# Research Area A1:

# New frontiers in arithmetic and algebraic geometry

## Pls: Faltings, Huybrechts, Scholze

## Contributions by Caraiani, Tian, Ziegler

## Topic and goals

Central to modern arithmetic and algebraic geometry are the concepts of Shimura varieties, algebraic cycles, modular forms, and cohomology theories, with many interactions among them and links to other areas. For example, points on Shimura varieties parametrize motives, which can be understood either in terms of cohomology theories or in terms of algebraic cycles; functions on Shimura varieties are modular forms; and the cohomology of Shimura varieties realizes Langlands correspondences. We want to further our understanding, leading to new results on the Tate and Hodge conjectures, diophantine height estimates, the Langlands conjectures, algebraic K-theory, and the geometry of K3 surfaces.

## State of the art, our expertise

p-adic geometry. In the p-adic geometry of Shimura varieties, a prominent new object is the perfectoid space [Sch12] arising as the inverse limit over all levels at p. It admits a novel period map to a flag variety, known as the Hodge–Tate period map. Applications of this construction include the construction of Galois representations for torsion classes [Sch15], bounds for the cohomological degrees in which torsion appears [CS17], and local-global compatibility results for these Galois representations, leading up to potential automorphy results for elliptic curves over CM fields.

Cohomology theories. Recent work of Bhatt–Morrow–Scholze [BMS15] constructs a cohomology theory unifying étale, crystalline and de Rham cohomology, taking values in p-adic shtukas. With coefficients, Faltings has defined a logarithmic version of the category MF of Fontaine [Fal16]. One main goal will be to develop a general and solid theory of coefficients unifying these points of view. Globally, a universal cohomology theory should be given by a shtuka relative to the curve Spec Z. The completion of this theory along the diagonal should be related to the q-deformation of de Rham cohomology conjectured in [Sch17]. The regulator from algebraic K-theory and Bloch complexes to étale cohomology has recently been made explicit by Calegari–Garoufalidis–Zagier; the answer involves a q-deformation of the polylogarithm.

Arakelov theory. Arakelov theory of curves still has many open problems. While a number of equalities between Green’s functions can be found in the literature, much less is known for inequalities which would be useful for applications. For example the asymptotic behavior under degenerations to semistable curves was only established recently in a thesis under Faltings’ supervision [Wil17].

Cycles. In the mod p fibers of Shimura varieties, interesting algebraic cycles are given by the irreducible components of the basic locus. An explicit description of these cycles as in [TX16] usually gives a geometric interpretation of the Jacquet–Langlands correspondence of automorphic forms, and, as discovered recently, those cycles contribute to the generic part of the Tate classes on Shimura varieties over finite fields [HTX17]. On the other hand, morphisms of Shimura data give another source of special cycles on Shimura varieties. The interrelation between these two kinds of algebraic cycles encodes interesting information on congruences of automorphic forms.

K3 surfaces. The theory of K3 surfaces, a particularly rich class of varieties, is an excellent testing ground for central questions in arithmetic and algebraic geometry [Huy16]. For products of K3 surfaces, the Hodge and Tate conjectures are both still open. A motivic interpretation for classes corresponding to Hodge isometries has recently been given [Huy]. This has led to a conceptual proof of a long standing conjecture of Shafarevich and of the Hodge conjecture for squares of K3 surfaces of CM type, originally due to Buskin. Twisted K3 surfaces also play a central role in the intensively studied interactions between K3 surfaces and cubic fourfolds [Huy17]. As for the Kuga–Satake construction, the link is via Hodge theory and awaits a more direct geometric description.

## Research program

p-adic geometry. The theory of Shimura varieties has a p-adic analogue given by Rapoport–Zink spaces, and their conjectural generalizations of local Shimura varieties. Further generalizations are possible in the new geometric category of diamonds, where one can go beyond the case of minuscule cocharacters, leading to general moduli spaces of p-adic shtukas. Here, the theory becomes as general as in the equal characteristic case, allowing for several legs, leading to spaces living over a product of several copies of Spec Zp. A long-term goal is to develop the theory of diamonds with the aim of establishing the local Langlands correspondence for general reductive groups, linking this research area to RA A2. For Shimura varieties, Caraiani and Scholze want to build on the methods and results of [CS17] to obtain a detailed understanding of torsion classes in the cohomology and their associated Galois representations. For applications to automorphy lifting theorems, it is necessary to understand the p-adic Hodge-theoretic properties of these classes. In general, it is already difficult to formulate the desired properties. In favorable situations such as the Fontaine–Laffaille or ordinary cases, results have recently been obtained. One goal will be to handle the cases of potentially crystalline and trianguline representations that seem to be critical for general potential automorphy theorems, with applications to the meromorphic continuation of many L-functions. Moreover, the related moduli spaces of objects of the category MF studied by Faltings have been used as local models for the p-adic geometry of some Shimura varieties parametrizing abelian varieties with higher Hodge cycles, and in particular for the construction of good integral models. We hope to extend this to general Shimura varieties and general local Shimura varieties by using moduli spaces of p-adic shtukas, including in the semistable reduction case, where Faltings’ work on a logarithmic version of MF has recently played a role.

Cohomology theories. We want to generalize the construction of cohomology theories by Bhatt– Morrow–Scholze [BMS15], taking values in p-adic shtukas, to the case of coefficients and general base rings. This should also include the case of logarithmic singularities and establish a link to Fontaine’s category MF. Our goal is to establish basic functorialities of these coefficients such as stability under direct image of proper log-smooth maps. In the semistable case, the corresponding category MFlog was constructed recently by Faltings and allows one to construct semistable models for Spin Shimura varieties with certain level structures. We aim to combine the resulting theories for varying p, as in the q-deformation of de Rham cohomology conjectured by Scholze [Sch17], which would give a glimpse of the notion of shtukas over number fields. A different part of the picture would be an R-linear cohomology theory for varieties over Fp, which should be related to vector bundles on the twistor-P 1 ; see the figure. A proposal for a theory in the case of curves was made by Zagier via Eisenstein series. Another part of the picture should be an explicit description of algebraic K-theory such as the higher Bloch complex as conjectured by Zagier [Zag91].

Arakelov theory. Recently, Faltings obtained an upper bound for the Arakelov norm on the sheaf of differentials on a compact Riemann surface X. The result is phrased in terms of the hyperbolic and the Strebel metric. However, for arithmetic applications a lower bound is needed. There has not been much progress in this direction recently, as the focus has somewhat shifted to higherdimensional arithmetic varieties. The known techniques based on theta-functions have provided little geometric insight and new methods will have to be developed. Moduli spaces of curves from a differential geometric point of view are also studied in RA A3.

Cycles. In the case of Shimura varieties, we are interested in finding explicit formulae for padic regulators of special cycles, which should be related to special values of L-functions of automorphic forms. In this process, methods from algebraic geometry, p-adic Hodge theory and automorphic representations interact with each other in an essential way. It is also remarkable that analogous objects (such as local shtukas) and similar methods as in the Langlands program for the function field case play increasingly important roles in recent developments of the subject. K3 surfaces. Derived categories have played a crucial role in the understanding of cycles on K3 surfaces and on products thereof. We plan to explore their potential to understand all Hodge classes and therefore to prove the Hodge and Tate conjecture in this case. The methods should have an impact on cycles on products of cubic fourfolds, which may turn out to be crucial in answering the classical question on rationality and irrationality of this particular class of varieties. From the point of view of moduli spaces and Shimura varieties, the relation between K3 surfaces and cubic fourfolds shall be studied further with potential applications to mirror symmetry, linking this RA to IRU D2. There are strong links to RA A3, where hyperkähler manifolds as higher-dimensional analogues of K3 surfaces are studied. A long-term goal is a geometric understanding of the Kuga–Satake construction. Proving the Hodge conjecture in this case would have formidable consequences, e.g., establishing Kimura finite-dimensionality for all K3 surfaces.

## Summary

Cohomology of algebro-geometric and arithmetic structures, in its various known and unknown incarnations, is at the heart of this RA. We address foundational aspects and study applications to the Langlands program and to the construction of special cycles on Shimura varieties, moduli spaces, and varieties related to K3 surfaces.

## Bibliography

[BMS15] B. Bhatt, M. Morrow, and P. Scholze. Integral p-adic Hodge theory–announcement. *Math. Res. Lett., 22(6):1601–1612, 2015*.

[CS17] A. Caraiani and P. Scholze. On the generic part of the cohomology of compact unitary Shimura varieties. *Ann. of Math. (2), 186(3):649–766, 2017*.

[Fal16] G. Faltings. The category MF in the semistable case. *Izv. Ross. Akad. Nauk Ser. Mat., 80(5):41–60, 2016*.

[HTX17] D. Helm, Y. Tian, and L. Xiao. Tate cycles on some unitary Shimura varieties mod p.* Algebra Number Theory, 11(10):2213–2288, 2017*.

[Huy] D. Huybrechts. Motives of isogenous K3 surfaces. Comment. Math. Helv. to appear.

[Huy16] D. Huybrechts. Lectures on K3 surfaces, volume 158 of Cambridge *Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2016*.

[Huy17] D. Huybrechts. The K3 category of a cubic fourfold. *Compos. Math., 153(3):586–620, 2017*.

[Sch12] P. Scholze. Perfectoid spaces.* Publ. Math. Inst. Hautes Études Sci., 116:245–313, 2012*.

[Sch15] P. Scholze. On torsion in the cohomology of locally symmetric varieties. *Ann. of Math. (2), 182(3):945–1066, 2015*.

[Sch17] P. Scholze. Canonical q-deformations in arithmetic geometry.* Ann. Fac. Sci. Toulouse Math. (6), 26(5):1163–1192, 2017*.

[TX16] Y. Tian and L. Xiao. On Goren–Oort stratification for quaternionic Shimura varieties.* Compos. Math., 152(10):2134–2220, 2016*.

[Wil17] R. Wilms. New explicit formulas for Faltings’ delta-invariant. *Invent. Math., 209(2):481–539, 2017*.

[Zag91] D. Zagier. Polylogarithms, Dedekind zeta functions and the algebraic K-theory of fields. *In Arithmetic Algebraic Geometry (Texel, 1989), volume 89 of Progr. Math., pages 391–430. Birkhäuser, Boston, 1991*.