Arithmetic and geometry: ten years in Alpbach
Arithmetic and Geometry presents highlights of recent work in arithmetic algebraic geometry by some of the world's leading mathematicians. Together, these 2016 lectures—which were delivered in celebration of the tenth anniversary of the annual summer workshops in Alpbach, Austria—provide an int...
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| Other Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Princeton and Oxford
Princeton University Press
2019
|
| Series: | Annals of Mathematics Studies
202 |
| Subjects: | |
| Summary: | Arithmetic and Geometry presents highlights of recent work in arithmetic algebraic geometry by some of the world's leading mathematicians. Together, these 2016 lectures—which were delivered in celebration of the tenth anniversary of the annual summer workshops in Alpbach, Austria—provide an introduction to high-level research on three topics: Shimura varieties, hyperelliptic continued fractions and generalized Jacobians, and Faltings height and L-functions. The book consists of notes, written by young researchers, on three sets of lectures or minicourses given at Alpbach.The first course, taught by Peter Scholze, contains his recent results dealing with the local Langlands conjecture. The fundamental question is whether for a given datum there exists a so-called local Shimura variety. In some cases, they exist in the category of rigid analytic spaces; in others, one has to use Scholze's perfectoid spaces.The second course, taught by Umberto Zannier, addresses the famous Pell equation—not in the classical setting but rather with the so-called polynomial Pell equation, where the integers are replaced by polynomials in one variable with complex coefficients, which leads to the study of hyperelliptic continued fractions and generalized Jacobians.The third course, taught by Shou-Wu Zhang, originates in the Chowla–Selberg formula, which was taken up by Gross and Zagier to relate values of the L-function for elliptic curves with the height of Heegner points on the curves. Zhang, X. Yuan, and Wei Zhang prove the Gross–Zagier formula on Shimura curves and verify the Colmez conjecture on average |
| Physical Description: | viii, 174 Seiten ein Diagramm |
| ISBN: | 9780691193786 9780691193779 |
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| 520 | |a Arithmetic and Geometry presents highlights of recent work in arithmetic algebraic geometry by some of the world's leading mathematicians. Together, these 2016 lectures—which were delivered in celebration of the tenth anniversary of the annual summer workshops in Alpbach, Austria—provide an introduction to high-level research on three topics: Shimura varieties, hyperelliptic continued fractions and generalized Jacobians, and Faltings height and L-functions. The book consists of notes, written by young researchers, on three sets of lectures or minicourses given at Alpbach.The first course, taught by Peter Scholze, contains his recent results dealing with the local Langlands conjecture. The fundamental question is whether for a given datum there exists a so-called local Shimura variety. In some cases, they exist in the category of rigid analytic spaces; in others, one has to use Scholze's perfectoid spaces.The second course, taught by Umberto Zannier, addresses the famous Pell equation—not in the classical setting but rather with the so-called polynomial Pell equation, where the integers are replaced by polynomials in one variable with complex coefficients, which leads to the study of hyperelliptic continued fractions and generalized Jacobians.The third course, taught by Shou-Wu Zhang, originates in the Chowla–Selberg formula, which was taken up by Gross and Zagier to relate values of the L-function for elliptic curves with the height of Heegner points on the curves. Zhang, X. Yuan, and Wei Zhang prove the Gross–Zagier formula on Shimura curves and verify the Colmez conjecture on average | ||
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| illustrated | Not Illustrated |
| indexdate | 2024-07-10T08:40:07Z |
| institution | BVB |
| isbn | 9780691193786 9780691193779 |
| language | English |
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| physical | viii, 174 Seiten ein Diagramm |
| publishDate | 2019 |
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| publisher | Princeton University Press |
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| series | Annals of Mathematics Studies |
| series2 | Annals of Mathematics Studies |
| spelling | Arithmetic and geometry ten years in Alpbach edited by Gisbert Wüstholz and Clemens Fuchs Princeton and Oxford Princeton University Press 2019 © 2019 viii, 174 Seiten ein Diagramm txt rdacontent n rdamedia nc rdacarrier Annals of Mathematics Studies 202 Arithmetic and Geometry presents highlights of recent work in arithmetic algebraic geometry by some of the world's leading mathematicians. Together, these 2016 lectures—which were delivered in celebration of the tenth anniversary of the annual summer workshops in Alpbach, Austria—provide an introduction to high-level research on three topics: Shimura varieties, hyperelliptic continued fractions and generalized Jacobians, and Faltings height and L-functions. The book consists of notes, written by young researchers, on three sets of lectures or minicourses given at Alpbach.The first course, taught by Peter Scholze, contains his recent results dealing with the local Langlands conjecture. The fundamental question is whether for a given datum there exists a so-called local Shimura variety. In some cases, they exist in the category of rigid analytic spaces; in others, one has to use Scholze's perfectoid spaces.The second course, taught by Umberto Zannier, addresses the famous Pell equation—not in the classical setting but rather with the so-called polynomial Pell equation, where the integers are replaced by polynomials in one variable with complex coefficients, which leads to the study of hyperelliptic continued fractions and generalized Jacobians.The third course, taught by Shou-Wu Zhang, originates in the Chowla–Selberg formula, which was taken up by Gross and Zagier to relate values of the L-function for elliptic curves with the height of Heegner points on the curves. Zhang, X. Yuan, and Wei Zhang prove the Gross–Zagier formula on Shimura curves and verify the Colmez conjecture on average MATHEMATICS / Arithmetic bisacsh Arithmetical algebraic geometry Congresses Wüstholz, Gisbert 1948- (DE-588)129338710 edt Fuchs, Clemens 1976- (DE-588)1045504378 edt Erscheint auch als Online-Ausgabe 978-0-691-19754-8 Annals of Mathematics Studies 202 (DE-604)BV000000991 202 |
| spellingShingle | Arithmetic and geometry ten years in Alpbach Annals of Mathematics Studies MATHEMATICS / Arithmetic bisacsh Arithmetical algebraic geometry Congresses |
| title | Arithmetic and geometry ten years in Alpbach |
| title_auth | Arithmetic and geometry ten years in Alpbach |
| title_exact_search | Arithmetic and geometry ten years in Alpbach |
| title_full | Arithmetic and geometry ten years in Alpbach edited by Gisbert Wüstholz and Clemens Fuchs |
| title_fullStr | Arithmetic and geometry ten years in Alpbach edited by Gisbert Wüstholz and Clemens Fuchs |
| title_full_unstemmed | Arithmetic and geometry ten years in Alpbach edited by Gisbert Wüstholz and Clemens Fuchs |
| title_short | Arithmetic and geometry |
| title_sort | arithmetic and geometry ten years in alpbach |
| title_sub | ten years in Alpbach |
| topic | MATHEMATICS / Arithmetic bisacsh Arithmetical algebraic geometry Congresses |
| topic_facet | MATHEMATICS / Arithmetic Arithmetical algebraic geometry Congresses |
| volume_link | (DE-604)BV000000991 |
| work_keys_str_mv | AT wustholzgisbert arithmeticandgeometrytenyearsinalpbach AT fuchsclemens arithmeticandgeometrytenyearsinalpbach |