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Discriminants, resultants, and multidimensional determinants Gelfand, I. M. ; Kapranov, M. M. ; Zelevinsky , A. V. | Birkhäuser 1994

Ouvrage

V

- 523 p.
ISBN 978-0-8176-3660-9

Mathematics

Localisation : Ouvrage RdC (GELF)

A-discriminant # A-résultant # déterminant multidimensionnel # hyperdéterminant # polytope # variété de Chow

11R29 ; 14C05 ; 15A15 ; 51M20

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Class field theory :
from theory to practice
Gras, Georges | Springer 2003

Ouvrage

V

- 491 p.
ISBN 978-3-540-44133-5

Springer monographs in mathematics

Localisation : Ouvrage RdC (GRAS)

théorie du corps de classe # idèle # corps de classe de rayon # morphisme d'Artin # symbole de Hasse # symbole de norme résiduelle # théorie des genres

11Rxx ; 11R29 ; 11R70 ; 11R34 ; 11S31 ; 11Y40

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Collected papers of Taro Morishima Morishima, Taro ; Karomatsu, Y. | Queen'S University 1990

Ouvrage

V

- 227 p.

Queen's papers in pure and applied mathematics , 0084

Localisation : Oeuvres complètes RdC (MORI)

classe d'ideaux # conjecture de Fermat # corps de classe # nombre de classe # théorie des corps cyclotomiques # théorème de Fermat # Morishima # oeuvres complètes

11Axx ; 11R29 ; 11T22 ; 13A15

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Computational algebraic theory Pohst, Michael E. | Birkhäuser 1993

Ouvrage

V

- 88 p.
ISBN 978-3-7643-2913-6

DMV Seminar , 0021

Localisation : Séminaire RdC

théorie algébrique des nombres # polynôme # géométrie des nombres # calcul en groupe d'unité # calcul en groupe de classe # factorisation de Pollard

11Y40 ; 11-02 ; 11Rxx ; 11A41 ; 11H06 ; 11R27 ; 11R29

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Die berechnung der klassenzahl abelscher körper über quadratischen zahlkörpern Meyer, Curt | Akademie Verlag 1957

Ouvrage

V

- 132 p.

Mathematische lehrbücher und monographien , 0005

Localisation : Ouvrage RdC (MEYE)

calcul du nombre de classe # classe d'anneau # classe de rayon # corps abélien # corps de nombre quadratique # fonction l # forme de nombre de classe analytique # forme limite de Kronecker # loi de formation arithmétique et comportement analytique des # sommation des séries l # théorie algébrique des nombres # théorie des nombres # type de corps et structure

11-XX ; 11R29 ; 11R42 ; 11Rxx

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Stark's conjectures :
recent work and new directions#an international conference on... held at Johns Hopkins University#Aug. 5-9
Burns, David ; Popescu, Cristian ; Sands, Jonathan ; Solomon, David | American Mathematical Society 2004

Congrès

V

- 221 p.
ISBN 978-0-8218-3480-0

Contemporary mathematics , 0358

Localisation : Collection 1er étage

théorie des nombres # conjecture de Stark # L-fonction # théorie d'Isawara

11-06 ; 11G40 ; 11R23 ; 11R27 ; 11R29 ; 11R33 ; 11R42 ; 11S40 ; 11Y40

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Class number statistics for imaginary quadratic fields Kurlberg, Pär | CIRM H

Multi angle

y

Research talks

The number $F(h)$ of imaginary quadratic fields with class number $h$ is of classical interest: Gauss’ class number problem asks for a determination of those fields counted by $F(h)$. The unconditional computation of $F(h)$ for $h \le 100$ was completed by M. Watkins, and K. Soundararajan has more recently made conjectures about the order of magnitude of $F(h)$ as $h \to \infty$ and determined its average order.
For odd $h$ we refine Soundararajan’s conjecture to a conjectural asymptotic formula and also consider the subtler problem of determining the number $F(G)$ of imaginary quadratic fields with class group isomorphic to a given finite abelian group $G$.
Using Watkins’ tables, one can show that some abelian groups do not occur as the class group of any imaginary quadratic field (for instance $(\mathbb{Z}/3\mathbb{Z})^3$ does not). This observation is explained in part by the Cohen-Lenstra heuristics, which have often been used to study the distribution of the p-part of an imaginary quadratic class group. We combine the Cohen-Lenstra heuristics with a refinement of Soundararajan’s conjecture to make precise predictions about the asymptotic nature of the entire imaginary quadratic class group, in particular addressing the above-mentioned phenomenon of “missing” class groups, for families of $p$-groups as $p$ tends to infinity. For instance, it appears that no groups of the form $(\mathbb{Z}/p\mathbb{Z})^3$ and $p$ prime occurs as a class group of a quadratic imaginary field.
Conditionally on the Generalized Riemann Hypothesis, we extend Watkins’ data, tabulating $F(h)$ for odd $h \le 10^6$ and $F(G)$ for $G$ a $p$-group of odd order with $|G| \le 10^6$. (To do this, we examine the class numbers of all negative prime fundamental discriminants $-q$, for $q \le 1.1881 \cdot 10^{15}.$) The numerical evidence matches quite well with our conjectures.
This is joint work with S. Holmin, N. Jones, C. McLeman, and K. Petersen.
The number $F(h)$ of imaginary quadratic fields with class number $h$ is of classical interest: Gauss’ class number problem asks for a determination of those fields counted by $F(h)$. The unconditional computation of $F(h)$ for $h \le 100$ was completed by M. Watkins, and K. Soundararajan has more recently made conjectures about the order of magnitude of $F(h)$ as $h \to \infty$ and determined its average order.
For odd $h$ we refine So...

11R11 ; 11R29

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Congruent number problem and BSD conjecture Zhang, Shou-Wu | CIRM H

Multi angle

y

Research talks

A thousand years old problem is to determine when a square free integer $n$ is a congruent number ,i,e, the areas of right angled triangles with sides of rational lengths. This problem has a some beautiful connection with the BSD conjecture for elliptic curves $E_n : ny^2 = x^3 - x$. In fact by BSD, all $n= 5, 6, 7$ mod $8$ should be congruent numbers, and most of $n=1, 2, 3$ mod $8$ should not be congruent numbers. Recently, Alex Smith has proved that at least 41.9% of $n=1,2,3$ satisfy (refined) BSD in rank $0$, and at least 55.9% of $n=5,6,7$ mod $8$ satisfy (weak) BSD in rank $1$. This implies in particular that at last 41.9% of $n=1,2,3$ mod $8$ are not congruent numbers, and 55.9% of $n=5, 6, 7$ mod $8$ are congruent numbers. I will explain the ingredients used in Smith's proof: including the classical work of Heath-Brown and Monsky on the distribution F_2 rank of Selmer group of E_n, the complex formula for central value and derivative of L-fucntions of Waldspurger and Gross-Zagier and their extension by Yuan-Zhang-Zhang, and their mod 2 version by Tian-Yuan-Zhang. A thousand years old problem is to determine when a square free integer $n$ is a congruent number ,i,e, the areas of right angled triangles with sides of rational lengths. This problem has a some beautiful connection with the BSD conjecture for elliptic curves $E_n : ny^2 = x^3 - x$. In fact by BSD, all $n= 5, 6, 7$ mod $8$ should be congruent numbers, and most of $n=1, 2, 3$ mod $8$ should not be congruent numbers. Recently, Alex Smith has ...

11G40 ; 11D25 ; 11R29

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