Chapter XI: The Arithmetic Riemann–Roch Theorem and the Jacquet–Langlands Correspondence

2020 ◽  
pp. 403-432
Author(s):  
Gerard Freixas i Montplet
Keyword(s):  
Langlands Correspondence ◽  
Roch Theorem

THE JACQUET–LANGLANDS CORRESPONDENCE AND THE ARITHMETIC RIEMANN–ROCH THEOREM FOR POINTED CURVES

2012 ◽  
Vol 08 (01) ◽  
pp. 1-29 ◽  
Author(s):  
GERARD FREIXAS I MONTPLET
Keyword(s):  
Modular Forms ◽  
Shimura Curves ◽  
Intersection Numbers ◽  
Shimura Curve ◽  
Langlands Correspondence ◽  
Modular Curve ◽  
Roch Theorem ◽  
Petersson Norms

We show how the Jacquet–Langlands correspondence and the arithmetic Riemann–Roch theorem for pointed curves, relate the arithmetic self-intersection numbers of the sheaves of modular forms — with their Petersson norms — on modular and Shimura curves: these are equal modulo ∑l∈S ℚ log l, where S is a controlled set of primes. These quantities were previously considered by Bost and Kühn (modular curve case) and Kudla–Rapoport–Yang and Maillot–Roessler (Shimura curve case). By the work of Maillot and Roessler, our result settles a question raised by Soulé.

A torsion Jacquet-Langlands correspondence

Astérisque ◽  
2019 ◽  
Vol 409 ◽  
pp. 1-226 ◽  
Author(s):  
Frank CALEGARI ◽  
Akshay VENKATESH
Keyword(s):  
Langlands Correspondence

scholarly journals A Riemann–Roch theorem

2006 ◽  
Vol 301 (1) ◽  
pp. 148-164 ◽  
Author(s):  
Mrinal Kanti Das ◽  
Satya Mandal
Keyword(s):  
Roch Theorem

scholarly journals On the Notion of Conductor in the Local Geometric Langlands Correspondence

2017 ◽  
Vol 69 (1) ◽  
pp. 107-129
Author(s):  
Masoud Kamgarpour
Keyword(s):  
Irreducible Representation ◽  
Galois Representation ◽  
Loop Group ◽  
Langlands Correspondence ◽  
Categorical Representation ◽  
Swan Conductor ◽  
Geometric Langlands ◽  
Meromorphic Connection ◽  
Local Langlands Correspondence ◽  
Geometric Analogue

AbstractUnder the local Langlands correspondence, the conductor of an irreducible representation of Gln(F) is greater than the Swan conductor of the corresponding Galois representation. In this paper, we establish the geometric analogue of this statement by showing that the conductor of a categorical representation of the loop group is greater than the irregularity of the corresponding meromorphic connection.

scholarly journals Cohomology and the Riemann-Roch Theorem

1953 ◽  
Vol 39 (7) ◽  
pp. 660-669
Author(s):  
D. C. Spencer
Keyword(s):  
Roch Theorem
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