By Nicholas M. Katz

It is now a few thirty years due to the fact Deligne first proved his normal equidistribution theorem, hence developing the basic consequence governing the statistical homes of certainly "pure" algebro-geometric households of personality sums over finite fields (and in their linked L-functions). approximately conversing, Deligne confirmed that this kind of kin obeys a "generalized Sato-Tate law," and that understanding which generalized Sato-Tate legislations applies to a given kinfolk quantities primarily to computing a undeniable advanced semisimple (not inevitably hooked up) algebraic crew, the "geometric monodromy crew" hooked up to that relatives.

Up to now, approximately all recommendations for settling on geometric monodromy teams have relied, no less than partially, on neighborhood details. In Moments, Monodromy, and Perversity , Nicholas Katz develops new suggestions, that are resolutely international in nature. they're in line with very important elements, neither of which existed on the time of Deligne's unique paintings at the topic. the 1st is the speculation of perverse sheaves, pioneered by means of Goresky and MacPherson within the topological surroundings after which brilliantly transposed to algebraic geometry by way of Beilinson, Bernstein, Deligne, and Gabber. the second one is Larsen's replacement, which pretty much characterizes classical teams through their fourth moments. those new thoughts, that are of serious curiosity of their personal correct, are first constructed after which used to calculate the geometric monodromy teams connected to a few particularly particular common households of (L-functions connected to) personality sums over finite fields.

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1. 1. 1, there exists ∫ in ä$…≠, such 34 Chapter 1 that M‚∫deg on X is isomorphic to its own Verdier dual. This ∫, unique up to sign, has |∫| = 1. So by part 4), applied to M‚∫deg, we have ∫2deg(E/k)‡x in X(E) M(E, x)2 = 1 + O((ùE)-1/2). By hypothesis, we have ‡x in X(E) M(E, x)2 = 1 + O((ùE)-œ/2). So we find ∫2deg(E/k) = 1 + O((ùE)-œ/2). Consider the complex power series in one variable T defined by ‡n≥0 ∫2nTn = 1/(1 - ∫ 2T). It satisfies 1/(1 - ∫2 T) -1 /(1 - T) = a series convergent in |T| < (ùk)œ/2.

M)‚äk, pr1*L‚pr2*K) is concentrated in degree ≤ m. 4) We will say that data (m ≥ 1, K, V, h, L, d ≥ 2, (Ï, †)) as above, which satisfies hypotheses 1) through 4), is "sstandard i n p u t ". 5 Given standard input (m ≥ 1, K, V, h, L, d ≥ 2, (Ï, †)), consider the object M = Twist(L, K, Ï, h) on the space Ï. We have the following results. 1) The Tate-twisted object M(dimÏ0/2) is perverse, and “-mixed of weight ≤ 0. m)‚äk, pr1*L‚pr2*K)wt=m(-dimÏ0/2). m)‚äk, pr1*L‚pr2*K) is “-mixed of weight ≤ m - œ, for some œ > 0.

1) Hci(X‚käk, M) vanishes for i outside the closed interval [-d, d]. 2) If in addition M is irreducible and nonconstant on X‚käk, then Hcd(X‚käk, M) = 0. proof of sublemma Immediate reduction to the case when M is perverse irreducible. If d = dim(X) is zero, the first assertion is obvious and the second is vacuous: X‚käk is a point and every perverse sheaf on X‚käk is constant. So suppose d ≥ 1, and let M have dimension of support d(M) ≥ 0. Look at the spectral sequence E2p,q = Hcp(X‚käk, Óq(M)) à Hcp+q(X‚käk, M).