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By Chowdhury K.C.

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Both Gr0(M) and Gr0(N) are geometrically semisimple (because they are “-pure of weight 0). Write their pullbacks Gr0(M)geom and Gr0(N)geom to X‚käk as sums of perverse irreducibles with multiplicities, say Gr0(M)geom = ‡i miVi, Gr0(N)geom = ‡i niVi, with {Vi}i a finite set of pairwise non-isomorphic perverse irreducibles on X‚käk, and with non-negative integers mi and ni. 3, we have limsupE ‡E |N|2 = ‡i (ni)2, limsupE ‡E |M|2 = ‡i (mi)2, limsupE |‡E NäM| = ‡i nimi. In view of the above estimates, these three limsup's are all equal to limsupE ‡E |F|2.

Therefore we get 48 Chapter 1 ‡i (ni - mi)2 = 0, so ni = mi for each i, as required. QED Here is an arithmetic sharpening of this uniqueness result. 2 Let X/k be a separated scheme of finite type, of dimension d ≥ 0, F an abstract trace function on X/k, and M and N two perverse sheaves on X. Suppose that both M and N are “-mixed of weight ≤ 0, that both are semisimple objects in the category of perverse sheaves on X, and that F is an approximate trace function for both M and N. Then Gr0(M) ¶ Gr0(N) as perverse sheaves on X.

The innermost sum is O((ùE)-1/2) unless å(W1) = å(W2) = å. For each å, denote by ∏(å) the set of those indices i such that å(Wi) = å. Then we get ‡x in X(E) M(E, x)äN(E, x) = ‡å ‡i, j in ∏(å) aibjTrace(FrobE | Hc2dim(Zå)(Zå‚käk, „i‚ä„j)) + O((ùE)-1/2). = Trace(FrobE | ·å·i,j in ∏(å) Hc2dim(Zå)(Zå‚käk, „i‚ä„j)aibj) + O((ùE)-1/2). The direct sum T := ·å·i,j in ∏(å) Hc2dim(Zå)(Zå‚käk, „i‚ä„j)aibj is “-pure of weight zero. s. on a finite-dimensional ^-space “T := T‚^ such that for any finite extension E/k, “Trace(FrobE | T) = Trace(Adeg(E/k) | “T).

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