Citations: Presses Universitaires de France - Serre, d'Arithm'etique (ResearchIndex)

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J.P. Serre: Cours d'arithmetique, Presses Universitaires de France, 1970. 137

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onl near Analvss. Theory, Methods Applicalions. 'qol I.. - On Discrete-Time.. (Correct)

....Marchesini, was unable to include this paper in the proceedings. 55 56 E.D. SONTAG AND Y. ROUCFIALEAU 1. ALGEBRAIC PRELIMINARIES We review now some notions from commutative algebra and algebraic geometry. The results can be found in any of the standard references; see, for instance, Dieudonn6 [1], Mumford [4] Macdonald [3] Although the literature usually treats only the case in which solutions are taken in algebraically closed fields, the generalization to the present setup (namely, arbitrary infinite fields) does not present any major difficulty. We shall therefore omit the proofs ....

DIEUDONNI J., Cours de Gomtrie ,417brique (2 vols), Presses Universitaires de France (1974).

Operations and Cooperations in Elliptic Cohomology, Part I.. - Baker (1995) (Correct)

....at infinity; of course we have S(K) k M(K) k . Thus there are two strictly commutative graded rings S(K) and M(K) with a homomorphism of graded rings S(K) Gamma M(K) The following classical result describes the structure of such rings. Elementary accounts of this result can be found in [25, 39]; for a discussion of rigidity under base change, see [22] Theorem 1.1. If 1=6 2 K, then as graded rings we have S(K) K[E 4 ; E 6 ] M(K) S(K) Delta ] K[E 4 ; E 6 ; Delta where E 2n 2 S(K) 2n M(K) 2n is the 2nth Eisenstein function and Delta = 1 1728 4 Gamma E ....

J-P. Serre, Cours d'Arithm'etique, Presses Universitaires de France, Paris, 1970.

Loc Merel - Oc Mer El (Correct)

....Y ] # SL 2 (Z) # k is equal to 0 if k 2 and is equal to the image of # if k = 2. Let # j## SL 2 (Z) P j [j] # C k 2 [X, Y ] # SL 2 (Z) # k # C k 2 [X, Y ] # SL 2 (Z) # k . We have P j # = P (j#) and P j # = P (j#) for all j # # SL 2 (Z) Since # and # generate SL 2 (Z) see [10]) we have P (jg) P j = P j g for all j # # SL 2 (Z) and all g # SL 2 (Z) This implies that P j is a constant polynomial, i.e. P = 0 or k = 2. If k = 2, then P j is a complex number independent of j (SL 2 (Z) operates transitively on # SL 2 (Z) So # j## SL 2 (Z) P j [j] is in the image ....

Serre J-P. Cours d'arithmetique. Presses Universitaires de France, 1970.

Symmetries On Manifolds, Deformations And Rational Homotopy: A.. - Raußen (Correct)

....Ass. 5.1. The results of Sullivan and Barge (cf. in particular, 3] Thm. 4 and Prop. 2.4.6) show: Proposition 7.4. 1) For n 6 1 mod 4, there are no surgery obstructions. 2) For n = 4k 1, there is a unique surgery obstruction, i.e. the torsion (N) 2 W (Q) the rational Witt ring, cf. [31]. Moreover, N) W ) where (W ) are the torsions of certain forms de ned on a quotient of H (W; F CP ) resp. on the Q[e] torsion module T orH (X [e] de ned in [27] The technically most dicult problem is to get hold on these torsion elements In brief: we ....

J. P. Serre, Cours d'arithmetique, Presses Universitaires de France, Paris, 1970.

An extension of Satoh's algorithm and its implementation - Fouquet, Gaudry, Harley (2000) (6 citations) (Correct)

....n . Clearly, once x n is known, one can recover x k for k n easily by projection down to Z=p k Z. We set = 1 . Note that can be considered as projecting to F p . 2 The invertible elements of Z p are those not in pZ p i.e. those with (x) 6= 0. For further details on p adic integers, see [Ser70, Chapter II]. 2.2 The ring Z q Let q = p d with p prime. Let f(t) be a monic polynomial in Z p [t] of degree d such that the polynomial (f) obtained by projecting the coefficients is irreducible in F p [t] Definition 2.2 The ring Z q is Z p [t] modulo (the ideal generated by) f(t) The following diagram ....

J. P. Serre. Cours d'arithm'etique. Presses universitaires de France, 1970.

A Family of Caps in Projective 4-Space in Odd Characteristic - Bierbrauer, Edel (1997) (Correct)

.... 8 : 5q 2 Gamma 2q Gamma 7) 2 if q j 1(mod 8) 5q 2 Gamma 8q Gamma 13) 2 if 3 q j 3(mod 8) 5q 2 Gamma 6q Gamma 11) 2 if q j 5(mod 8) 5q 2 Gamma 4q Gamma 9) 2 if q j 7(mod 8) It follows from one of the by laws of quadratic reciprocity (see for example [5]) that 2 is a square if q j Sigma1(mod 8) and 2 is a nonsquare if q j Sigma3(mod 8) We use homogeneous coordinates for PG(4; q) A typical point will be written as x = x 1 : x 2 : x 3 : x 4 : x 5 ) Denote by e i the vector which has x i = 1; x j = 0 for j 6= i: We wish to thank Prof. ....

J.P. Serre: Cours d'arithm'etique, Presses Universitaires de France 1970.

The Genus Of The Barnes-Wall Lattice - Scharlau, Venkov (1994) (1 citation) (Correct)

.... The mass of G is equal to M(G) 691 2 30 Delta 3 10 Delta 5 4 Delta 7 2 Delta 11 Delta 13 Delta 1 2 Delta 11 Delta 13 Delta 17 2 Delta 31 Delta 43 Delta 127: The first factor is the mass of even unimodular 16 dimensional lattices which can be looked up for instance in [Se], Chap. V, x2. The second factor is a correction factor coming from the determinant times the quotient of the 2 adic densities of the genera in question. It is readily derived e.g. from [Pf] Satz 1, Hilfssatz 8. The mass can also be obtained from [C S2] In the notation of that paper, our genus ....

J.-P. Serre: Cours d'Arithm`etique. Presses Universitaires de France (1970).

Double-Periodic Solutions Of Quasilinear Cauchy-Riemann Equations - Kuksin (Correct)

....: u(z) # (i#) J(u(z) u(z) # # # (z, #) # T#. 1. 2) We recall that each complex 2 torus is biholomorphically equivalent to a torus T 2 # = C (Z #Z) where # is the uniquely defined element of the fundamental domain for the action of the modular group on the complex upper half plane [13]. If z = x iy is the natural complex coordinate on T 2 # , then the relation (1.2) takes the form of the nonlinear Cauchy Riemann equation #u #x J(u) #u #y = 0, u # M, 1.3) since for # = i one has u # (i) #u #y, u # (i 2 ) u # (1) #u #x. In particular, if (M, J f ) ....

Serre J.-P., Cours d'arithmetique, Presses Universitaires de France, Paris, 1970.

Extremal Lattices - Scharlau, Schulze-Pillot (1998) (2 citations) (Correct)

....each occurring extremal lattice, including the well known ones. We shall use without further explanation a few basic notions and facts about lattices (some of them were already mentioned) The reader may consult [Que95] or [SchVen94] for these, and the books [O Me71] Kit93] Kne73] MiHu73] or [Ser70] for general background about integral quadratic forms. In [Ser70] the reader also finds an exposition of the basic theory of modular forms, including its application to the simplest case of lattices, namely those of level one. As a condensed introduction to modular forms including a basic stock ....

....shall use without further explanation a few basic notions and facts about lattices (some of them were already mentioned) The reader may consult [Que95] or [SchVen94] for these, and the books [O Me71] Kit93] Kne73] MiHu73] or [Ser70] for general background about integral quadratic forms. In [Ser70], the reader also finds an exposition of the basic theory of modular forms, including its application to the simplest case of lattices, namely those of level one. As a condensed introduction to modular forms including a basic stock of widely used explicit formulas we recommend [Sko92] ....

J.-P. Serre: Cours D'Arithmetique, Presses Universitaires de France, Paris 1970.

Some Problems in Elementary Number Theory and Modular Forms - Giuseppe Melfi (1998) (Correct)

....the full modular group yield well known arithmetical identities [43, p. 152] In order to find further identities, we require modular forms for certain congruence subgroups. Ample surveys of the theory of modular forms and functions can be found in literature. In particular we quote [20] 24] [43] and [44] for our purposes. In this chapter we mainly show how to construct suitable modular forms for certain congruence subgroups. These modular forms will be used in the next chapter to prove some remarkable arithmetical identities. 4.1 Modular forms and functions In this section we recall ....

.... = a 0;A = lim i1 (c d) Gamma2k f a b c d : Definition 9 If a 0;A = 0 for every A 2 Gamma; then f is called a cusp form for G: In other words, a cusp form is a modular form that vanishes at every finite cusp a=c 2 Q and at the cusp 1: For integers k 2; as is well known [43], the Eisenstein series E 2k ( 1 2 i(1 Gamma 2k) 1 X n=1 oe 2k Gamma1 (n)q n are modular forms of weight 2k for Gamma, and Delta( q 1 Y n=1 (1 Gamma q n ) 24 is a cusp form of weight 12 for Gamma: We shall also denote E 2k;m ( E 2k (m ) As we shall see, for k 2; ....

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J. P. Serre, Cours d'arithm'etique, 3 rd edition, Presses universitaires de France 1988.

Cyclic Group Actions on Manifolds From Deformations of Rational.. - Raußen (Correct)

....[e] from Ass. 2. The results of Sullivan and Barge (cf. in particular, 3] Thm. 4 and Prop. 2.4.6) show: Proposition 4.1 1. For n 6j 1 mod 4, there are no surgery obstructions. 2. For n = 4k 1, there is a unique surgery obstruction, i.e. the torsion (N) 2 W (Q) the rational Witt ring, cf. [20]. Moreover, N) W ) Gamma ( where (W ) are the torsions of certain forms defined on a quotient of H (W; F Theta CP ) resp. on T orH (X [e] that will be defined below. Barge [3] describes the quadratic form associated with a general Poincar ecomplex with boundary ....

.... , and let (z 1 ; z 2 ) y 1 [ z 2 ; X [e] with [X [e] 2 H 4k (X [e] F [e] Q a (chosen) fundamental class for X [e] Here and in the sequel, b [e] denotes inclusion X [e] X [e] F [e] To analyse the obstruction (N ) we make use of the split exact sequence (cf. [20]) 0 Z = W (Z) W (Q) ffi Gamma W (Q; Z) 0: of Witt groups; the splitting is given by the signature map oe : W (Q) Z. We can thus divide our analysis in two steps: 1. Find conditions that ensure that ffi (N) ffi ( 0 2 W (Q; Z) 2. Find conditions that ensure that oe (W ) ....

J.P. Serre, Cours d'arithm'etique, Presses Universitaires de France, Paris, 1970.

[13], Prop. 2.8, 2.11 and Lemma 3.1. The latter provides us.. - Theta Using (Correct)

....and Barge (cf. in particular, 3] Thm. 4 and Prop. 2.4.6) show: ACTIONS ON MANIFOLDS FROM DEFORMATIONS 11 Proposition 4.1. 1. For n 6j 1 mod 4, there are no surgery obstructions. 2. For n = 4k 1, there is a unique surgery obstruction, the torsion (a) 2 W (Q) the rational Witt ring, cf. [14]. Moreover, a) W ) Gamma ( with certain forms on the right hand side defined on a quotient of H (W; F Theta CP ) resp. on T orH (X [e] and to be explained below. For a general n manifold with boundary (Z; Z) Lefschetz duality gives rise to the map H (Z; Z) H (Z) ....

.... 2 2 T orH (X [e] choose y 1 2 H (X [e] F [e] with j (y 1 ) z 1 , and let (z 1 ; z 2 ) y 1 [z 2 ; X [e] with [X [e] 2 H 4k (X [e] F [e] Q a (chosen) fundamental class for X [e] To analyse the obstruction (a) we make use of the split exact sequence (cf. [14]) 0 Z = W (Z) W (Q) ffi Gamma W (Q; Z) 0: 4.1) where the splitting is given by the signature map oe : W (Q) Z: Since (W; F Theta CP ) is a manifold with boundary, ffi( W ) 0. We can thus divide our analysis in two steps: 1. Find conditions that ensure that ffi ( 0 2 ....

J. P. Serre, Cours d'arithm'etique, Presses Universitaires de France, Paris, 1970.

On Some Modular Identities - Melfi (Correct)

.... Gamma k) 1 9 oe 3 (n) for every n j 2 mod 3: In a celebrated paper [6] Ramanujan, using elementary arguments, proved nine identities of the type (1) with m = 1: Ramanujan s nine identities can be also obtained in a natural way from the theory of modular forms for the full modular group (see [9]) A short elementary proof of Ramanujan s identities is due to Skoruppa [12] We remark that one of the formulae we prove in Theorem 2, namely (10) below, is explicitly mentioned by Ramanujan himself in [6] Unfortunately, he never provided either of the two proofs he announced. The first proof ....

J. P. Serre, Cours d'arithm'etique, Presses Universitaires de France, Paris, 1970.

Finite Geometries - Ma Jurgen Bierbrauer (Correct)

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J.P. Serre: Cours d'arithmetique, Presses Universitaires de France, 1970. 137

Sign of the monodromy for Liouville integrable systems - Cushman, San (Correct)

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J.-P. Serre, Cours d'arithmtique, Presses Universitaires de France, 1970.

Hecke Algebras Acting on Elliptic Cohomology - Baker (1 citation) (Correct)

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J-P. Serre, Cours d'Arithmetique, Presses Universitaires de France, 1977.

The Weight Distribution of C_5(1,n) - Lam, Sica (2001) (Correct)

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J.-P. Serre, Cours d'arithmetique, Presses Universitaires de France (1988.

Finite Geometries - Bierbrauer (2002) (Correct)

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J.P. Serre: Cours d'arithm'etique, Presses Universitaires de France, 1970. 105

Affine Rings - Nielsen (Correct)