1388:. Over fields of non-zero characteristic, formal group laws are not equivalent to Lie algebras. In fact, in this case it is well-known that passing from an algebraic group to its Lie algebra often throws away too much information, but passing instead to the formal group law often keeps enough information. So in some sense formal group laws are the "right" substitute for Lie algebras in characteristic
3313:
Over an algebraically closed field, the substack of one-dimensional formal groups is either a point (in characteristic zero) or an infinite chain of stacky points parametrizing heights. In characteristic zero, the closure of each point contains all points of greater height. This difference gives
1291:
877:, by taking coordinates at the identity and writing down the formal power series expansion of the product map. The additive and multiplicative formal group laws are obtained in this way from the additive and multiplicative algebraic groups. Another important special case of this is the
3341:(usually with some extra conditions added, such as being pointed or connected). This is more or less dual to the notion above. In the smooth case, choosing coordinates is equivalent to taking a distinguished basis of the formal group ring.
1083:
2264:
3297:
The moduli space of formal group laws is a disjoint union of infinite-dimensional affine spaces, whose components are parametrized by dimension, and whose points are parametrized by admissible coefficients of the power series
1773:
2053:
1114:
2145:
1384:
0, formal group laws are essentially the same as finite-dimensional Lie algebras: more precisely, the functor from finite-dimensional formal group laws to finite-dimensional Lie algebras is an
2615:= 1, and the group-like elements form a group under multiplication. In the case of the Hopf algebra of a formal group law over a ring, the group like elements are exactly those of the form
3500:
3036:. At first sight it seems to be incredibly complicated: the relations between its generators are very messy. However Lazard proved that it has a very simple structure: it is just a
3280:
asserts the existence of lifts of deformations and can apply to formal schemes that are larger than points. A smooth formal group scheme is a special case of a formal group scheme.
1899:
2150:
1364:
from Lie groups or algebraic groups to Lie algebras can be factorized into a functor from Lie groups to formal group laws, followed by taking the Lie algebra of the formal group:
2501:; the point is that all the formal power series now converge because they are being applied to nilpotent elements, so there are only a finite number of nonzero terms. This makes
3268:
756:
3183:
1949:
678:
202:
should be something like the formal power series expansion of the product of a Lie group, where we choose coordinates so that the identity of the Lie group is the origin.
2731:
951:
3326:. For example, the Serre-Tate theorem implies that the deformations of a group scheme are strongly controlled by those of its formal group, especially in the case of
2890:
1665:
1954:
3286:
The (non-strict) isomorphisms between formal group laws induced by change of parameters make up the elements of the group of coordinate changes on the formal group.
857:
there is no such homomorphism as defining it requires non-integral rational numbers, and the additive and multiplicative formal groups are usually not isomorphic.
3150:
3115:
1814:
1849:
608:+ terms of higher degree. Two formal group laws with an isomorphism between them are essentially the same; they differ only by a "change of coordinates".
44:
sometimes means the same as formal group law, and sometimes means one of several generalizations. Formal groups are intermediate between Lie groups (or
1286:{\displaystyle \int _{0}^{x}{dt \over {\sqrt {1-t^{4}}}}+\int _{0}^{y}{dt \over {\sqrt {1-t^{4}}}}=\int _{0}^{F(x,y)}{dt \over {\sqrt {1-t^{4}}}}.}
486:, as this turns out to follow automatically from the definition of a formal group law. In other words we can always find a (unique) power series
4315:
4215:
4153:
4041:
3966:
3823:
3787:
2996:, with the relations that are forced by the associativity and commutativity laws for formal group laws. More or less by definition, the ring
3334:, this control is complete, and this is quite different from the characteristic zero situation where the formal group has no deformations.
2290:
of a Lie algebra, both of which are also cocommutative Hopf algebras. In general cocommutative Hopf algebras behave very much like groups.
2293:
For simplicity we describe the 1-dimensional case; the higher-dimensional case is similar except that notation becomes more involved.
3294:, rather than just over commutative rings or fields, and families can be classified by maps from the base to a parametrizing object.
4188:
4121:
3868:
3739:
3129:
is the functor of points of a formal group. (left exactness of a functor is equivalent to commuting with finite projective limits).
4307:
4372:
3361:
2907:
There is a universal commutative one-dimensional formal group law over a universal commutative ring defined as follows. We let
2441:
from it. So 1-dimensional formal group laws are essentially the same as Hopf algebras whose coalgebra structure is given above.
3672:
Note that the formula for the logarithm in terms of the invariant differential given in dimension one does not assume that
2058:
1865:
3327:
3283:
Given a smooth formal group, one can construct a formal group law and a field by choosing a uniformizing set of sections.
3599:
3331:
2287:
3643:
4362:
4033:
915:
1094:
4232:
2902:
2767:
420:
4367:
4236:
3626:
1416:, then it is strictly isomorphic to the additive formal group law. In other words, there is a strict isomorphism
1385:
3622:
3416:
1381:
2437:
Conversely, given a Hopf algebra whose coalgebra structure is given above, we can recover a formal group law
3925:
3595:
53:
3614:
3086:
4241:
4055:
3323:
3195:
4342:
3306:
of smooth formal groups is a quotient of this space by a canonical action of the infinite-dimensional
699:
3901:
3291:
3069:
is naturally isomorphic as a graded ring to Lazard's universal ring, explaining the unusual grading.
2329:
3314:
formal groups a rich geometric theory in positive and mixed characteristic, with connections to the
3159:
2529:
1904:
1377:
1078:{\textstyle F(x,y)=\left.\left(x{\sqrt {1-y^{4}}}+y{\sqrt {1-x^{4}}}\right)\right/\!(1+x^{2}y^{2})}
630:
483:
25:
474:
has no nonzero torsion nilpotents (i.e., no nonzero elements that are both torsion and nilpotent).
4266:
4180:
4080:
3999:
3122:
939:
845:, there is an isomorphism from the additive formal group law to the multiplicative one, given by
767:
57:
4293:
2699:
3274:
if the converse holds; others reserve the term "formal group" for objects locally of this form.
4311:
4258:
4211:
4184:
4149:
4117:
4072:
4037:
3962:
3893:
3864:
3829:
3819:
3783:
3618:
3066:
3013:
2591:
is 1-dimensional; the general case is similar. For any cocommutative Hopf algebra, an element
1098:
2862:
2552:) to be the inverse limit of the corresponding groups. For example, this allows us to define
2259:{\displaystyle f(t)=\int \omega (t)=t+{\frac {c_{1}}{2}}t^{2}+{\frac {c_{2}}{3}}t^{3}+\dots }
1598:
are often constructed by writing down their logarithm as a power series with coefficients in
431:-algebra and use the exponential and logarithm to write any one-dimensional formal group law
4329:
4282:
4250:
4203:
4139:
4107:
4064:
3980:
3954:
3944:
3934:
3909:
3897:
3874:
3837:
3793:
3653:
3315:
73:
4325:
4278:
4225:
4163:
4092:
3976:
2688:
is a homomorphism between one-dimensional formal group laws over a field of characteristic
4333:
4321:
4286:
4274:
4221:
4199:
4159:
4135:
4099:
4088:
4050:
3984:
3972:
3948:
3913:
3878:
3860:
3841:
3797:
3779:
3037:
885:
866:
842:
479:
45:
33:
3732:
3818:. Encyclopaedia of Mathematical Sciences. Vol. 49 (Second ed.). p. 168.
3376:
3135:
3100:
3062:
2852:
2775:
943:
880:
1778:
4356:
3939:
3920:
3811:
3648:
3118:
3090:
2856:
2566:
168:
3757:
1819:
3338:
3303:
3153:
3082:
2279:
80:
4027:
3638:
3610:
2337:
1768:{\displaystyle \omega (t)={\frac {\partial F}{\partial x}}(0,t)^{-1}dt\in R]dt,}
1639:
1606:, and then proving that the coefficients of the corresponding formal group over
95:
49:
17:
3961:. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press.
2048:{\displaystyle F^{*}\omega :=p(F(t,s)){\frac {\partial F}{\partial x}}(t,s)dt.}
2859:. Supersingularity can be detected by the vanishing of the Eisenstein series
2283:
1623:
4262:
4076:
3833:
2855:
has height either one or two, depending on whether the curve is ordinary or
2778:
they have the same height, and the height can be any positive integer or ∞.
2486:
870:
29:
2696:
is either zero, or the first nonzero term in its power series expansion is
3573: + higher-degree terms. This gives an action of the ring
3307:
2367:(so the dual of this coalgebra is just the ring of formal power series).
4270:
4207:
4144:
4112:
4084:
3290:
Formal groups and formal group laws can also be defined over arbitrary
2734:
2506:
1361:
3625:
and an essential component in the construction of Morava E-theory in
2755:
of a one-dimensional formal group law over a field of characteristic
4254:
4068:
1618:. When working in positive characteristic, one typically replaces
3859:. Encycl. Math. Sci. Vol. 62 (2nd printing of 1st ed.).
1090:
4198:, Lecture Notes in Mathematics, vol. 443, Berlin, New York:
3040:(over the integers) on generators of degrees 2, 4, 6, ... (where
914:) is a formal group law coming from the addition formula for the
861:
More generally, we can construct a formal group law of dimension
4134:, Lecture Notes in Mathematics, vol. 74, Berlin, New York:
3192:
The formal completion of a smooth group scheme is isomorphic to
1658:
is one-dimensional, one can write its logarithm in terms of the
2278:
The formal group ring of a formal group law is a cocommutative
789:. If we "change coordinates" to make 0 the identity by putting
1594:
has positive characteristic. Formal group laws over a ring
478:
There is no need for an axiom analogous to the existence of
2648:. In particular we can identify the group-like elements of
1038:
977:
2748:. The height of the zero homomorphism is defined to be ∞.
938:)), and is also the formula for addition of velocities in
4239:(1965), "Formal complex multiplication in local fields",
2660:, and the group structure on the group-like elements of
3337:
A formal group is sometimes defined as a cocommutative
3621:. It is also a major ingredient in some approaches to
2140:{\textstyle \omega (t)=(1+c_{1}t+c_{2}t^{2}+\dots )dt}
2061:
1907:
1822:
1781:
954:
3541:
satisfying these conditions are strictly isomorphic.
3419:
3198:
3189:
at the identity, has the structure of a formal group.
3162:
3138:
3103:
2983:
to be the commutative ring generated by the elements
2865:
2759: > 0 is defined to be the height of its
2702:
2153:
1957:
1868:
1668:
1117:
702:
633:
459:
is necessarily commutative. More generally, we have:
762:
This rule can be understood as follows. The product
3494:
3262:
3177:
3144:
3109:
2884:
2725:
2579:can also be described using the formal group ring
2258:
2139:
2047:
1943:
1893:
1843:
1808:
1767:
1285:
1077:
750:
672:
1408:-dimensional formal group law over a commutative
1042:
4106:, Lecture Notes in Mathematics, vol. 302,
3537:. All the group laws for different choices of
3386:is the unique (1-dimensional) formal group law
2668:is then identified with the group structure on
1582:can be constructed by extension of scalars to
1396:The logarithm of a commutative formal group law
2766:Two one-dimensional formal group laws over an
466:. Every one-dimensional formal group law over
4303:Grundlehren der mathematischen Wissenschaften
3561:of the Lubin–Tate formal group law such that
1370:Lie groups → Formal group laws → Lie algebras
8:
4301:
1622:with a mixed characteristic ring that has a
588:A homomorphism with an inverse is called an
4171:Étude infinitésimale des schémas en groupes
2300:is a (1-dimensional) formal group law over
2274:The formal group ring of a formal group law
1590:, but this will send everything to zero if
3521: + higher-degree terms and
4143:
4111:
3938:
3606:
3495:{\displaystyle e(F(x,y))=F(e(x),e(y)).\ }
3418:
3322:-divisible groups, Dieudonné theory, and
3270:. Some people call a formal group scheme
3245:
3226:
3199:
3197:
3164:
3163:
3161:
3137:
3102:
3008:, one-dimensional formal group laws over
2870:
2864:
2715:
2710:
2701:
2244:
2229:
2223:
2214:
2199:
2193:
2152:
2116:
2106:
2090:
2060:
2001:
1962:
1956:
1906:
1873:
1867:
1821:
1780:
1720:
1684:
1667:
1315:, defined in terms of the quadratic part
1270:
1258:
1248:
1227:
1222:
1205:
1193:
1183:
1177:
1172:
1155:
1143:
1133:
1127:
1122:
1116:
1066:
1056:
1025:
1013:
999:
987:
953:
701:
632:
4029:Stable homotopy and generalised homology
3609:, in a successful effort to isolate the
28:behaving as if it were the product of a
3896:(1967). "Local class field theory". In
3665:
1311:-dimensional Lie algebra over the ring
1307:-dimensional formal group law gives an
37:
3000:has the following universal property:
1578:does not contain the rationals, a map
1102:
3585:There is a similar construction with
2587:. For simplicity we will assume that
1894:{\displaystyle F^{*}\omega =\omega ,}
766:in the (multiplicative group of the)
198:. The idea of the definition is that
7:
3908:. Academic Press. pp. 128–161.
3816:Introduction to Modern Number Theory
3605:This construction was introduced by
3582:on the Lubin–Tate formal group law.
3065:proved that the coefficient ring of
2812:The multiplicative formal group law
2055:If one then considers the expansion
4183:Publisher: Academic Pr (June 1978)
3758:"The Geometry of Lubin-Tate Spaces"
3921:"Local class field theory is easy"
3263:{\displaystyle \mathrm {Spf} (R])}
3206:
3203:
3200:
2316:) is a cocommutative Hopf algebra
2012:
2004:
1695:
1687:
1420:from the additive formal group to
14:
3509:to be any power series such that
2979:and we define the universal ring
2774: > 0 are isomorphic
853:. Over general commutative rings
4000:"Lubin-Tate Theory (Lecture 21)"
3776:An introduction to Hopf algebras
3745:from the original on 2022-09-12.
3613:part of the classical theory of
2540:is an inverse limit of discrete
2520:We can extend the definition of
2493:. The product is given by using
1566:) + log(1 +
751:{\displaystyle F(x,y)=x+y+xy.\ }
70:one-dimensional formal group law
3998:Lurie, Jacob (April 27, 2010).
3557:there is a unique endomorphism
3328:supersingular abelian varieties
2840:th power map is (1 +
2656:with the nilpotent elements of
2374:is given by the coefficient of
688:multiplicative formal group law
387:The formal group law is called
4177:Formal Groups and Applications
4104:Lectures on p-divisible groups
3718:Formal groups and applications
3703:Formal groups and applications
3688:Formal groups and applications
3483:
3480:
3474:
3465:
3459:
3453:
3444:
3441:
3429:
3423:
3257:
3254:
3251:
3219:
3216:
3210:
3178:{\displaystyle {\widehat {G}}}
3032:constructed above is known as
2785:The additive formal group law
2453:-dimensional formal group law
2181:
2175:
2163:
2157:
2128:
2077:
2071:
2065:
2033:
2021:
1998:
1995:
1983:
1977:
1944:{\textstyle \omega (t)=p(t)dt}
1932:
1926:
1917:
1911:
1851:-module of rank 1 on a symbol
1838:
1835:
1829:
1826:
1797:
1794:
1788:
1785:
1753:
1750:
1744:
1741:
1717:
1704:
1678:
1672:
1243:
1231:
1072:
1043:
970:
958:
718:
706:
649:
637:
470:is commutative if and only if
1:
4053:(1946), "Formal Lie groups",
3814:; Panchishkin, A. A. (2007).
3774:Underwood, Robert G. (2011).
3332:supersingular elliptic curves
2445:Formal group laws as functors
2355:The coproduct Δ is given by Δ
1550:), because log(1 +
673:{\displaystyle F(x,y)=x+y.\ }
210:-dimensional formal group law
3940:10.1016/0001-8708(75)90156-5
3919:Hazewinkel, Michiel (1975).
3505:More generally we can allow
3356:Lubin–Tate formal group laws
3125:, then it is representable (
2575:The group-valued functor of
2288:universal enveloping algebra
1951:, then one has by definition
175:The simplest example is the
4179:(Pure and Applied Math 78)
4034:University of Chicago Press
3384:Lubin–Tate formal group law
3362:Lubin–Tate formal group law
3185:, the formal completion of
2851:The formal group law of an
1085:is a formal group law over
916:hyperbolic tangent function
4389:
4298:Algebraische Zahlentheorie
3359:
3344:Some authors use the term
2900:
2768:algebraically closed field
2726:{\displaystyle ax^{p^{h}}}
2477:) whose underlying set is
1650:The invariant differential
1322:of the formal group law.
32:. They were introduced by
4306:. Vol. 322. Berlin:
4196:Commutative formal groups
3627:chromatic homotopy theory
3594:replaced by any complete
3004:For any commutative ring
2266:defines the logarithm of
1386:equivalence of categories
879:formal group (law) of an
619:additive formal group law
177:additive formal group law
4026:Adams, J. Frank (1974),
3959:Local class field theory
3623:local class field theory
3061: − 1)).
2692: > 0. Then
2544:algebras, we can define
2497:to multiply elements of
2320:constructed as follows.
513:from a formal group law
298:+ terms of higher degree
125:+ terms of higher degree
24:is (roughly speaking) a
4373:Algebraic number theory
4194:Lazard, Michel (1975),
3926:Advances in Mathematics
3906:Algebraic Number Theory
3857:Algebraic Number Theory
3644:Artin–Hasse exponential
3607:Lubin & Tate (1965)
3596:discrete valuation ring
3310:of coordinate changes.
3034:Lazard's universal ring
2903:Lazard's universal ring
2885:{\displaystyle E_{p-1}}
54:algebraic number theory
4302:
3615:complex multiplication
3496:
3406:is an endomorphism of
3324:Galois representations
3264:
3179:
3146:
3111:
2886:
2733:for some non-negative
2727:
2469:, we can form a group
2414:is the coefficient of
2286:of a group and to the
2260:
2141:
2049:
1945:
1895:
1845:
1810:
1769:
1660:invariant differential
1562:) = log(1 +
1546:) = log(1 +
1287:
1079:
752:
674:
521:to a formal group law
275:variables, such that
4242:Annals of Mathematics
4130:Fröhlich, A. (1968),
4056:Annals of Mathematics
3855:Koch, Helmut (1997).
3731:Mavraki, Niki Myrto.
3716:Hazewinkel, Michiel.
3701:Hazewinkel, Michiel.
3686:Hazewinkel, Michiel.
3497:
3302:. The corresponding
3265:
3180:
3147:
3112:
3028:The commutative ring
2887:
2836:has height 1, as its
2805:has height ∞, as its
2728:
2565:) with values in the
2261:
2142:
2050:
1946:
1896:
1846:
1811:
1770:
1288:
1093:, in the form of the
1080:
753:
675:
541:variables, such that
423:, then one can embed
3417:
3196:
3160:
3136:
3101:
2863:
2744:of the homomorphism
2700:
2536:. In particular, if
2381:The identity is 1 =
2151:
2059:
1955:
1905:
1866:
1820:
1779:
1666:
1115:
952:
813:, then we find that
700:
631:
3600:residue class field
3121:to groups which is
2959:for indeterminates
2761:multiplication by p
2403:The coefficient of
2314:covariant bialgebra
1630:, such as the ring
1392: > 0.
1247:
1182:
1132:
212:is a collection of
205:More generally, an
52:. They are used in
26:formal power series
4363:Algebraic topology
4208:10.1007/BFb0070554
4181:Michiel Hazewinkel
4145:10.1007/BFb0074373
4113:10.1007/BFb0060741
3902:Fröhlich, Albrecht
3894:Serre, Jean-Pierre
3863:. pp. 62–63.
3756:Weinstein, Jared.
3619:elliptic functions
3492:
3260:
3175:
3142:
3117:is a functor from
3107:
3014:ring homomorphisms
2882:
2844:) − 1 =
2809:th power map is 0.
2770:of characteristic
2723:
2461:and a commutative
2256:
2137:
2045:
1941:
1901:where if we write
1891:
1862:in the sense that
1841:
1806:
1765:
1283:
1218:
1168:
1118:
1075:
940:special relativity
748:
670:
594:strict isomorphism
592:, and is called a
58:algebraic topology
4317:978-3-540-65399-8
4245:, Second Series,
4217:978-3-540-07145-7
4155:978-3-540-04244-0
4059:, Second Series,
4043:978-0-226-00524-9
3968:978-0-19-504030-2
3955:Iwasawa, Kenkichi
3825:978-3-540-20364-3
3789:978-0-387-72765-3
3544:For each element
3491:
3410:, in other words
3278:Formal smoothness
3172:
3145:{\displaystyle G}
3110:{\displaystyle G}
3067:complex cobordism
2509:from commutative
2340:with a basis 1 =
2308:(also called its
2306:formal group ring
2282:analogous to the
2238:
2208:
2019:
1809:{\textstyle R]dt}
1702:
1642:, and reduces to
1514:The logarithm of
1479:The logarithm of
1404:is a commutative
1278:
1276:
1213:
1211:
1163:
1161:
1099:elliptic integral
1031:
1005:
747:
669:
4380:
4368:Algebraic groups
4349:
4347:
4337:
4305:
4294:Neukirch, Jürgen
4289:
4228:
4166:
4147:
4126:
4115:
4100:Demazure, Michel
4095:
4051:Bochner, Salomon
4046:
4018:
4017:
4015:
4013:
4004:
3995:
3989:
3988:
3952:
3942:
3917:
3889:
3883:
3882:
3852:
3846:
3845:
3808:
3802:
3801:
3771:
3765:
3764:
3762:
3753:
3747:
3746:
3744:
3737:
3728:
3722:
3721:
3713:
3707:
3706:
3698:
3692:
3691:
3683:
3677:
3670:
3654:Addition theorem
3501:
3499:
3498:
3493:
3489:
3350:formal group law
3316:Steenrod algebra
3269:
3267:
3266:
3261:
3250:
3249:
3231:
3230:
3209:
3184:
3182:
3181:
3176:
3174:
3173:
3165:
3151:
3149:
3148:
3143:
3116:
3114:
3113:
3108:
2891:
2889:
2888:
2883:
2881:
2880:
2732:
2730:
2729:
2724:
2722:
2721:
2720:
2719:
2636: + ...
2265:
2263:
2262:
2257:
2249:
2248:
2239:
2234:
2233:
2224:
2219:
2218:
2209:
2204:
2203:
2194:
2146:
2144:
2143:
2138:
2121:
2120:
2111:
2110:
2095:
2094:
2054:
2052:
2051:
2046:
2020:
2018:
2010:
2002:
1967:
1966:
1950:
1948:
1947:
1942:
1900:
1898:
1897:
1892:
1878:
1877:
1850:
1848:
1847:
1842:
1815:
1813:
1812:
1807:
1774:
1772:
1771:
1766:
1728:
1727:
1703:
1701:
1693:
1685:
1614:actually lie in
1292:
1290:
1289:
1284:
1279:
1277:
1275:
1274:
1259:
1257:
1249:
1246:
1226:
1214:
1212:
1210:
1209:
1194:
1192:
1184:
1181:
1176:
1164:
1162:
1160:
1159:
1144:
1142:
1134:
1131:
1126:
1095:addition formula
1084:
1082:
1081:
1076:
1071:
1070:
1061:
1060:
1041:
1037:
1033:
1032:
1030:
1029:
1014:
1006:
1004:
1003:
988:
852:
843:rational numbers
809:= 1 +
801:= 1 +
793:= 1 +
757:
755:
754:
749:
745:
679:
677:
676:
671:
667:
537:power series in
529:is a collection
480:inverse elements
74:commutative ring
46:algebraic groups
22:formal group law
4388:
4387:
4383:
4382:
4381:
4379:
4378:
4377:
4353:
4352:
4345:
4343:"Formal groups"
4341:Strickland, N.
4340:
4318:
4308:Springer-Verlag
4292:
4255:10.2307/1970622
4233:Lubin, Jonathan
4231:
4218:
4200:Springer-Verlag
4193:
4173:SGA 3 Exp. VIIB
4156:
4136:Springer-Verlag
4129:
4124:
4098:
4069:10.2307/1969242
4049:
4044:
4025:
4022:
4021:
4011:
4009:
4002:
3997:
3996:
3992:
3969:
3953:
3918:
3898:Cassels, J.W.S.
3892:
3890:
3886:
3871:
3861:Springer-Verlag
3854:
3853:
3849:
3826:
3810:
3809:
3805:
3790:
3782:. p. 121.
3780:Springer-Verlag
3773:
3772:
3768:
3760:
3755:
3754:
3750:
3742:
3735:
3733:"Formal groups"
3730:
3729:
3725:
3715:
3714:
3710:
3700:
3699:
3695:
3685:
3684:
3680:
3676:is commutative.
3671:
3667:
3662:
3635:
3593:
3581:
3556:
3533: mod
3415:
3414:
3375:be the ring of
3374:
3364:
3358:
3241:
3222:
3194:
3193:
3158:
3157:
3134:
3133:
3099:
3098:
3075:
3052:
3038:polynomial ring
2995:
2974:
2949:
2905:
2899:
2866:
2861:
2860:
2711:
2706:
2698:
2697:
2682:
2564:
2447:
2407:in the product
2276:
2240:
2225:
2210:
2195:
2149:
2148:
2112:
2102:
2086:
2057:
2056:
2011:
2003:
1958:
1953:
1952:
1903:
1902:
1869:
1864:
1863:
1844:{\textstyle R]}
1818:
1817:
1777:
1776:
1716:
1694:
1686:
1664:
1663:
1652:
1398:
1347:
1332:
1321:
1301:
1266:
1250:
1201:
1185:
1151:
1135:
1113:
1112:
1062:
1052:
1021:
995:
983:
979:
976:
950:
949:
886:abelian variety
867:algebraic group
851:) − 1
846:
698:
697:
629:
628:
614:
596:if in addition
383:
374:
363:
354:
344:where we write
270:
261:
254:
247:
238:
231:
224:
66:
12:
11:
5:
4386:
4384:
4376:
4375:
4370:
4365:
4355:
4354:
4351:
4350:
4338:
4316:
4290:
4249:(2): 380–387,
4229:
4216:
4191:
4174:
4167:
4154:
4127:
4122:
4096:
4063:(2): 192–201,
4047:
4042:
4020:
4019:
3990:
3967:
3933:(2): 148–181.
3884:
3869:
3847:
3824:
3803:
3788:
3766:
3748:
3723:
3708:
3693:
3678:
3664:
3663:
3661:
3658:
3657:
3656:
3651:
3646:
3641:
3634:
3631:
3589:
3577:
3552:
3503:
3502:
3488:
3485:
3482:
3479:
3476:
3473:
3470:
3467:
3464:
3461:
3458:
3455:
3452:
3449:
3446:
3443:
3440:
3437:
3434:
3431:
3428:
3425:
3422:
3380:-adic integers
3370:
3360:Main article:
3357:
3354:
3288:
3287:
3284:
3281:
3275:
3259:
3256:
3253:
3248:
3244:
3240:
3237:
3234:
3229:
3225:
3221:
3218:
3215:
3212:
3208:
3205:
3202:
3190:
3171:
3168:
3141:
3130:
3119:Artin algebras
3106:
3091:formal schemes
3074:
3071:
3063:Daniel Quillen
3044:
3026:
3025:
3012:correspond to
2987:
2977:
2976:
2966:
2957:
2956:
2941:
2924:
2923:
2901:Main article:
2898:
2895:
2894:
2893:
2879:
2876:
2873:
2869:
2853:elliptic curve
2849:
2810:
2776:if and only if
2718:
2714:
2709:
2705:
2681:
2678:
2638:
2637:
2560:
2485:is the set of
2446:
2443:
2435:
2434:
2401:
2386:
2379:
2368:
2353:
2275:
2272:
2255:
2252:
2247:
2243:
2237:
2232:
2228:
2222:
2217:
2213:
2207:
2202:
2198:
2192:
2189:
2186:
2183:
2180:
2177:
2174:
2171:
2168:
2165:
2162:
2159:
2156:
2136:
2133:
2130:
2127:
2124:
2119:
2115:
2109:
2105:
2101:
2098:
2093:
2089:
2085:
2082:
2079:
2076:
2073:
2070:
2067:
2064:
2044:
2041:
2038:
2035:
2032:
2029:
2026:
2023:
2017:
2014:
2009:
2006:
2000:
1997:
1994:
1991:
1988:
1985:
1982:
1979:
1976:
1973:
1970:
1965:
1961:
1940:
1937:
1934:
1931:
1928:
1925:
1922:
1919:
1916:
1913:
1910:
1890:
1887:
1884:
1881:
1876:
1872:
1840:
1837:
1834:
1831:
1828:
1825:
1805:
1802:
1799:
1796:
1793:
1790:
1787:
1784:
1764:
1761:
1758:
1755:
1752:
1749:
1746:
1743:
1740:
1737:
1734:
1731:
1726:
1723:
1719:
1715:
1712:
1709:
1706:
1700:
1697:
1692:
1689:
1683:
1680:
1677:
1674:
1671:
1651:
1648:
1572:
1571:
1512:
1473:
1472:
1471:
1470:
1397:
1394:
1382:characteristic
1374:
1373:
1372:
1371:
1358:
1357:
1345:
1330:
1319:
1300:
1297:
1296:
1295:
1294:
1293:
1282:
1273:
1269:
1265:
1262:
1256:
1253:
1245:
1242:
1239:
1236:
1233:
1230:
1225:
1221:
1217:
1208:
1204:
1200:
1197:
1191:
1188:
1180:
1175:
1171:
1167:
1158:
1154:
1150:
1147:
1141:
1138:
1130:
1125:
1121:
1107:
1106:
1074:
1069:
1065:
1059:
1055:
1051:
1048:
1045:
1040:
1036:
1028:
1024:
1020:
1017:
1012:
1009:
1002:
998:
994:
991:
986:
982:
978:
975:
972:
969:
966:
963:
960:
957:
947:
944:speed of light
889:
881:elliptic curve
839:
838:
760:
759:
758:
744:
741:
738:
735:
732:
729:
726:
723:
720:
717:
714:
711:
708:
705:
692:
691:
683:
682:
681:
680:
666:
663:
660:
657:
654:
651:
648:
645:
642:
639:
636:
623:
622:
613:
610:
586:
585:
584:
583:
476:
475:
384:), and so on.
379:
372:
359:
352:
342:
341:
299:
266:
259:
252:
243:
236:
229:
220:
173:
172:
126:
65:
62:
34:S. Bochner
13:
10:
9:
6:
4:
3:
2:
4385:
4374:
4371:
4369:
4366:
4364:
4361:
4360:
4358:
4344:
4339:
4335:
4331:
4327:
4323:
4319:
4313:
4309:
4304:
4299:
4295:
4291:
4288:
4284:
4280:
4276:
4272:
4268:
4264:
4260:
4256:
4252:
4248:
4244:
4243:
4238:
4234:
4230:
4227:
4223:
4219:
4213:
4209:
4205:
4201:
4197:
4192:
4190:
4189:0-12-335150-2
4186:
4182:
4178:
4175:
4172:
4168:
4165:
4161:
4157:
4151:
4146:
4141:
4137:
4133:
4132:Formal groups
4128:
4125:
4123:0-387-06092-8
4119:
4114:
4109:
4105:
4101:
4097:
4094:
4090:
4086:
4082:
4078:
4074:
4070:
4066:
4062:
4058:
4057:
4052:
4048:
4045:
4039:
4035:
4031:
4030:
4024:
4023:
4008:
4001:
3994:
3991:
3986:
3982:
3978:
3974:
3970:
3964:
3960:
3956:
3950:
3946:
3941:
3936:
3932:
3928:
3927:
3922:
3915:
3911:
3907:
3903:
3899:
3895:
3888:
3885:
3880:
3876:
3872:
3870:3-540-63003-1
3866:
3862:
3858:
3851:
3848:
3843:
3839:
3835:
3831:
3827:
3821:
3817:
3813:
3812:Manin, Yu. I.
3807:
3804:
3799:
3795:
3791:
3785:
3781:
3777:
3770:
3767:
3759:
3752:
3749:
3741:
3734:
3727:
3724:
3719:
3712:
3709:
3704:
3697:
3694:
3689:
3682:
3679:
3675:
3669:
3666:
3659:
3655:
3652:
3650:
3649:Group functor
3647:
3645:
3642:
3640:
3637:
3636:
3632:
3630:
3628:
3624:
3620:
3616:
3612:
3608:
3603:
3601:
3597:
3592:
3588:
3583:
3580:
3576:
3572:
3568:
3564:
3560:
3555:
3551:
3547:
3542:
3540:
3536:
3532:
3528:
3524:
3520:
3516:
3512:
3508:
3486:
3477:
3471:
3468:
3462:
3456:
3450:
3447:
3438:
3435:
3432:
3426:
3420:
3413:
3412:
3411:
3409:
3405:
3402: +
3401:
3397:
3393:
3389:
3385:
3381:
3379:
3373:
3369:
3363:
3355:
3353:
3351:
3347:
3342:
3340:
3335:
3333:
3329:
3325:
3321:
3317:
3311:
3309:
3305:
3301:
3295:
3293:
3285:
3282:
3279:
3276:
3273:
3246:
3242:
3238:
3235:
3232:
3227:
3223:
3213:
3191:
3188:
3169:
3166:
3155:
3139:
3131:
3128:
3124:
3120:
3104:
3096:
3095:
3094:
3092:
3088:
3084:
3080:
3073:Formal groups
3072:
3070:
3068:
3064:
3060:
3057: +
3056:
3053:has degree 2(
3051:
3047:
3043:
3039:
3035:
3031:
3023:
3019:
3015:
3011:
3007:
3003:
3002:
3001:
2999:
2994:
2990:
2986:
2982:
2973:
2969:
2965:
2962:
2961:
2960:
2955:
2952:
2948:
2944:
2940:
2936:
2932:
2929:
2928:
2927:
2921:
2917:
2913:
2910:
2909:
2908:
2904:
2896:
2877:
2874:
2871:
2867:
2858:
2857:supersingular
2854:
2850:
2847:
2843:
2839:
2835:
2832: +
2831:
2828: +
2827:
2823:
2819:
2815:
2811:
2808:
2804:
2801: +
2800:
2796:
2792:
2788:
2784:
2783:
2782:
2779:
2777:
2773:
2769:
2764:
2762:
2758:
2754:
2749:
2747:
2743:
2740:, called the
2739:
2736:
2716:
2712:
2707:
2703:
2695:
2691:
2687:
2684:Suppose that
2679:
2677:
2675:
2671:
2667:
2663:
2659:
2655:
2651:
2647:
2643:
2635:
2632:
2629: +
2628:
2625:
2622: +
2621:
2618:
2617:
2616:
2614:
2610:
2606:
2602:
2598:
2594:
2590:
2586:
2582:
2578:
2573:
2571:
2570:-adic numbers
2569:
2563:
2559:
2555:
2551:
2547:
2543:
2539:
2535:
2533:
2527:
2523:
2518:
2516:
2512:
2508:
2504:
2500:
2496:
2492:
2488:
2484:
2480:
2476:
2472:
2468:
2464:
2460:
2456:
2452:
2444:
2442:
2440:
2432:
2428:
2424:
2420:
2417:
2413:
2410:
2406:
2402:
2399:
2396:to (−1)
2395:
2391:
2388:The antipode
2387:
2384:
2380:
2377:
2373:
2369:
2366:
2363: ⊗
2362:
2358:
2354:
2351:
2347:
2343:
2339:
2335:
2331:
2327:
2323:
2322:
2321:
2319:
2315:
2311:
2307:
2303:
2299:
2296:Suppose that
2294:
2291:
2289:
2285:
2281:
2273:
2271:
2269:
2253:
2250:
2245:
2241:
2235:
2230:
2226:
2220:
2215:
2211:
2205:
2200:
2196:
2190:
2187:
2184:
2178:
2172:
2169:
2166:
2160:
2154:
2147:, the formula
2134:
2131:
2125:
2122:
2117:
2113:
2107:
2103:
2099:
2096:
2091:
2087:
2083:
2080:
2074:
2068:
2062:
2042:
2039:
2036:
2030:
2027:
2024:
2015:
2007:
1992:
1989:
1986:
1980:
1974:
1971:
1968:
1963:
1959:
1938:
1935:
1929:
1923:
1920:
1914:
1908:
1888:
1885:
1882:
1879:
1874:
1870:
1861:
1858:
1854:
1832:
1823:
1803:
1800:
1791:
1782:
1762:
1759:
1756:
1747:
1738:
1735:
1732:
1729:
1724:
1721:
1713:
1710:
1707:
1698:
1690:
1681:
1675:
1669:
1661:
1657:
1649:
1647:
1645:
1641:
1637:
1633:
1629:
1625:
1621:
1617:
1613:
1609:
1605:
1601:
1597:
1593:
1589:
1585:
1581:
1577:
1569:
1565:
1561:
1558: +
1557:
1554: +
1553:
1549:
1545:
1541:
1537:
1534: +
1533:
1530: +
1529:
1525:
1521:
1517:
1513:
1510:
1506:
1502:
1498:
1495: +
1494:
1490:
1486:
1482:
1478:
1477:
1476:
1468:
1464:
1460:
1456:
1452:
1448:
1444:
1440:
1437:
1436:
1435:
1434:
1433:
1431:
1427:
1424:, called the
1423:
1419:
1415:
1411:
1407:
1403:
1395:
1393:
1391:
1387:
1383:
1379:
1369:
1368:
1367:
1366:
1365:
1363:
1355:
1351:
1344:
1340:
1336:
1329:
1325:
1324:
1323:
1318:
1314:
1310:
1306:
1298:
1280:
1271:
1267:
1263:
1260:
1254:
1251:
1240:
1237:
1234:
1228:
1223:
1219:
1215:
1206:
1202:
1198:
1195:
1189:
1186:
1178:
1173:
1169:
1165:
1156:
1152:
1148:
1145:
1139:
1136:
1128:
1123:
1119:
1111:
1110:
1109:
1108:
1104:
1100:
1096:
1092:
1088:
1067:
1063:
1057:
1053:
1049:
1046:
1034:
1026:
1022:
1018:
1015:
1010:
1007:
1000:
996:
992:
989:
984:
980:
973:
967:
964:
961:
955:
948:
945:
941:
937:
933:
929:
925:
922: +
921:
917:
913:
909:
905:
901:
897:
893:
890:
887:
883:
882:
876:
873:of dimension
872:
868:
864:
860:
859:
858:
856:
850:
844:
836:
833: +
832:
829: +
828:
824:
820:
816:
812:
808:
804:
800:
796:
792:
788:
784:
780:
776:
772:
769:
765:
761:
742:
739:
736:
733:
730:
727:
724:
721:
715:
712:
709:
703:
696:
695:
694:
693:
689:
685:
684:
664:
661:
658:
655:
652:
646:
643:
640:
634:
627:
626:
625:
624:
620:
616:
615:
611:
609:
607:
603:
599:
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591:
581:
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569:
565:
561:
557:
553:
549:
546:
545:
544:
543:
542:
540:
536:
532:
528:
525:of dimension
524:
520:
517:of dimension
516:
512:
507:
505:
501:
497:
493:
489:
485:
481:
473:
469:
465:
462:
461:
460:
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418:
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406:
402:
398:
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385:
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367:
362:
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351:
347:
339:
335:
331:
327:
323:
319:
315:
311:
307:
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297:
293:
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285:
281:
278:
277:
276:
274:
269:
265:
258:
251:
246:
242:
235:
228:
223:
219:
216:power series
215:
211:
209:
203:
201:
197:
193:
189:
185:
181:
178:
170:
169:associativity
166:
162:
158:
154:
150:
146:
142:
138:
134:
130:
127:
124:
120:
116:
112:
108:
105:
104:
103:
102:, such that
101:
97:
93:
89:
85:
82:
78:
75:
71:
63:
61:
59:
55:
51:
47:
43:
39:
35:
31:
27:
23:
19:
4297:
4246:
4240:
4195:
4176:
4170:
4169:P. Gabriel,
4131:
4103:
4060:
4054:
4028:
4010:. Retrieved
4006:
3993:
3958:
3930:
3924:
3905:
3887:
3856:
3850:
3815:
3806:
3775:
3769:
3751:
3726:
3717:
3711:
3702:
3696:
3687:
3681:
3673:
3668:
3604:
3598:with finite
3590:
3586:
3584:
3578:
3574:
3570:
3566:
3562:
3558:
3553:
3549:
3545:
3543:
3538:
3534:
3530:
3526:
3522:
3518:
3514:
3510:
3506:
3504:
3407:
3403:
3399:
3395:
3391:
3387:
3383:
3377:
3371:
3367:
3365:
3349:
3346:formal group
3345:
3343:
3339:Hopf algebra
3336:
3319:
3312:
3304:moduli stack
3299:
3296:
3289:
3277:
3271:
3186:
3154:group scheme
3126:
3083:group object
3079:formal group
3078:
3076:
3058:
3054:
3049:
3045:
3041:
3033:
3029:
3027:
3021:
3017:
3009:
3005:
2997:
2992:
2988:
2984:
2980:
2978:
2971:
2967:
2963:
2958:
2953:
2950:
2946:
2942:
2938:
2934:
2930:
2925:
2919:
2915:
2911:
2906:
2845:
2841:
2837:
2833:
2829:
2825:
2821:
2817:
2813:
2806:
2802:
2798:
2794:
2790:
2786:
2780:
2771:
2765:
2760:
2756:
2752:
2750:
2745:
2741:
2737:
2693:
2689:
2685:
2683:
2673:
2669:
2665:
2661:
2657:
2653:
2649:
2645:
2641:
2639:
2633:
2630:
2626:
2623:
2619:
2612:
2608:
2604:
2600:
2596:
2592:
2588:
2584:
2580:
2576:
2574:
2567:
2561:
2557:
2553:
2549:
2545:
2541:
2537:
2531:
2530:topological
2525:
2521:
2519:
2514:
2510:
2502:
2498:
2494:
2490:
2489:elements of
2482:
2478:
2474:
2470:
2466:
2462:
2458:
2454:
2450:
2448:
2438:
2436:
2430:
2426:
2422:
2418:
2415:
2411:
2408:
2404:
2397:
2393:
2389:
2382:
2375:
2371:
2364:
2360:
2356:
2349:
2345:
2341:
2333:
2325:
2317:
2313:
2310:hyperalgebra
2309:
2305:
2301:
2297:
2295:
2292:
2280:Hopf algebra
2277:
2267:
1859:
1856:
1855:. Then ω is
1852:
1816:is the free
1659:
1655:
1653:
1646:at the end.
1643:
1640:Witt vectors
1635:
1631:
1627:
1619:
1615:
1611:
1607:
1603:
1599:
1595:
1591:
1587:
1583:
1579:
1575:
1573:
1567:
1563:
1559:
1555:
1551:
1547:
1543:
1539:
1535:
1531:
1527:
1523:
1519:
1515:
1508:
1504:
1500:
1496:
1492:
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1484:
1480:
1474:
1466:
1462:
1458:
1454:
1450:
1446:
1442:
1438:
1429:
1425:
1421:
1417:
1413:
1409:
1405:
1401:
1399:
1389:
1375:
1360:The natural
1359:
1353:
1349:
1342:
1338:
1334:
1327:
1316:
1312:
1308:
1304:
1302:
1299:Lie algebras
1086:
946:equal to 1).
935:
931:
927:
923:
919:
911:
907:
903:
899:
895:
891:
878:
874:
862:
854:
848:
840:
834:
830:
826:
822:
818:
814:
810:
806:
802:
798:
794:
790:
786:
782:
778:
774:
773:is given by
770:
763:
687:
618:
605:
601:
597:
593:
589:
587:
579:
575:
571:
567:
563:
559:
555:
551:
547:
538:
534:
530:
526:
522:
518:
514:
511:homomorphism
510:
508:
503:
499:
495:
491:
487:
477:
471:
467:
463:
456:
452:
448:
447:) = exp(log(
444:
440:
436:
432:
428:
424:
416:
412:
408:
404:
400:
396:
392:
388:
386:
380:
376:
369:
365:
360:
356:
349:
345:
343:
337:
333:
329:
325:
321:
317:
313:
309:
305:
301:
295:
291:
287:
283:
279:
272:
267:
263:
256:
249:
244:
240:
233:
226:
221:
217:
213:
207:
206:
204:
199:
195:
191:
187:
183:
179:
176:
174:
164:
160:
156:
152:
148:
144:
140:
136:
132:
128:
122:
118:
114:
110:
106:
99:
96:coefficients
91:
87:
83:
81:power series
76:
69:
67:
50:Lie algebras
42:formal group
41:
40:). The term
21:
15:
4007:harvard.edu
3639:Witt vector
3611:local field
2897:Lazard ring
2517:to groups.
2370:The counit
1857:translation
1432:, so that
690:is given by
621:is given by
590:isomorphism
421:torsionfree
389:commutative
64:Definitions
18:mathematics
4357:Categories
4334:0956.11021
4287:0128.26501
4237:Tate, John
3985:0604.12014
3949:0312.12022
3914:0153.07403
3879:0819.11044
3842:1079.11002
3798:1234.16022
3778:. Berlin:
3720:. §11.1.6.
3705:. §14.2.3.
3660:References
3390:such that
3123:left exact
2781:Examples:
2597:group-like
2595:is called
2528:) to some
2513:-algebras
2284:group ring
1662:ω(t). Let
1624:surjection
1475:Examples:
1103:Strickland
942:(with the
490:such that
4263:0003-486X
4077:0003-486X
3834:0938-0396
3236:…
3170:^
2875:−
2644:elements
2642:nilpotent
2534:-algebras
2487:nilpotent
2465:-algebra
2449:Given an
2254:…
2173:ω
2170:∫
2126:…
2063:ω
2013:∂
2005:∂
1969:ω
1964:∗
1909:ω
1886:ω
1880:ω
1875:∗
1860:invariant
1736:∈
1722:−
1696:∂
1688:∂
1670:ω
1426:logarithm
1412:-algebra
1264:−
1220:∫
1199:−
1170:∫
1149:−
1120:∫
1089:found by
1019:−
993:−
871:Lie group
865:from any
841:Over the
30:Lie group
4296:(1999).
4102:(1972),
4012:June 23,
3957:(1986).
3904:(eds.).
3740:Archived
3633:See also
3348:to mean
3308:groupoid
3087:category
3020:to
2937:+ Σ
934:), tanh(
612:Examples
506:)) = 0.
451:) + log(
4326:1697859
4279:0172878
4271:1970622
4226:0393050
4164:0242837
4093:0015397
4085:1969242
3977:0863740
3690:. §6.1.
3366:We let
3330:. For
3292:schemes
3085:in the
2735:integer
2507:functor
2505:into a
2312:or its
1362:functor
1097:for an
918:: tanh(
910:)/(1 +
464:Theorem
455:)), so
427:into a
375:, ...,
355:, ...,
262:, ...,
239:, ...,
94:) with
72:over a
36: (
4332:
4324:
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4277:
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3877:
3867:
3840:
3832:
3822:
3796:
3786:
3490:
3382:. The
3272:smooth
2753:height
2742:height
2680:Height
2481:where
2392:takes
2330:module
2324:As an
2304:. Its
1775:where
1378:fields
930:(tanh(
805:, and
746:
668:
484:groups
415:). If
271:) in 2
48:) and
4346:(PDF)
4267:JSTOR
4081:JSTOR
4003:(PDF)
3891:e.g.
3761:(PDF)
3743:(PDF)
3736:(PDF)
3156:then
3152:is a
3081:is a
3016:from
2763:map.
2611:and ε
2457:over
2352:, ...
1654:When
1638:) of
1453:)) =
1376:Over
1091:Euler
902:) = (
566:)) =
368:for (
348:for (
320:)) =
147:)) =
79:is a
4312:ISBN
4259:ISSN
4212:ISBN
4185:ISBN
4150:ISBN
4118:ISBN
4073:ISSN
4038:ISBN
4014:2023
3963:ISBN
3865:ISBN
3830:ISSN
3820:ISBN
3784:ISBN
3569:) =
3529:) =
3517:) =
3398:) =
2824:) =
2797:) =
2751:The
2640:for
2599:if Δ
2338:free
1526:) =
1507:) =
1491:) =
1461:) +
1341:) −
1303:Any
926:) =
884:(or
847:exp(
825:) =
785:) =
768:ring
686:The
617:The
604:) =
482:for
403:) =
290:) =
190:) =
167:) (
117:) =
56:and
38:1946
20:, a
4330:Zbl
4283:Zbl
4251:doi
4204:doi
4140:doi
4108:doi
4065:doi
3981:Zbl
3945:Zbl
3935:doi
3910:Zbl
3875:Zbl
3838:Zbl
3794:Zbl
3617:of
3548:in
3132:If
3097:If
3089:of
2926:be
2676:).
2583:of
2421:in
2359:= Σ
2336:is
1626:to
1574:If
1538:is
1499:is
1428:of
1400:If
1380:of
869:or
582:)).
558:),
533:of
435:as
419:is
391:if
364:),
336:),
163:),
98:in
16:In
4359::
4328:.
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4320:.
4310:.
4300:.
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