Christopher Deninger

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Christopher Deninger Biography

Christopher Deninger (born 8 April 1958) is a German mathematician at the University of Münster.

Deninger obtained his doctorate from the University of Cologne in 1982, under the supervision of Curt Meyer. In 1992 he shared a Gottfried Wilhelm Leibniz Prize with Michael Rapoport, Peter Schneider and Thomas Zink. In 1998 he was a plenary speaker at the International Congress of Mathematicians in 1998 in Berlin. In 2012 he became a fellow of the American Mathematical Society.

from various points of view. In 1984, they computed the e-invariant of X in terms of ζ(−n), which leads to a construction of elements in the stable homotopy groups of spheres of arbitrarily large order. In 1988, they used methods of analytic number theory to give estimates on the dimension of the cohomology of nilpotent Lie algebras.

are known since Euler. In a landmark paper, Beilinson (1984) had proposed a set of far-reaching conjectures describing the special values of L-functions, i.e., the values of L-functions at integers. In very rough terms, Beilinson’s conjectures assert that for a smooth projective algebraic variety X over Q, motivic cohomology of X should be closely related to Deligne cohomology of X. In addition, the relation between these two cohomology theories should explain, according to Beilinson’s conjecture, the pole orders and the values of

In a series of papers between 1984 and 1987, Deninger studied extensions of Artin–Verdier duality. Broadly speaking, Artin–Verdier duality, a consequence of class field theory, is an arithmetic analogue of Poincaré duality, a duality for sheaf cohomology on a compact manifold. In this parallel, the (spectrum of the) ring of integers in a number field corresponds to a 3-manifold. Following work of Mazur, Deninger (1984) extended Artin–Verdier duality to function fields. Deninger then extended these results in various directions, such as non-torsion sheaves (1986), arithmetic surfaces (1987), as well as higher-dimensional local fields (with Wingberg, 1986). The appearance of Bloch’s motivic complexes considered in the latter papers influenced work of several authors including Geisser (2010), who identified Bloch’s complexes to be the dualizing complexes over higher-dimensional schemes.

at integers s. Bloch and Beilinson proved essential parts of this conjecture for h(X) in the case where X is an elliptic curve with complex multiplication and s=2. In 1988, Deninger & Wingberg gave an exposition of that result. In 1989 and 1990, Deninger extended this result to certain elliptic curves considered by Shimura, at all s≥2. Deninger & Nart (1995) expressed the height pairing, a key ingredient of Beilinson’s conjecture, as a natural pairing of Ext-groups in a certain category of motives. In 1995, Deninger studied Massey products in Deligne cohomology and conjectured therefrom a formula for the special value for the L-function of an elliptic curve at s=3, which was subsequently confirmed by Goncharov (1996). As of 2018, Beilinson’s conjecture is still wide open, and Deninger’s contributions remain some of the few cases where Beilinson’s conjecture has been successfully attacked (surveys on the topic include Deninger & Scholl (1991), Nekovář (1994)).

More general L-functions are also defined by Euler products, involving, at each finite place, the determinant of the Frobenius endomorphism acting on l-adic cohomology of some variety X / Q, while the Euler factor for the infinite place are, according to Serre, products of Gamma functions depending on the Hodge structures attached to X / Q. Deninger (1991) harvtxt error: no target: CITEREFDeninger1991 (help) expressed these Γ-factors in terms of regularized determinants and moved on, in 1992 and in greater generality in 1994, to unify the Euler factors of L-functions at both finite and infinite places using regularized determinants. For example, for the Euler factors of the Riemann zeta-function this uniform description reads

This program was surveyed by Deninger in his talks at the European Congress of Mathematicians in 1992, at the International Congress of Mathematicians in 1998, and also by Leichtnam (2005). In 2002, Deninger constructed a foliated space which corresponds to an elliptic curve over a finite field, and Hesselholt (2016) showed that the Hasse-Weil zeta-function of a smooth proper variety over Fp can be expressed using regularized determinants involving topological Hochschild homology. In addition, the analogy between knots and primes has been fruitfully studied in arithmetic topology. However, as of 2018, the construction of a foliated space corresponding to Spec Z remains elusive.

Here M is a motive, such as the motives h(X) occurring in Beilinson’s conjecture, and F(M) is conceived to be the sheaf on Y attached to the motive M. The operator Θ is the infinitesimal generator of the flow given by the R-action. The Riemann hypothesis would be, according to this program, a consequence of properties parallel to the positivity of the intersection pairing in Hodge theory. A version of the Lefschetz trace formula on this site, which would be part of this conjectural setup, has been proven by other means by Deninger (1993). In 2010, Deninger proved that classical conjectures of Beilinson and Bloch concerning the intersection theory of algebraic cycles would be further consequences of his program.

Moreover, it had been known that Mahler measures of certain polynomials were known to be expressible in terms of special values of certain L-functions. In 1997, Deninger observed that the integrand in the definition of the Mahler measure has a natural explanation in terms of Deligne cohomology. Using known cases of the Beilinson conjecture, he deduced that m(f) is the image of the symbol {f, t1, …, tn} under the Beilinson regulator, where the variety is the complement in the n-dimensional torus of the zero set of f. This led to a conceptual explanation for the afore-mentioned formulas for Mahler measures. Besser & Deninger (1999) and Deninger later in 2009 carried over these ideas to the p-adic world, by replacing the Beilinson regulator map to Deligne cohomology by a regulator map to syntomic cohomology, and the logarithm appearing in the definition of the entropy by a p-adic logarithm.

The classical fact from Hodge theory that any cohomology class on a Kähler manifold admits a unique harmonic had been generalized by Álvarez López & Kordyukov (2001) to Riemannian foliations. Deninger & Singhof (2001) show that foliations on the above space X, which satisfy only slightly weaker conditions, do not admit such Hodge theoretic properties. In another joint paper from 2001, they established a dynamical Lefschetz trace formula: it relates the trace of an operator on harmonic forms the local traces appearing at the closed orbits (on certain foliated spaces with an R-action). This result serves as a corroboration of Deninger’s program mentioned above in the sense that it verifies a prediction made by this program on the analytic side, i.e., the one concerning dynamics on foliated spaces.

In Deninger & Werner (2005) established a p-adic analogue thereof: for a smooth projective algebraic curve over Cp, obtained by base change from X / Q ¯ p {\displaystyle X/{\overline {\mathbf {Q} }}_{p}} , they constructed an action of the etale fundamental group π1(X) on the fibers on certain vector bundles, including those of degree 0 and having potentially strongly semistable reduction. In another paper of 2005, they related the resulting representations of the fundamental group of the curve X with representations of the Tate module of the Jacobian variety of X. In 2007 and 2010 they continued this work by showing that such vector bundles form a Tannakian category which amounts to identifying this class of vector bundles as a category of representations of a certain group.

In 2006 and 2007, Deninger and Klaus Schmidt pushed the parallel between entropy and Mahler measures beyond abelian groups, namely residually finite, countable discrete amenable groups Γ. They showed that the Γ-action on Xf is expansive if and only if f is invertible in the L-convolution algebra of Γ. Moreover, the logarithm of the Fuglede-Kadison determinant on the von Neumann algebra NΓ associated to Γ (which replaces the Mahler measure for Z) agrees with the entropy of the above action.

A series of joint papers with Annette Werner examines vector bundles on p-adic curves. A classical result motivating this study is the Narasimhan–Seshadri theorem, a cornerstone of the Simpson correspondence. It asserts that a vector bundle on a compact Riemann surface X is stable if it arises from a unitary representation of the fundamental group π1(X).

Joachim Cuntz and Deninger worked together on Witt vectors. In two papers around 2014, they simplified the theory by giving a presentation of the ring of Witt vectors in terms of a completion of the monoid algebra ZR. This approach avoids the universal polynomials used in the classical definition of the addition of Witt vectors.

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