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Albert-Ludwigs-Universität FreiburgMathematisches InstitutAbteilung für Reine MathematikArbeitsgruppe Zahlentheorie/
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I am working in arithmetic geometry: number theoretic questions are turned into geometric ones: The solutions of a polynomial equation are reinterpreted as points on an algebraic variety defined by these equations. Integral solutions become points with integral coordinates. This is allows us to apply the full strength of modern geometry to study our problems. A particular powerful tool is cohomology. It is a machine that produces invariants. The best known is the genus of a compact surface, the number of ``holes´´. A priori there is no reason that these topological invariants should have implications for number theoretic questions--but they do.
There is a whole range of cohomology theories in algebraic geometry and number theory. Motives and motivic cohomology play a special role as the universal cohomology theory. They are closely related to algebraic K-theory, an invariant built from the category of vector bundles on a manifold or algebraic variety.
In all, my research falls into the MSC groups 14, 11, and 19, with varying emphasis.
Below, I want to explain my research projects, starting with the most recent. They are interconnected. For example, the results on polylogarithms were essential input into the work on special values of L-functions by myself and others. The project on differential forms started in order to address a technical problem in the construction of the period isomorphism, but now has a life on its own.
Periods have more conceptual interpretations as the entries of the period matrix between singular cohomology and algebraic de Rham cohomology of algebraic varieties over Q. Grothendieck conjectured that all algebraic relations between period numbers should come from geometric relations like functoriality and product formulas. This can be made precise in the language of motives. He only published a very vague hint of these conjectures, but the ideas circulated and have been clarified completely by Andre. While Grothendieck was thinking mostly in terms of smooth projec tive varieties (and so-called pure motives), Kontsevich came to similar conjectures also for relative cohomomlogy of arbitrary varieties (and so-called mixed motives).
My first involvement into these questions was via a Master project in 2004/05. Benjamin Friedrich was able to relate the naive definition of period numbers as integrals over semi-algebraic sets to the conceptual one via cohomology. His thesis can be found on arxiv, see here. The comparison is not complete and the thesis was not published at the time. We were missing a crucial piece of the puzzle: Nori had found an unconditional definition of a category of mixed motives over a field embeddable into C, but not published it. In joint work with Müller-Stach [32], we later cleared up these foundations. The two preprints and a lot of back ground material were put into the book [33]. While we have a very good conceptual understanding of the period conjecture, actual evidence is scant. The most general available result is the Analytic Subgroup Theorem of Wüstholz. In [35] we use it to deduce the curve case of the period conjecture: all linear relations are induced by functoriality of marked curves or, equivalently, by functoriality of Deligne 1-motives.Differential forms and their sheaf cohomology are a very valuable tool in algebraic geometry, in particular in birational geometry and singularity theory. In contrast to the theories used in the other projects, the invariant is not homotopy invariant. Basically by definition, the notion is not as well-behaved in the sigular case. Various replacements have been used over time.
In this project we explore h-differentials, i.e., differential forms in the h-topology. This topolology was introduced by Voevodsky in his first approach to defining triangulated motives. In contrast to most Grothendieck topologies, it is not subcanonical: the structure sheaf is not a sheaf for the h-topology. Passing to algebraic geometry with the h-topology really brings us into a new world--a better one, I would argue. In [28], we handled the case of characteristic 0. It turns out that the notion is very well-behaved, in particular, it agrees with the standard notion in the smooth case. Locally for the h-topology every variety is smooth, so this allows us to compute. In the singular case, we get back ad-hoc notions like reflexive differentials and Du Bois differentials in the geometric situations where those notions have been used.
The situation is a lot more open in positive characteristic. Frobenius covers are h-covers, so it is clear that we have to use weaker replacements. In joint work with Kebekus and Kelly [30], [31], we identified a promising candiate, equivalently the eh-, cdh or rh-topology. Differential forms in these topologies agree. They agree with ordinary differentials on smooth varieties. Under resolution of singularities, this is also true for their sheaf cohomology. However, we have not been able to find an unconditional argument.
L-functions of algebraic varieties or more generally of motives are meromorphic functions encoding local (i.e., prime by prime) data. In mysterious ways they are also related to global data. The most important example is the Riemann zeta function, or its analogue for other number fields, the Dedekind zeta-functions. Also very well studied is the L-function of an elliptic curve. Beilinson conjectured formulas for the vanishing order and the leading coefficient of these L-functions in integral points up to a rational factor. If his conjecture holds, then these numbers are periods, relating it to the project above. Bloch and Kato refined his conjecture into an integral statement. Very few cases are known. Notably, the most famous case is about the value of the L-function of an elliptic curve at the point 1: the conjecture of Birch and Swinnerton-Dyer.
Together with Kings, I have worked on the case of Artin motives, i.e., Dedekind zeta-functions and Artin L-functions. Many problems disappear in this case: we know that the functions are indeed meromorphic on the whole complex plane and understand their poles. Motivic cohomology is known to finite dimensional. But actually the best general result is still modest: the case of abelian number fields, or equivalently, Dirichlet L-functions in [22]. Our proof follows the one of Bloch and Kato for the case of the Riemann zeta function. Again we need a comparison result for regulators obtained via comparing polylogarithms, see [19]. We also need a slight generalisation of the Main Conjecture of Iwasawa Theory, a theorem in this case.
As these tools are not available in the non-abelian case, we proposed a completely different approach in [24]. Note that the Beilinson conjecture, i.e., the formula of up to a rational factor, is known in this case by work of Borel. Beilinson showed how to connect Borel's regulator to what we call the Beilinson regulator, i.e., the Chern class in Deligne cohomology. We introduce a p-adic analogue of the Borel regulator that is meant to correct for the unknown rational factor. We were able to relate it to the Soule regulator i.e., the Chern class in p-adic Galois cohomology, see [22]. Even more surprisingly, we together with Naumann were able to control integral structures in this comparison. In the 2018 PhD thesis of my student Maximilian Schmidtke researchgate the precise comparison of the factors of the Bloch-Kato conjecture and our version was worked out. However, the actual proof is still wide open.
The polylogarithm functions are a sequence of multivalued holomorphic functions on P^1 minus three points. In the case k=1 it is -log(1-z). Deligne recognised that these functions encode the properties of a variation of Hodge structure on P^1 minus three points. He also constructed l-adic analogues. Actually, the object is of motivic origin as Beilinson explained. The natural place for this object is in a category of motivic sheaves. However, these categories were only constructed a lot later.
In [15], Wildeshaus and I provided the technical background to carry out the construction of the motivic polylog as an element of motivic cohomology of a complicated geometric object and to compare it to explicit classes in Deligne cohomology and p-adic cohomology. By specialising at roots of unity, we obtain a necessary input into the proof of the Tamagawa number conjecture for the Riemann zeta function and later for abelian number fields.
The same comparison result can also be deduced from degenerations of the elliptic polylogarithm. This was shown by Kings and myself in [17] and [18]. While less direct, this approach is actually technically less demanding.
We picked up the objects lately. Now that categories of motivic sheaves have beend constructed, we have been able to define the polylogarithm object not only for the multiplicative group (the classical case) and elliptic curves, but for all families of commutative group schemes, see [20].
The idea of motives is due to Grothendieck. There should be an abelian category and a cohomology theory with values in this category which is universal among all cohomology theories that behave like singular cohomology, the so-called Weil cohomologies. Moreover, this abelian category should be very close to algebraic geometry. Grothendieck gave a conjectural description in terms of algebraic cycles in the smooth projective case. It has the expected properties under the Hodge conjecture or the Tate conjecture.
For general varieties, we expect (following Beilinson, Jannsen and others) a bigger category of mixed motives. They should have an intrinsic weight filtration such that the associated gradeds are pure motives. This turned out to be even more elusive. By now we have several candidate categories. Absolute Hodge motives came first. Deligne and Jannsen proposed to replace actual cycles by compatible systems of cohomogy classes. This yields a well-behaved abelian category, which conjecturally agrees with the one we want. In my thesis [5] I constructed a triangulated category with a t-structure which has these mixed realisations as heart. This allowed me to construct Chern classes on algebraich K-theory and Abel-Jacobi maps, another structure we expected to find.
Hanamura, Levine and Voevoedsky independently constructed triangulated categories of mixed motives. They miss the expected t-structure, but they have the connection to algebraic K-groups conjectured by Beilinson. They turned out to be equivalent, giving us a strong clue that this is the right object. The setting of Voevodsky is the most flexible and the one used most. By work of Ayoub and Cisinski-Deglise there is even a full 6 functor formalism of motivic sheaves. The papers [8] and its corrigendum [9] make the connection between Voevodsky motives and absolute Hodge motives. In particular all Chern classes factor.
The paper [10] with Kahn concentrates on the subcategory of mixed Tate motives, the subcategory generated by motives of linear spaces. The slice filtration turns out to be valuable tool in this setting. In joint work with Ancona, Enright-Ward and Pepin-Lehalleur, we used it to compute the motives of commutative group shemes in [12] and [13]. This was input into the most general construction of motivic polylogarithms mentioned above.
The third type of construction of motives depends on fixing a cohomology theory, usually singular cohomology. It has the advantage of producing an abelian category. This was done by Andre the in the pure case and with a completely different construction by Nori in the mixed case. The 2016 thesis of my student Daniel Harrer, see researchgate makes the connection between Voevodsky and Nori motives.
The field of motives has a reputation of being full of conjecture, deservedly so. However, real progress has been made and we now also have many results and a set of useful tools. The most spectacular success was Voevodsky's proof of the Milnor conjecture obtained by comparing motivic cohomology for the Nisnevich topology and the etale topology. The use of polylogarithms in the proof of conjectures on special values of L-functions as above is another. Deligne and Goncharov used our knowledge of categories of mixed Tate motives to bound the dimensions of certain sets of periods spaces generated by multiple zeta values. Finally, the recent proof of the period conjecture on linear relations between periods of curves (given by Wüstoholz and myself in [35]) depends on the use of 1-motives.My master thesis, which is the basis of [1] and [2], gave details on a result of Beilinson: he introduces a notion of adles for noetherian schemes generalising the well-known construction of the (finite) adeles in algebraic number theory. It is a actually a complex and computes coherent cohomology. The paper has been well received. I have been a passive observer of the field since then, but I am not working on adeles myself.
The paper [3] on homological algebra was input into [15]. Finally [4] is addressed at a general non-mathematical audience.