%POSTFIX%%.14.2. Realizations: absolute co*
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