Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg最新文献

Using trace formulas for Hecke operators, Eichler first provided a positive solution about basis problems of elliptic cusp forms by quadratic forms. J-L. Waldspurger established that elliptic cusp forms of arbitrary level are spanned by theta series by means of different and interesting ideas and methods. This result is given by Zagier’s analytic theorems, the Siegel main theorem of quadratic forms and the theory of Hecke operators. We intend to generalize Waldspurger’s results and determine theta series which span the space of Hilbert new forms over arbitrary totally real algebraic number fields following Waldspurger’s methods.

Let ({mathcal {T}}) be an algebraic triangulated category and ({mathcal {C}}) an extension-closed subcategory with ({{,textrm{Hom},}}({mathcal {C}}, Sigma ^{<0} {mathcal {C}})=0). Then ({mathcal {C}}) has an exact structure induced from exact triangles in ({mathcal {T}}). Keller and Vossieck say that there exists a triangle functor (operatorname {D}^{b}({mathcal {C}}) rightarrow {mathcal {T}}) extending the inclusion ({mathcal {C}} subseteq {mathcal {T}}). We provide the missing details for a complete proof.

We compute a formula for the discriminant of tautological bundles on symmetric powers of a complex smooth projective curve. It follows that the Bogomolov inequality does not give a new restriction to stability of these tautological bundles. It only rules out tautological bundles which are already known to have the structure sheaf as a destabilising subbundle.

Let (0 < a le 1/2) and define the quadrilateral zeta function by (2Q(s,a):= zeta (s,a) + zeta (s,1-a) + mathrm{{Li}}_s (e^{2pi ia}) + mathrm{{Li}}_s(e^{2pi i(1-a)})), where (zeta (s,a)) is the Hurwitz zeta function and (mathrm{{Li}}_s (e^{2pi ia})) is the periodic zeta function. In the present paper, we show that there exists a unique real number (a_0 in (0,1/2)) such that all real zeros of Q(s, a) are simple and are located only at the negative even integers just like (zeta (s)) if and only if (a_0 < a le 1/2). Moreover, we prove that Q(s, a) has infinitely many complex zeros in the region of absolute convergence and the critical strip when (a in {mathbb {Q}} cap (0,1/2) setminus {1/6, 1/4, 1/3}). The Lerch formula, Hadamard product formula, Riemann-von Mangoldt formula for Q(s, a) are also shown.

We give a description of the intermediate Jacobian fibration attached to a general complex cubic fourfold X containing a plane as a Lagrangian subfibration of a moduli space of torsion sheaves on the K3 surface associated to X up to a cover. To do so, we propose a general construction of Lagrangian fibrations in Prym varieties as subfibrations of Beauville–Mukai systems over some loci of nodal curves in linear systems on K3 surfaces.

The classical Dedekind sums s(d, c) can be represented as sums over the partial quotients of the continued fraction expansion of the rational (frac{d}{c}). Hardy sums, the analog integer-valued sums arising in the transformation of the logarithms of (theta )-functions under a subgroup of the modular group, have been shown to satisfy many properties which mirror the properties of the classical Dedekind sums. The representation as sums of partial quotients has, however, been missing so far. We define non-classical continued fractions and prove that Hardy sums can be expressed as a sums of partial quotients of these continued fractions. As an application, we prove that the graph of the Hardy sums is dense in (textbf{R}times textbf{Z}).

At the international congress of mathematicians in 1900, Hilbert claimed that the Riemann zeta function (zeta (s)) is not the solution of any algebraic ordinary differential equations on its region of analyticity. Let T be an infinite order linear differential operator introduced by Van Gorder in 2015. Recently, Prado and Klinger-Logan [9] showed that the Hurwitz zeta function (zeta (s,a)) formally satisfies the following linear differential equation

Then in [6], by defining (T_{p}^{a}), a p-adic analogue of Van Gorder’s operator T,  we constructed the following convergent infinite order linear differential equation satisfied by the p-adic Hurwitz-type Euler zeta function (zeta _{p,E}(s,a))

In this paper, we consider this problem in the positive characteristic case. That is, by introducing (zeta _{infty }(s_{0},s,a,n)), a Hurwitz type refinement of Goss zeta function, and an infinite order linear difference operator L, we establish the following difference equation

We extend previous work by constructing a universal abelian tensor category (textbf{T}_t) generated by two objects X, Y equipped with finite filtrations (0subsetneq X_0subsetneq ...subsetneq X_{t+1}= X) and (0subsetneq Y_0subsetneq ... subsetneq Y_{t+1}= Y), and with a pairing (Xotimes Yrightarrow mathbbm {1}), where (mathbbm {1}) is the monoidal unit. This category is modeled as a category of representations of a Mackey Lie algebra (mathfrak {gl}^M(V,V_*)) of cardinality (2^{aleph _t}), associated to a diagonalizable pairing between two vector spaces (V,V_*) of dimension (aleph _t) over an algebraically closed field ({{mathbb {K}}}) of characteristic 0. As a preliminary step, we study a tensor category ({{mathbb {T}}}_t) generated by the algebraic duals (V^*) and ((V_*)^*). The injective hull of the trivial module ({{mathbb {K}}}) in ({{mathbb {T}}}_t) is a commutative algebra I, and the category (textbf{T}_t) consists of all free I-modules in ({{mathbb {T}}}_t). An essential novelty in our work is the explicit computation of Ext-spaces between simples in both categories (textbf{T}_t) and ({{mathbb {T}}}_t), which had been an open problem already for (t=0). This provides a direct link from the theory of universal tensor categories to Littlewood-Richardson-type combinatorics.

A celebrated theorem of Delange gives a sufficient condition for an arithmetic function to be the sum of the associated Ramanujan expansion with the coefficients provided by a previous result of Wintner. By applying the Delange theorem to the correlation of the von Mangoldt function with its incomplete form, we deduce an inequality involving the counting function of the prime numbers in arithmetic progressions. A remarkable aspect is that such an inequality is equivalent to the famous conjectural formula by Hardy and Littlewood for the twin primes.

Let (zeta (s)), (s={sigma }+it), be the Riemann zeta function. We use Fourier analysis to obtain, after a preliminary study of quadratic Riemann sums, a precise formula of the local integrals (int _n^{n+1} |zeta ({sigma }+it ) |^2 textrm{d}t), for (frac{1}{2}<{sigma }<1). We also study related (mathcal {S}^{2})-Stepanov norms of (zeta (s)) in connection with the strong Voronin Universality Theorem.