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Appendix: Deligne’s Fibre Functor

Nicholas M. Katz.

in Convolution and Equidistribution

January 2012; p ublished online October 2017 .

Chapter. Subjects: Number Theory. 2352 words.

This chapter takes up the proof of Theorem 3.1. It shows that N ↦ ω‎(N) := H⁰(𝔸¹/k¯, j 0! N) is a fiber functor on the Tannakian category Ƿsubscript geom of those...

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Applying Arakelov theory

Bas Edixhoven and Robin de Jong.

in Computational Aspects of Modular Forms and Galois Representations

June 2011; p ublished online October 2017 .

Chapter. Subjects: Number Theory. 4774 words.

This chapter starts applying Arakelov theory in order to derive a bound for the height of the coefficients of the polynomials Pno hexa conversion found as in (8.2.10). It proceeds in a few...

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Approximating <i>V<sub>f</sub> </i> over the complex numbers

Jean-Marc Couveignes.

in Computational Aspects of Modular Forms and Galois Representations

June 2011; p ublished online October 2017 .

Chapter. Subjects: Number Theory. 21287 words.

This chapter addresses the problem of computing torsion divisors on modular curves with an application to the explicit calculation of modular representations. The final result of the...

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Assumptions on the Schwartz Function

Xinyi Yuan, Shou-Wu Zhang and Wei Zhang.

in The Gross-Zagier Formula on Shimura Curves

December 2012; p ublished online October 2017 .

Chapter. Subjects: Number Theory. 4101 words.

This chapter introduces two classes of degenerate Schwartz functions which significantly simplify the computations and arguments of both the analytic kernel and the geometric kernel...

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Autodualities and Signs

Nicholas M. Katz.

in Convolution and Equidistribution

January 2012; p ublished online October 2017 .

Chapter. Subjects: Number Theory. 1310 words.

This chapter takes up the proofs of Theorems 9.1 and 9.2. Theorem 9.1: Suppose that N in G arith is geometrically irreducible, ι‎-pure of weight zero, and...

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Bounds for Arakelov invariants of modular curves

Bas Edixhoven and Robin de Jong.

in Computational Aspects of Modular Forms and Galois Representations

June 2011; p ublished online October 2017 .

Chapter. Subjects: Number Theory. 10879 words.

This chapter gives bounds for all quantities on the right-hand side in the inequality in Theorems 9.1.1 and 9.2.5, in the context of the modular curves X₁(5l) with l > 5 prime, using the...

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The Case of <i>SL</i>(2); the Examples of Evans and Rudnick

Nicholas M. Katz.

in Convolution and Equidistribution

January 2012; p ublished online October 2017 .

Chapter. Subjects: Number Theory. 3212 words.

This chapter treats both of these examples, as well as all the examples to come, using the Euler–Poincaré formula, cf. [Ray, Thm. 1] or [Ka-GKM, 2.3.1] or [Ka-SE, 4.6, (v) atop p. 113] or...

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Computational Aspects of Modular Forms and Galois Representations

Edited by Bas Edixhoven and Jean-Marc Couveignes.

June 2011; p ublished online October 2017 .

Book. Subjects: Number Theory. 440 pages.

Modular forms are tremendously important in various areas of mathematics, from number theory and algebraic geometry to combinatorics and lattices. Their Fourier coefficients, with...

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Computations with modular forms and Galois representations

Johan Bosman.

in Computational Aspects of Modular Forms and Galois Representations

June 2011; p ublished online October 2017 .

Chapter. Subjects: Number Theory. 8248 words.

This chapter discusses several aspects of the practical side of computing with modular forms and Galois representations. It starts by discussing computations with modular forms, and from...

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Computing coefficients of modular forms

Bas Edixhoven.

in Computational Aspects of Modular Forms and Galois Representations

June 2011; p ublished online October 2017 .

Chapter. Subjects: Number Theory. 5627 words.

This chapter applies the main result on the computation of Galois representations attached to modular forms of level one to the computation of coefficients of modular forms. It treats the...

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