Transactions of the American Mathematical Society, ISSN 0002-9947, 02/2019, Volume 371, Issue 2, pp. 1119 - 1149

We give an explicit description of fundamental domains associated with the p-adic uniformisation of families of Shimura curves of discriminant Dp and level N...

Shimura curves | Mumford curves | p-adic fundamental domains | MATHEMATICS | UNIFORMIZATION | CONSTRUCTION | POINTS

Shimura curves | Mumford curves | p-adic fundamental domains | MATHEMATICS | UNIFORMIZATION | CONSTRUCTION | POINTS

Journal Article

2013, Annals of mathematics studies, ISBN 9780691155913, Volume no. 184, viii, 256

This comprehensive account of the Gross-Zagier formula on Shimura curves over totally real fields relates the heights of Heegner points on abelian varieties to the derivatives of L-series...

Automorphic forms | Quaternions | Arithmetical algebraic geometry | Shimura varieties | Mathematics | PBH | Number Theory

Automorphic forms | Quaternions | Arithmetical algebraic geometry | Shimura varieties | Mathematics | PBH | Number Theory

Book

1992, Mathematical notes, ISBN 0691085595, Volume 40., xv, 427

Book

Communications in Contemporary Mathematics, ISSN 0219-1997, 03/2019, Volume 21, Issue 2, p. 1850009

We study Shimura curves of PEL type in A g generically contained in the Prym locus...

Shimura varieties | Prym locus | Prym varieties | MATHEMATICS | MATHEMATICS, APPLIED | DIMENSION | 2ND GAUSSIAN MAP | VARIETIES | MODULI SPACE | GENERIC TORELLI THEOREM

Shimura varieties | Prym locus | Prym varieties | MATHEMATICS | MATHEMATICS, APPLIED | DIMENSION | 2ND GAUSSIAN MAP | VARIETIES | MODULI SPACE | GENERIC TORELLI THEOREM

Journal Article

Journal of Algebra, ISSN 0021-8693, 01/2016, Volume 445, pp. 458 - 502

The Langlands Programme predicts that a weight 2 newform f over a number field K with integer Hecke eigenvalues generally should have an associated elliptic curve Ef over K. In [19...

Langlands Programme | Modular forms | Elliptic curves | QUADRATIC FIELDS | SHIMURA CURVES | ARITHMETIC GROUPS | VARIETIES | HECKE EIGENVALUES | DARMON POINTS | FORMS | MATHEMATICS | COHOMOLOGY | SYSTEMS | SYMBOLS | Algebra | Algorithms

Langlands Programme | Modular forms | Elliptic curves | QUADRATIC FIELDS | SHIMURA CURVES | ARITHMETIC GROUPS | VARIETIES | HECKE EIGENVALUES | DARMON POINTS | FORMS | MATHEMATICS | COHOMOLOGY | SYSTEMS | SYMBOLS | Algebra | Algorithms

Journal Article

Journal of Number Theory, ISSN 0022-314X, 04/2016, Volume 161, pp. 175 - 203

This paper is devoted to abelian varieties arising from generalized Legendre curves...

Hypergeometric functions | Galois representations | Triangle groups | Shimura curves | Abelian varieties | Counting points over finite fields | MATHEMATICS | FIELDS | REPRESENTATIONS | ATKIN | VALUES | FORMULAS | HYPERGEOMETRIC-FUNCTIONS | Algebra | Mathematics - Number Theory

Hypergeometric functions | Galois representations | Triangle groups | Shimura curves | Abelian varieties | Counting points over finite fields | MATHEMATICS | FIELDS | REPRESENTATIONS | ATKIN | VALUES | FORMULAS | HYPERGEOMETRIC-FUNCTIONS | Algebra | Mathematics - Number Theory

Journal Article

Journal of the Institute of Mathematics of Jussieu, ISSN 1474-7480, 11/2017, Volume 16, Issue 5, pp. 1075 - 1101

.... We then show that for modular and Shimura curves this ${\mathcal{L}}_{\unicode[STIX]{x1D714}_{1},\unicode[STIX]{x1D714}}$ -sentence has a unique model in every infinite cardinality...

open image theorems | categoricity | modular curves | Shimura varieties | MATHEMATICS | CONJECTURES | COVERS | Geometry | Mathematics

open image theorems | categoricity | modular curves | Shimura varieties | MATHEMATICS | CONJECTURES | COVERS | Geometry | Mathematics

Journal Article

Acta Arithmetica, ISSN 0065-1036, 2018, Volume 183, Issue 3, pp. 237 - 256

Journal Article

2004, CRM monograph series, ISBN 0821833596, Volume 22, xvi, 196

Book

Publicacions Matematiques, ISSN 0214-1493, 2018, Volume 62, Issue 2, pp. 355 - 396

We study Heegner points on elliptic curves, or more generally modular abelian varieties, coming from uniformization by Shimura curves attached to a rather general type of quaternionic orders...

Shimura curves | Heegner points | BSD conjecture | L-functions | ROOT NUMBERS | ADIC L-FUNCTIONS | SERIES | REPRESENTATIONS | MATHEMATICS | MODULAR-CURVES | JACOBIANS | UNIFORMIZATION | ORDERS | OPERATORS | Aritmètica | Classificació AMS | Arithmetical algebraic geometry | Geometria algèbrica | Anàlisi matemàtica | Matemàtiques i estadística | 11 Number theory | 11G Arithmetic algebraic geometry (Diophantine geometry) | Àrees temàtiques de la UPC

Shimura curves | Heegner points | BSD conjecture | L-functions | ROOT NUMBERS | ADIC L-FUNCTIONS | SERIES | REPRESENTATIONS | MATHEMATICS | MODULAR-CURVES | JACOBIANS | UNIFORMIZATION | ORDERS | OPERATORS | Aritmètica | Classificació AMS | Arithmetical algebraic geometry | Geometria algèbrica | Anàlisi matemàtica | Matemàtiques i estadística | 11 Number theory | 11G Arithmetic algebraic geometry (Diophantine geometry) | Àrees temàtiques de la UPC

Journal Article

Journal of number theory, ISSN 0022-314X, 2018, Volume 190, pp. 187 - 211

.... In particular, when f is a weight 2 modular form attached to an elliptic curve E/Q having multiplicative reduction at p, and p is ramified in K, we show an analogue of the exceptional zeroes...

Iwasawa theory | p-adic L-functions | Elliptic curves | MATHEMATICS | ABELIAN-VARIETIES | SHIMURA CURVES | VALUES | HEEGNER POINTS | MODULAR-FORMS | CONJECTURE

Iwasawa theory | p-adic L-functions | Elliptic curves | MATHEMATICS | ABELIAN-VARIETIES | SHIMURA CURVES | VALUES | HEEGNER POINTS | MODULAR-FORMS | CONJECTURE

Journal Article

Algebra and Number Theory, ISSN 1937-0652, 2012, Volume 6, Issue 3, pp. 455 - 485

In earlier work, the second named author described how to extract Darmon-style L-invariants from modular forms on Shimura curves that are special at p...

Shimura curves | L-invariants | Hida families | Stark-Heegner points | FORMS | MATHEMATICS | COHOMOLOGY

Shimura curves | L-invariants | Hida families | Stark-Heegner points | FORMS | MATHEMATICS | COHOMOLOGY

Journal Article

Journal of the Mathematical Society of Japan, ISSN 0025-5645, 2016, Volume 68, Issue 2, pp. 609 - 635

Journal Article

Mathematische Zeitschrift, ISSN 0025-5874, 2/2013, Volume 273, Issue 1, pp. 173 - 210

.... This paper considers the finite list of arithmetic (1; e)-groups. We define canonical models for the associated quotients by relating these to genus 1 Shimura curves...

Shimura curves | Arithmetic groups | Explicit methods | 14Q05 | Mathematics, general | Uniformization | Mathematics | Primary 11G18 | Secondary 14G35 | MATHEMATICS | REPRESENTATIONS | HILBERT MODULAR-FORMS | ORDERS | GENUS | FUCHSIAN-GROUPS | Algebraic Geometry

Shimura curves | Arithmetic groups | Explicit methods | 14Q05 | Mathematics, general | Uniformization | Mathematics | Primary 11G18 | Secondary 14G35 | MATHEMATICS | REPRESENTATIONS | HILBERT MODULAR-FORMS | ORDERS | GENUS | FUCHSIAN-GROUPS | Algebraic Geometry

Journal Article

Mathematische Zeitschrift, ISSN 0025-5874, 8/2012, Volume 271, Issue 3, pp. 757 - 779

... is not arithmetic the possible conjugators are rare and easy to classify. In the arithmetic case, i.e. for Shimura curves, the problem is much more complicated...

Shimura curves | Dessins d’enfants | Serre–Bruhat–Tits trees | Mathematics | 14G35 | 30F10 | 30F35 | Primary 20H10 | Secondary 11R52 | Congruence subgroups | Mathematics, general | Uniformization | Fuchsian groups | 20H10 | Serre-Bruhat-Tits trees | Dessins d'enfants | MATHEMATICS | SURFACES

Shimura curves | Dessins d’enfants | Serre–Bruhat–Tits trees | Mathematics | 14G35 | 30F10 | 30F35 | Primary 20H10 | Secondary 11R52 | Congruence subgroups | Mathematics, general | Uniformization | Fuchsian groups | 20H10 | Serre-Bruhat-Tits trees | Dessins d'enfants | MATHEMATICS | SURFACES

Journal Article

Designs, codes, and cryptography, ISSN 1573-7586, 2018, Volume 87, Issue 2-3, pp. 517 - 525

...}$$ F p 2 where p denotes a prime number $$\ge 5$$ ≥ 5 . In this aim, we use the symmetric Chudnovsky-type generalized algorithm applied on sufficiently dense families of modular curves defined over $$\mathbb {F}_{p^2...

Information and Communication, Circuits | Finite field | 14Q20 | Modular curve | Cryptology | Algorithm | Data Structures and Information Theory | Tensor rank | 11Y16 | Discrete Mathematics in Computer Science | Computer Science | 12Y05 | 14Q05 | Coding and Information Theory | Tower of function fields | Shimura curve | Algebraic function field | MATHEMATICS, APPLIED | COMPLEXITY | EXTENSION | COMPUTER SCIENCE, THEORY & METHODS | ALGEBRAIC FUNCTION-FIELDS | Rankings | Algorithms | Information management | Electric power transmission | Mathematics | Number Theory

Information and Communication, Circuits | Finite field | 14Q20 | Modular curve | Cryptology | Algorithm | Data Structures and Information Theory | Tensor rank | 11Y16 | Discrete Mathematics in Computer Science | Computer Science | 12Y05 | 14Q05 | Coding and Information Theory | Tower of function fields | Shimura curve | Algebraic function field | MATHEMATICS, APPLIED | COMPLEXITY | EXTENSION | COMPUTER SCIENCE, THEORY & METHODS | ALGEBRAIC FUNCTION-FIELDS | Rankings | Algorithms | Information management | Electric power transmission | Mathematics | Number Theory

Journal Article

2000, ISBN 9810243375, x, 361

Book

Compositio mathematica, ISSN 0010-437X, 12/2014, Volume 150, Issue 12, pp. 1963 - 2002

We consider two families of arithmetic divisors defined on integral models of Shimura curves...

Arakelov geometry | Shimura curves | modular forms | special cycles | Shimura lift | FORMS | MATHEMATICS | INTERSECTION THEORY | Theorems | Number theory | Integrals | Series (mathematics) | Mathematical models | Modular | Arithmetic | Lift

Arakelov geometry | Shimura curves | modular forms | special cycles | Shimura lift | FORMS | MATHEMATICS | INTERSECTION THEORY | Theorems | Number theory | Integrals | Series (mathematics) | Mathematical models | Modular | Arithmetic | Lift

Journal Article

Science China Mathematics, ISSN 1674-7283, 2010, Volume 53, Issue 3, pp. 573 - 592

... of the Gross-Zagier formula, Euler systems on Shimura curves, and rational points on elliptic curves.

Shimura curves | CM-points | L-series | MATHEMATICS | MATHEMATICS, APPLIED | CHARACTERS

Shimura curves | CM-points | L-series | MATHEMATICS | MATHEMATICS, APPLIED | CHARACTERS

Journal Article

Geometriae Dedicata, ISSN 0046-5755, 4/2016, Volume 181, Issue 1, pp. 177 - 192

... ′ is a smooth complex projective curve of genus $$g' \ge 1$$ g ′ ≥ 1 and $$g= g(C)$$ g = g ( C ) . We give the complete list of all such families that satisfy a simple...

Geometry | Torelli map | 14H15 | Moduli spaces of curves | Galois covers | Mathematics | 14H40 | Shimura varieties | 14G35 | 32G20 (primary) and 14K22 (secondary) | MATHEMATICS | MODULI SPACE | SURFACES

Geometry | Torelli map | 14H15 | Moduli spaces of curves | Galois covers | Mathematics | 14H40 | Shimura varieties | 14G35 | 32G20 (primary) and 14K22 (secondary) | MATHEMATICS | MODULI SPACE | SURFACES

Journal Article

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