Christopher Deninger

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

Deninger's research focuses on arithmetic geometry, including applications to L-functions.

Deninger obtained his doctorate from the University of Cologne in 1982, under the supervision of Curt Meyer.

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, who identified Bloch's complexes to be the dualizing complexes over higher-dimensional schemes.

Another group of Deninger's papers studies L-functions and their special values.

A classical example of an L-function is the Riemann zeta function ζ(s), for which formulas such as are known since Euler.

In a landmark paper, 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 at integers s.

Bloch and Beilinson proved essential parts of this conjecture for h1(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.

In 1992 he shared a Gottfried Wilhelm Leibniz Prize with Michael Rapoport, Peter Schneider and Thomas Zink.

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

Here p is either a prime number or infinity, corresponding to the non-Archimedean Euler factors and the Archimedean Euler factor respectively, and Rp is the space of finite real valued Fourier series on R/log(p)Z for a prime number p, and R∞ = R[exp(−2y)].

Finally, Θ is the derivative of the R-action given by shifting such functions.

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).

Deninger (1994) also exhibited a similar unifying approach for ε-factors (which express the ratio between completed L-functions at s and at 1−s).

These results led Deninger to propose a program concerning the existence of an "arithmetic site" Y associated to the compactification of Spec Z.

Among other properties, this site would be equipped with an action of R, and each prime number p would correspond to a closed orbit of the R-action of length log(p).

Moreover, analogies between formulas in analytic number theory and dynamics on foliated spaces led Deninger to conjecture the existence of a foliation on this site.

Moreover, this site is supposed to be endowed with an infinite-dimensional cohomology theory such that the L-function of a motive M is given by Here M is a motive, such as the motives hn(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.

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.

In 1998 he was a plenary speaker at the International Congress of Mathematicians in 1998 in Berlin.

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.

In 2012 he became a fellow of the American Mathematical Society.

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), ).

The Riemann ζ-function is defined using a product of Euler factors

for each prime number p.

In order to obtain a functional equation for ζ(s), one needs to multiply them with an additional term involving the Gamma function:

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.