View source for Beilinson conjectures

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b1102201.png" /> be a smooth projective variety (cf. [[Projective scheme|Projective scheme]]) defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b1102202.png" />. For such <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b1102203.png" /> one has, on the one hand, the algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b1102204.png" />-groups (cf. [[K-theory|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b1102205.png" />-theory]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b1102206.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b1102207.png" />, and on the other hand, various [[Cohomology|cohomology]] theories, such as Betti cohomology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b1102208.png" />, de Rham cohomology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b1102209.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022010.png" />-adic cohomology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022011.png" />. These cohomology theories can be considered as realizations of the (Chow) motive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022012.png" /> associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022013.png" />. There are comparison isomorphisms between them. Decomposing the motive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022015.png" />, one may fix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022017.png" />, and define, via the Frobenius action on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022018.png" />-adic cohomology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022019.png" /> (cf. [[#References|[a6]]]), the [[L-function|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022020.png" />-function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022021.png" />, an infinite product which converges absolutely for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022022.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022023.png" /> is a pure motive of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022024.png" />. Using the [[Hodge structure|Hodge structure]] on the cohomology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022025.png" /> of the [[Complex manifold|complex manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022026.png" />, one defines the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022027.png" />-factor  "at infinity" , <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022028.png" />, essentially as a product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022029.png" />-factors. Finally, one defines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022030.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022031.png" /> one has a conjectural analytic continuation and functional equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022032.png" />, for a suitable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022033.png" /> of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022034.png" />, and with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022035.png" /> the dual motive of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022036.png" />. Here, by [[Poincaré duality|Poincaré duality]] and hard Lefschetz, this means <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022037.png" />. In general, for an arbitrary motive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022038.png" /> of pure weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022039.png" />, one extends the above construction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022042.png" />. One should have <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022043.png" />.

On the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022044.png" />-groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022045.png" /> one has the action of the Adams operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022046.png" /> (cf. [[Cohomology operation|Cohomology operation]]). They all commute with each other. Write <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022047.png" /> for the subspace on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022048.png" /> acts as multiplication by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022050.png" />. A. Beilinson defines the absolute or motivic cohomology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022051.png" />. As a matter of fact, this can be defined for any regular or affine (simplicial) scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022052.png" />. It has many nice properties of a cohomology theory; in particular there is a motivic Chern character mapping (a sum of projections after tensoring with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022053.png" />) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022054.png" />. A classical theorem of A. Grothendieck says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022055.png" />. Beilinson has extended motivic cohomology to the category of (Chow) motives with coefficients in a number field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022056.png" />. Assuming that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022057.png" /> admits a regular model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022058.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022059.png" />, one defines

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022060.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022061.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022062.png" /></td> </tr></table>

This is independent of the regular model, provided that it exists. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022063.png" /> are conjectured to be finite-dimensional. Their construction may be applied to define groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022064.png" /> for any Chow motive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022065.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022066.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022067.png" />.

Another main ingredient of Beilinson's conjectures is Deligne (or Deligne–Beilinson) cohomology. This is defined for any quasi-projective variety (cf. [[Quasi-projective scheme|Quasi-projective scheme]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022068.png" /> defined over the complex numbers. For smooth projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022069.png" /> it is easy to define. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022070.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022071.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022072.png" /> and write <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022073.png" /> for the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022074.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022075.png" />. Consider the following complex of sheaves on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022076.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022077.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022078.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022079.png" />) is placed in degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022080.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022081.png" />). <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022082.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022083.png" />) denotes the sheaf of holomorphic functions (respectively, holomorphic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022084.png" />-forms) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022085.png" />. One defines the Deligne cohomology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022086.png" /> as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022087.png" /></td> </tr></table>

the hypercohomology of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022088.png" />. For arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022089.png" /> one uses a smooth compactification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022090.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022091.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022092.png" /> is a normal crossings divisor, and, using the associated logarithmic de Rham complex of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022093.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022094.png" />, it is possible to construct well-defined Deligne–Beilinson cohomology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022095.png" />. Thus, one obtains a good cohomology theory, with supports, Poincaré duality, even a homological counterpart, satisfying the axioms of a Poincaré duality theory in the sense of S. Bloch and A. Ogus. In particular, there is again a Chern character mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022096.png" />. For smooth projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022097.png" /> defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022098.png" />, one defines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b11022099.png" /> as the subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220100.png" /> invariant under the induced action of complex conjugation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220101.png" /> acting on the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220102.png" />, i.e., acting on differential forms by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220103.png" />. Similarly for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220105.png" />. Then, for an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220106.png" />, there is a short exact sequence

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220107.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220108.png" /></td> </tr></table>

The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220109.png" />-structures on the first two terms give rise to a natural <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220110.png" />-structure

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220111.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220112.png" /></td> </tr></table>

on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220113.png" />. In the general case of motives with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220114.png" />, one will have <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220115.png" />-structures, etc.

Taking things together, one sees that, for varieties over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220116.png" />, there are natural transformations, called regulators, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220117.png" />. Already the simplest explicit examples suggest one should restrict to  "integral motivic cohomology"  and one is led to Beilinson's regulator mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220118.png" />. It should be remarked that one can extend the formalism to the category of Chow motives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220119.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220120.png" /> with field of coefficients in the number field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220121.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220122.png" />-functions will take their values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220123.png" /> and the regulator mappings will be of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220124.png" />. This is even expected to work for Grothendieck motives, i.e., motives modulo homological (which, conjecturally, coincides with numerical) equivalence.

==Beilinson's first conjecture.==
To state Beilinson's conjectures on special values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220125.png" /> at integer arguments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220126.png" />, one distinguishes between three cases:

i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220127.png" />, which by the functional equation corresponds to the region of absolute convergence;

ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220128.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220129.png" /> even, which lies on the boundary of the critical strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220130.png" />;

iii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220131.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220132.png" /> odd, the centre of the critical strip. It is easily shown that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220133.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220134.png" />.

Beilinson's first conjecture reads as follows. Assume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220135.png" />. Then: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220136.png" /> is an [[Isomorphism|isomorphism]]; and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220137.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220138.png" /> is the first non-vanishing coefficient of the Taylor series expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220139.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220140.png" />.

In [[#References|[a1]]], Beilinson states this conjecture for general Chow motives with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220141.png" />.

Some special cases are as follows.

a) For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220142.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220143.png" /> a number field, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220144.png" />, one recovers the situation studied by A. Borel [[#References|[a4]]]. Beilinson showed that his regulator coincides with Borel's regulator (at least modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220145.png" />). Thus, by Borel's results, the first conjecture is true. Classically, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220146.png" /> one obtains the Dirichlet regulator and Dedekind's class number formula.

b) Bloch and Beilinson were the first to construct a regulator mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220147.png" /> (or even <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220148.png" />) for a [[Riemann surface|Riemann surface]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220149.png" />, and make a conjecture about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220150.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220151.png" /> an [[Elliptic curve|elliptic curve]] without complex multiplication, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220152.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220153.png" />, Bloch and D. Grayson made computer calculations which actually gave rise to a formulation of the first conjecture in terms of the integral model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220154.png" />. For elliptic curves with complex multiplication a weak form of the first conjecture was proved by Bloch and Beilinson.

c) Another conjecture which motivated Beilinson's first conjecture is due to P. Deligne [[#References|[a6]]]. It is stated in terms of motives and predicts that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220155.png" />-function of such a motive (cf. also [[Motives, theory of|Motives, theory of]]) at a so-called critical value of the argument would be equal (modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220156.png" />) to a well-defined period.

d) J.-F. Mestre and N. Schappacher gave numerical evidence for the case of the symmetric square of an elliptic curve without complex multiplication.

e) For Dirichlet motives, Beilinson proved the conjecture. For general Artin motives one recovers Gross' conjecture.

f) C. Deninger has obtained results for motives of Hecke characters of imaginary quadratic number fields.

g) Beilinson has proved partial results for (products of) modular curves.

h) K.-I. Kimura has given numerical evidence for (a projective curve related to) the Fermat curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220157.png" />.

Some further examples are known. They all deal with modular curves, Shimura curves, products of such curves, Hilbert modular surfaces, or products of elliptic modular surfaces. A general phenomenon occurs: in all these examples there exists a subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220158.png" /> giving rise, via the regulator mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220159.png" />, to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220160.png" />-structure on the corresponding Deligne–Beilinson cohomology with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220161.png" /> equal (up to a non-zero rational number) to the first non-vanishing coefficient of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220162.png" />-function at a suitable integer value of its argument.

==Beilinson's second conjecture.==
The second conjecture takes into account the possible pole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220163.png" /> at the Tate point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220164.png" />. One shows that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220165.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220166.png" /></td> </tr></table>

for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220167.png" />.

Beilinson's second conjecture reads as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220168.png" /> be even and write <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220169.png" />. Then:

i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220170.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220171.png" /> is the group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220172.png" />-codimensional algebraic cycles on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220173.png" /> modulo homological equivalence, i.e., the image of the morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220174.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220175.png" /> is the inclusion;

ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220176.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220177.png" />.

iii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220178.png" />.

This conjecture can also be stated in terms of motives.

a) For Artin motives it gives Stark's conjecture.

b) For a Hilbert modular surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220179.png" />, D. Ramakrishnan proved the existence of a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220180.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220181.png" /></td> </tr></table>

gives a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220182.png" />-structure on Deligne cohomology with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220183.png" /> equal (up to a non-zero rational number) to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220184.png" />.

==Beilinson's third conjecture.==
The third conjecture deals with the centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220185.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220186.png" /> odd, of the critical strip. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220187.png" /> be a smooth projective variety (cf. [[Projective scheme|Projective scheme]]) of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220188.png" />, and assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220189.png" /> admits a regular, proper model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220190.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220191.png" />. One has an isomorphism

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220192.png" /></td> </tr></table>

giving a period matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220193.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220194.png" />. Beilinson [[#References|[a3]]] showed that there exists a unique bilinear pairing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220195.png" /> of an arithmetic nature, i.e., closely related to the Gillet–Soulé arithmetic intersection pairing (on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220196.png" />), and generalizing Arakelov's intersection pairing on arithmetic surfaces.

Beilinson's third conjecture reads as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220197.png" /> be a smooth projective variety defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220198.png" />, and assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220199.png" /> has a regular, proper model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220200.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220201.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220202.png" />. Then:

i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220203.png" />;

ii) the pairing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220204.png" /> is non-degenerate.

iii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220205.png" />.

iv) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220206.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220207.png" />.

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220208.png" /> an elliptic curve, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220209.png" />, one recovers the Mordell–Weil theorem and the Birch–Swinnerton-Dyer conjectures for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220210.png" />.

==Generalizations.==
Deligne observed that, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220211.png" />, one can interprete <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220212.png" /> as a Yoneda extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220213.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220214.png" /> is the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220215.png" />-mixed Hodge structures with a real Frobenius. This made the search for a category of  "mixed motives"  over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220216.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220217.png" /> (or, even better, over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220218.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220219.png" />) very tempting. The regulator mapping in this setting would be just the Betti realization functor

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220220.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220221.png" /></td> </tr></table>

The category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220222.png" /> should contain Grothendieck's category of pure motives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220223.png" /> and allow the treatment of arbitrary varieties over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220224.png" />. Analogously, for other base fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220225.png" />, one should have categories <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220226.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220227.png" />, etc. Also, the role of the Chow groups in the theory of Grothendieck motives might be enlarged to include all the algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220228.png" />-groups of the variety. In this respect one may mention a very geometric construction by Bloch of generalized Chow groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220229.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220230.png" /> they coincide with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220231.png" />. They are integrally defined and satisfy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220232.png" />. A series of other conjectures, mainly about filtrations on Chow groups (Beilinson, J.P. Murre), emerges, and the ultimate formulation of Beilinson's conjectures appears in terms of derived categories, mixed motivic sheaves, mixed perverse sheaves, etc., cf. [[#References|[a3]]], [[#References|[a8]]].

In [[#References|[a2]]], Beilinson introduced the notion of absolute Hodge cohomology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220233.png" />. This generalizes Deligne–Beilinson cohomology by taking the weight filtration into account. It is a [[Derived functor|derived functor]] cohomology defined for any [[Scheme|scheme]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220234.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220235.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220236.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220237.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220238.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220239.png" /> denote the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220240.png" />-mixed Hodge structures. In this setting, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220241.png" />, one can define the Abel–Jacobi mappings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220242.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220243.png" />. For smooth projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220244.png" /> this gives the classical Abel–Jacobi mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220245.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220246.png" /> is Griffiths' intermediate [[Jacobian|Jacobian]].

The following conjecture generalizes the classical [[Hodge conjecture|Hodge conjecture]]. In this form it is due to Beilinson and U. Jannsen.

The Beilinson–Jannsen conjecture. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220247.png" /> be a smooth variety defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220248.png" />. Then, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220249.png" />, the regulator mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220250.png" /> has dense image.

In [[#References|[a5]]] there is a formulation of Beilinson's conjectures in terms of (mixed) motives, without the modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220251.png" /> ambiguity. There is also a very precise conjecture in terms of Tamagawa numbers, cf. [[#References|[a5]]], [[#References|[a7]]] and the contribution by J.-M. Fontaine and B. Perrin-Riou in [[#References|[a8]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Beilinson,   "Higher regulators and values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220252.png" />-functions"  ''J. Soviet Math.'' , '''30'''  (1985)  pp. 2036–2070  (In Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Beilinson,   "Notes on absolute Hodge cohomology" , ''Contemp. Math.'' , '''55''' , Amer. Math. Soc.  (1985)  pp. 35–68</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Beilinson,   "Height pairings for algebraic cycles" , ''Lecture Notes in Mathematics'' , '''1289''' , Springer  (1987)  pp. 1–26</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Borel,   "Cohomologie de <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220253.png" /> et valeurs de fonctions zeta aux points entiers"  ''Ann. Sci. Pisa''  (1976)  pp. 613–636</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  S. Bloch,   K. Kato,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220254.png" />-functions and Tamagawa numbers of motives" , ''The Grothendieck Festschrift I'' , ''Progress in Mathematics'' , '''86''' , Birkhäuser  (1990)  pp. 333–400</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  P. Deligne,   "Valeurs de fonctions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110220/b110220255.png" /> et périodes d'intégrales" , ''Proc. Symp. Pure Math.'' , '''33''' , Amer. Math. Soc.  (1979)  pp. 313–346</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  J.-M. Fontaine,   B. Perrin-Riou,   "Autour des conjectures de Bloch et Kato, I--III"  ''C.R. Acad. Sci. Paris'' , '''313'''  (1991)  pp. 189–196; 349–356; 421–428</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  "Motives"  U. Jannsen (ed.)  etAAsal. (ed.) , ''Proc. Symp. Pure Math.'' , '''55''' , Amer. Math. Soc.  (1994)</TD></TR></table>