48 Normat 58:1, 48 (2010)

Summary in English

Loren D. Olson, Abelprisen 2010 -

John Tate. (Norwegian).

In connection with the awarding of

the Abel Prize to John Tate, a short

presentation of his work is presented

along with some personal reminiscences

of Tate by the author. The emphasis

is put on Tate’s work on elliptic cur-

ves, especially what relates to Tate-

Shafarevic groups, L-series and their

connections to Mordell-Weil and the

Birch/Swinnerton-Dyer conjecture.

L. Gårding and P-A. Ivert, Antalet

egenvärden under en stor gräns (Swe-

dish)

The purpose is to give an elementa-

ry heuristic proof of a formula for the

number of eigenvalues of a vibrating

membrane in R

n

described by an ellip-

tic linear diﬀerential equation. The for-

mula, that goes back to Herman Weyl

and has been developed further by Cou-

rant, Carleman and Hörmander, says

that asymptotically the number N (λ) of

eigenvalues below λ is given by

(2π)

−n

ZZ

x∈Bp(x,ξ)<λ

dxdξ

where B is the region formed by the

membrane and p(x, ξ) is the characte-

ristic equation of the principal part of

the relevant diﬀerential operator. The

key idea (going back to Weyl and known

as maximum-minimum principle) is to

consider the energy of the vibrating

membranes, which can be expressed as

a quadratic form E(u, u) of the derivati-

ves of u and consider the successive mi-

nima of the quotients E(u, u)/(u, u) on

decreasing sequences of closed subspa-

ces deﬁned inductively and giving ri-

se to the eigenfunctions. More precise-

ly the successive minima will constitute

the eigenvalues, and the decreasing se-

quence of closed subspaces will be for-

med by the orthogonal complements to

the increasing sequences of subspaces

spanned by the eigenvectors.

The argument is carried through

for operators with constant coeﬃcients,

and the general case is indicated by so-

me handwaving.

John K. Dagsvik, Elliptiske Integra-

ler I (Norwegian).

The aim of the article is to give an

elementary proof of Abel’s generaliza-

tion of additions theorems for elliptic

integrals. In addition an elementary di-

scussion of the preceding achievements

of especially Euler, Legendre (the ﬁrst

to study such integrals systematically)

and Lagrange is presented. This invol-

ves recalling standard examples of el-

liptic integrals such as the computation

of arc-lengths of ellipses and lemnisca-

tes as well as calculating the period of a

pendulum.