Research Statement Jonathan A. Huang jahuang@math.umd.edu My primary research area is number theory, specifically in motives and motivic zeta functions. I am particularly interested in the relationship between zeta functions and the big ring of Witt vectors. Zeta functions play a central role in number theory; the theme of many conjectures is that the special values of certain zeta functions, which are analytically defined objects, can be expressed in terms of algebraic invariants. These algebraic invariants (l-adic cohomology, Weil-étale cohomology,...) satisfy usual properties, for example they are typically additive and multiplicative (via a Künneth formula, for instance). These are so-called Euler-Poincaré characteristics on the category of certain spaces (varieties over Spec k, schemes of finite type over Spec Z,...). If we view the zeta function as a map on the category of such spaces, then it is natural to ask if zeta functions themselves can be considered Euler-Poincaré characteristics. This would shed light on why the special values of zeta functions take the form of Euler characteristic formulas: the zeta function is itself an Euler characteristic. My research centers on understanding zeta functions as Euler-Poincaré characteristics taking values in the big Witt ring W (R). 1. Motivic zeta functions and the Grothendieck ring of varieties Let k be a field and Var k be the category of varieties over k (reduced schemes of finite type over Spec k). The Grothendieck ring of varieties K 0 (Var k ) is the abelian group generated by symbols [X] of isomorphism classes of X Var k subject to the scissor relation [X] = [Y ] + [X \ Y ] for Y any closed subvariety of X. K 0 (Var k ) is a commutative ring under the product [X] [Y ] = [X k Y ]. This is the universal value group of Euler-Poincaré characteristics on Var k ; ring homomorphisms on K 0 (Var k ) are called motivic measures. That is, for R a commutative ring with identity, we consider µ : K 0 (Var k ) R. Kapranov in [15] constructs a motivic zeta function ζ µ (X, t) for X Var k using these algebraic invariants µ(x). For X quasi-projective, it is the generating series for (the measure of) symmetric powers Sym n X: ζ µ (X, t) = µ(sym n X)t n R[[t]]. n=0 This is a group homomorphism on K 0 (Var k ) taking values in the group of invertible power series (1 + tr[[t]], ). To understand ζ µ as an Euler-Poincaré characteristic, we must study its (possible) structure as a ring homomorphism. We can endeavor to describe the product ζ µ (X k Y ) using a product structure on (1 + tr[[t]], ). As it turns out, the correct product structure is often given by the big Witt ring W (R). Consider, for example, the case of varieties over finite fields X Var Fq and the Hasse-Weil zeta function: Z(X, t) = (1 t deg(x) ) 1 1 + tz[[t]] x X This is clearly additive: for Y X a closed subvariety, Z(X, t) = Z(Y, t) Z(X \ Y, t) the product of power series, which is the additive operation in the group of invertible power series. It is also
multiplicative: for X, Y Var Fq, Z(X k Y, t) = Z(Y, t) W Z(Y, t) where W is the product in the Witt ring W (Z). Thus the zeta function Z(X, t) behaves as an Euler-Poincaré characteristic. In general, we can require the same for the motivic zeta functions associated to abstract motivic measures µ : K 0 (Var k ) R. Following Ramachandran [23], a motivic measure is called exponentiable if the associated motivic zeta function is a ring homomorphism taking values in the Witt ring, ζ µ : K 0 (Var k ) W (R). In the finite field case, the zeta function Z(X, t) is the motivic zeta function ζ µ# associated to the counting measure µ # (X) = #X(F q ) on X Var Fq ; hence µ # is exponentiable. A central question posed in [15] is whether ζ µ is always a rational function. Kapranov shows that this is true whenever X is a curve; meanwhile, Larsen-Lunts have constructed a measure µ LL such that in certain cases has an associated motivic zeta function ζ µll (X, t) which is not rational (Theorem 7.6 in [17]). For exponentiable measures µ, it is often easier to prove rationality of the zeta function ζ µ (see [25]). In the case µ : K 0 (Var k ) R takes values in a λ-ring R, Ramachandran and Tabuada [25] provide a condition whereby µ is exponentiatable. Some examples of λ-rings are: Z is (uniquely) a λ-ring (see [16]); for R a commutative ring with identity, the Grothendieck group K 0 (R) (see [9]); for C a symmetric monoidal additive pseudo-abelian category, K 0 (C) is a λ-ring (see Heinloth [12]). The importance of λ-ring-valued motivic measures has been observed in Larsen and Lunts [17], Heinloth [12] [13], Gusein-Zade, Lluengo, and Melle-Hernandez [11] [10], del Baño Rollin [5] [6], and Maxim and Schurmann [20]. However, although Witt-type phenomena of zeta functions appears in prior studies, identifying the Kapranov motivic zeta function as an object in the big Witt ring is only earnestly present in Ramachandran [23] and Ramachandran-Tabuada [25]. Our main focus is on motivic measures taking values in R a λ-ring; our main observation is that for any ring A, the big Witt ring W (A) is a λ-ring. 2. Exponentiation of motivic zeta functions The zeta function associated to an exponentiable motivic measure is a ring homomorphism into W (R), and thus itself defines a motivic measure. This measure is the induced zeta function measure µ Z (X) = ζ µ (X, t) and its associated the motivic zeta function ζ µz is a generating series for the zeta function of symmetric powers ζ µz (X, u) = Z(Sym n, t) u n 1 + uw (R)[[u]]. n=0 We may ask if this zeta function measure µ Z is exponentiable, i.e. if ζ µz takes values in W (W (R)). In my thesis [14], I study several such questions related to induced zeta function measures and λ-ring measures posed in [23] and [25]. 2.1. The finite field case. In the case of Var Fq, the Hasse-Weil zeta function may be realized as Z(X, t) = i P i(t) ( 1)i+1 with P i (t) = j (1 α ijt) 1 where the α ij are (inverse) eigenvalues of the Frobenius Φ acting on H i et,c(x; Q l ). When written in the Witt ring W (Z), this takes the form of an Euler-Poincaré characteristic as an alternating sum of Teichmüller elements [α]: Z(X, t) = i,j ( 1) i+1 [α ij ] W (Q l ). 2
Research Statement Jonathan Huang May 22, 2017 Thus the zeta function Z(X, t) is a motivic measure µ Z taking values in the ring A = W (Z); now consider its associated motivic zeta function ζ µz. We show that µ Z exponentiates and ζ µz takes values in W (A) = W (W (Z)) the double Witt ring. To achieve this, we provide a similar formula Theorem 1 (H.). The Weil zeta function of a variety X over F q, when considered as a motivic measure µ Z (X) = Z(X, t), has an associated motivic zeta function ζ µz taking the form ζ µz (X, u) = i,j ( 1) i+1 [[ [α ij ] ]] where the second sum is taking place in the Witt ring W (W (Q l )). This formula can be used to compute the zeta function Z(Sym n X, t) of the symmetric power of a variety X in terms of Z(X, t). Note that the case of symmetric powers of smooth projective curves was worked out by MacDonald in [19]. Our results generalize and reframe this in the setting of motivic zeta functions. 2.2. Lambda-ring measures. Let R be a λ-ring and µ : K 0 (Var k ) R a motivic measure. In this case, we have an opposite λ-ring map σ t : R W (R). While the Grothendieck ring of varieties is not known to have a λ-ring structure in general, it is a pre-λ-ring via the universal (Kapranov) motivic zeta function. A motivic measure µ is called a λ-ring measure µ : K 0 (Var k ) R if it is a pre-λ-ring map. In [25], it is shown that such measures are exponentiable, and K 0 (Var k ) µ R ζ µ σ t W (R) commutes. The case of the zeta function of varieties over finite fields may be restated in this setting using the opposite λ-ring structure map on W (Z). Given a λ-ring measure µ, consider the induced motivic zeta measure µ Z = ζ µ. This takes values in W (R), which is has a natural λ-ring structure. We show that for λ-ring measures µ, the induced motivic zeta measure µ Z is always exponentiable. Theorem 2 (H.). Let µ : K 0 (Var k ) F be a λ-ring measure and µ Z : K 0 (Var k ) W (F ) its induced motivic zeta function measure is also a λ-ring measure: ζ µz (X, u) = n ζ µ (Sym n X, t) u n = σ u (ζ µ (X, t)) and thus µ Z is exponentiable with ζ µz taking values in W (A) = W (W (R)). Moreover, this process may be iterated ad infinitum, i.e. ζ µz is itself a λ-measure. These results require a study of the λ-ring structure of the Witt ring W (R), particularly in the case where R itself is a λ-ring. 2.3. MacDonald formulae. The study of λ-ring measures and their induced zeta functions measures can be related to various MacDonald formulae. The classical MacDonald formula in [18] provides a closed product for the generating series of the Poincaré polynomial P (X, z) of symmetric powers Sym n X of a space X. Recall the Poincaré polynomial P (X, z) is defined using ranks of 3
cohomology with compact support b i (X) = rank Hc(X(C)) i as P (X, z) = 2m i=0 ( 1)i b i (X)z i where m is the dimension of X. The MacDonald formula is P (Sym n X, z)t n = (1 z1 t) b 1 (1 z 3 t) b3 (1 z 2m 1 t) b 2m 1 (1 z 1 t) b 1 (1 z 2 t) b 2 (1 z 2m t) b. 2m n=0 This can be reformulated in the setting of λ-ring measures. For k = C, the Poincaré polynomial defines a motivic measure µ P (X) = P (X, z), µ P : K 0 (Var C ) Z[z] where the polynomial ring Z[z] has a λ-ring structure induced from Z. The opposite λ-ring structure map on Z[z] acts as σ t (az) = (1 zt) a = a[z] W (Z[z]), and thus the classical MacDonald formula may be reformulated as ζ µp (X, t) = σ t (µ P ), i.e. µ P is a λ-ring measure. Note that using the language of Witt rings, P (X, z) = 2m i=0 ( 1) i b i (X)z i Z[z] ζ µp (X, t) = This is similar to the case of Z(X, t) and varieties over finite fields, where Z(X, t) = i,j ( 1) i+1 [α ij ] W (Q l ) ζ µz (X, u) = i,j 2m i=0 ( 1) i+1 b i [z i ] W (Z[z]) ( 1) i+1 [[ [α ij ] ]] W (W (Q l )) where Z(X, t) is a λ-ring measure. The formula in Theorem 1 should be viewed as a MacDonald formula for zeta functions of varieties over finite fields. 3. Ongoing and Future Research Projects We begin by discussing some possible extensions of the above results and immediate questions arising from these results. 3.1. Lambda-ring measures versus the counting measure. The zeta function for varieties over finite fields Z(X, t) is the Kapranov motivic zeta function associated to the counting measure µ # on Var Fq. We have shown that Z(X, t), as a measure, exponentiates; however, counting measure is not a λ-ring measure. There is a unique λ-ring structure on Z: for a Z, λ t (a) = (1 + t) a, λ n (a) = ( a n ), σ n (a) = ( a + n 1 n It is easy to verify that µ # (Sym n X) σ n (µ # (X)). For instance, X = A 1, Sym n A 1 = A n and µ # (A 1 ) = q whereas µ # (A n ) = q n σ n (q). As this is the only λ-ring structure on Z, counting measure can not be a pre-λ-ring map. The zeta function measure µ Z = Z(X, t) is a λ-ring measure while µ # is not. This should be possible for other abstract measures as well. Question 1. When and how do motivic zeta functions arise from measures µ : K 0 (Var k ) R which are not λ-ring measures? Are the induced zeta measures exponentiable? Note that for motivic zeta functions of motives, i.e. for char k > 0, those measures factoring through K 0 (Chow k ) through a motivic function h, we have the fact that K 0 (Chow k ) is a λ-ring (see [12])). On the other hand, the counting measure factors through a trace map on End Ql. We would like to explore and better understand this dichotomy. 4 ).
3.2. K-theory of endomorphisms. The big Witt ring W (R) is related to the K-theory of endomorphisms. Consider End R the category of pairs (P, f) where P is a finitely generated projective R-module and f : P P is an endomorphism, with morphisms ϕ : (P, f) (P, f ) respecting the endomorphisms f and f. The group K 0 (End R ) is a λ-ring, and there is an injective ring homomorphism due to Almkvist [1] (see also Grayson [7]) K 0 (End R )/J W (R), (P, f) det(id P tf) where J is the ideal generated by (R, 0) in K 0 (End R ). We wish to explore how this relates to exponentiable motivic measures and motivic zeta functions. For instance, Question 2. The zeta function of an exponentiable measure takes values in W (R). For which measures µ does ζ µ take values in the image K 0 (End R )/J in W (R)? In such cases, is the induced zeta function measure µ Z exponentiable? 3.3. Other related future research projects. We now briefly mention a couple of future research projects: Higher Euler characteristics. Following Ramachandran [24], another invariant which shows up in the study of special values of zeta functions, as well as in many other fields, is the so-called secondary Euler characteristic χ. Given a finite chain complex C, the usual Euler characteristic is defined as an alternating sum of ranks χ(c ) = i ( 1)i rank C i. When thought of as a map on categories of spaces, this is a homotopy invariant which is additive and multiplicative; thus is an Euler-Poincaré characteristic with universal value group given by the Grothendieck ring of varieties. If the Euler characteristic of C is 0, then we may define the secondary Euler characteristic χ (C ) = ( 1) i i rank C i. i This invariant appears in the definition of Ray-Singer analytic torsion [26], Milne s correcting factor for special values of zeta functions [21], and is related to the Adams operations on higher K-groups (see [8]). Notice that while χ is additive, it is not multiplicative. We wish to study the analogous universal value group of secondary Euler characteristics and possible relations to motivic zeta functions and their special values. Grothendieck spectrum of varieties. The Grothendieck ring of varieties K 0 (Var k ), arising from cutand-paste scissor relations on Var k, was extended by Zakharevich [27] and Campbell [3] to a K- theory spectrum of varieties K(Var k ). A recent preprint of Campbell, Wolfson and Zakharevich [4] takes the further step of lifting the Hasse-Weil zeta function on Var Fq to a map of K-theory spectra. Moreover, recent results of Blumberg, Gepner and Tabuada [2] extend the endomorphism functor End( ) to (small, idempotent-complete stable) -categories. I hope to explore exponentiable motivic measures and motivic zeta functions in this setting of the Grothendieck spectrum of varieties. Specifically, we may ask in what sense is the map on K-theory spectra exponentiable, and if the induced motivic zeta function measure also lifts to a map of ring spectra. 3.3.1. Arithmetic topology. One interesting relationship between topology and number theory is the analogy between knots and primes (sometimes called arithmetic topology or the MKR dictionary (Mazu-Kapranov-Reznokov) (see [22]). Its central observation is that circles S 1 with π 1 (S 1 ) = Z have an étale homotopic analog in primes Spec F p with π1 et(spec F p) = Ẑ. Thus given an orientable, closed 3-manifold M and a ring of integers O k of a number field k, knots S 1 M are analogous 5
to primes Spec F p Spec O i. One of the ideas that arises from this is the analogy between the Ray-Singer conjecture on topological torsion invariants and the main conjecture in Iwasawa theory; an infinite cyclic covering M/M corresponds to cyclotomic Z p -extensions k /k. I am interested in exploring these concepts further; specifically in the context of the determinant functor ETNC formulation of the Iwasawa main conjecture and Kurihara s refined main conjecture on higher Fitting ideals. References 1. G. Almkvist, Endomorphisms of finitely generated projective modules over a commutative ring, Ark. Mat. 11 (1973), 263 301. 2. A. Blumberg, D. Gepner, and G. Tabuada, K-theory of endomorphisms via noncommutative motives, Trans. Amer. Math. Soc. 368 (2016), 1435 1465. 3. J. Campbell, The K-theory spectrum of varieties, Available at arxiv:1505.03136. 4. J. Campbell, J. Woflson, and I. Zakharevich, Derived l-adic zeta functions, Available at arxiv:1703.09855. 5. S. del Baño Rollin, On the Chow motive of some moduli spaces, J. Reine Angew. Math. 532 (2001), 105 132. 6., On the motive of moduli spaces of rank two vector bundles over a curve, Compos. Math. 131 (2002), no. 1, 1 30. 7. D. Grayson, Grothendieck rings and Witt vectors, Comm. Algebra 6 (1978), no. 3, 249 255. 8., Adams operatiosn on higher K-theory, K-theory 6 (1992), 97 111. 9. A. Grothenieck, La théorie des classes de Chern, Bull. Soc. Math. France 86 (1958), 137 154. 10. S. M. Gusein-Zade, I. Luengo, and A. Melle-Hernandez, Power structure over the Grothendieck ring of varieties, Michigan Math. J. 54 (2006), no. 2, 353 359. 11., On the pre-λ-ring structure on the Grothendieck ring of stacks and the powers structures over it, Bull. Lond. Math. Soc. 45 (2013), no. 3, 520 528. 12. F. Heinloth, The universal Euler characteristic for varieties of characteristic zero, Compos. Math. 140 (2004), no. 4, 1011 1032. 13., A note on functional equations for zeta functions with values in Chow motives, Ann. Inst. Fourier (Grenoble) 57 (2007), no. 6, 1927 1945. 14. J. Huang, Exponentiation of motivic zeta functions, Ph.D. thesis, University of Maryland, College Park, 2017. 15. M. Kapranov, The elliptic curve in S-duality theory and Eisenstein series for Kac-Moody groups, Available at arxiv:math/0001005v2. 16. D. Knutson, λ-rings and the Representation Theory of the Symmetric Group, Lecture Notes in Mathematics, vol. 308, Springer-Verlag, Berlin-New York, 1973. 17. M. Larsen and V. Lunts, Rationality criteria for motivic zeta functions, Compos. Math. 140 (2004), no. 6, 1537 1560. 18. I. G. MacDonald, The Poincaré polynomial of a symmetric product, Proc. Cambridge Philos. Soc. 58 (1962), 563 568. 19., Symmetric products of an algebraic curve, Topology 1 (1962), 319 343. 20. L. Maxim and J. Schürmann, Twisted genera of symmetric products, Selecta Math. (N.S.) 18 (2012), no. 1, 283 317. 21. James Milne, Values of zeta functions of varieties over finite fields, American Journal of Mathematics 108 (1986), 297 360. 22. Masanori Morishita, Milnor invariants and l-class groups, Progress in Mathematics 265 (2008), 669 683. 23. N. Ramachandran, Zeta functions, Grothendieck groups, and the Witt ring, Bull. Sci. Math. 139 (2015), no. 6, 599 627. 24., Higher Euler characteristics: variations on a theme of Euler, Homology Homotopy Appl. 18 (2016), no. 1, 231 246. 25. N. Ramachandran and G. Tabuada, Exponentiable motivic measures, J. Ramanujan Math. Soc. 30 (2015), no. 4, 349 360. 26. D. B. Ray and I. M. Singer, R-torsion and the laplacian on riemannian manifolds, Advances in Math. 7 (1971), 145 210. 27. I. Zakharevich, The K-theory of assemblers, Adv. Math. 304 (2017), 1176 1218. 6