Notes on gross-zagier formula pdf

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WebGross-Zagier-Zhang formula over function elds B: incoherent quaternion algebra over A F, i.e. division at odd number of places. B 1is division. Fix A E,!B. M U=F: Drinfeld modular curves, indexed by U ˆB . Hecke action: B acts on lim U M . A=F: abelian variety such that (up to twist by a character) ˇ A:= lim! U Hom(J(M U);A) C = JL B (˙): t ... WebGROSS-ZAGIER REVISITED BRIAN CONRAD Contents 1. Introduction 1 2. Some properties of abelian schemes and modular curves 3 3. The Serre-Tate theorem and the Grothendieck …

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WebWith this setup, Gross and Zagier proved: Theorem: L ′ ( E / K, 1) = h ^ ( y K) ∬ E ( C) ω ∧ i ω D, where ω is the invariant differential on E (suitably normalized), and D is the discriminant … WebThe goal of the Gross-Zagier formula is to express L0(1;E=K) in terms of these previously de ned height functions. 5 The Gross-Zagier Formula and Applications Theorem 5.1. With all … reading music festival logo

riemannian geometry - Gross-Zagier formulae outside of number …

WebCUBE SUM PROBLEM AND AN EXPLICIT GROSS-ZAGIER FORMULA LI CAI, JIE SHU, AND YE TIAN Contents 1. Introduction and Main Results 1 2. Nontriviality of Heegner Points 6 3. An Explicit Gross-Zagier Formula 12 ... Note that the torsion part of the Mordell-Weil group C(n)(Q) is trivial unless 2n is a cube, which is not a cube sum Web2 Gross-Zagier Subgroups In this section, we x our notation and conventions, and de ne the Manin constant. Then we recall the statement of the full Birch and Swinnerton-Dyer conjecture over an imaginary quadratic eld K. We give a new de nition of Gross-Zagier subgroups of E(K) and prove that they all satisfy a Gross-Zagier style formula. When r ... WebIn this notes, we are trying to give an intersection formula of local heights for nonarchimedean places. It closely follows ﬁrst six sections of part III in Gross-Zagier[3]. If there is any mistake in this notes, it is due to the author’s limited ... Gross-Zagierrevisited. InHeegner points and Rankin L-series, volume49ofMath. Sci. Res. Inst ... reading museum bayeux tapestry tour

The Gross-Zagier formula: a brief introduction

WebGross–Zagier Formula for GL(2), II SHOU-WU ZHANG Contents 1. Introduction and Notation 191 2. Automorphic Forms 195 3. Weights and Levels 197 4. Automorphic L-Series 199 5. … WebThis general formula applies to arbitrary cuspidal automorphic representations of GL2over totally real ﬁelds holomorphic of weight 2. In this paper, over global function ﬁelds of … reading museum shopWeb0(N), Gross–Kohnen–Zagier prove [2] that certain generating series of Heegner points are modular forms of weight 3/2 with values in Jacobian as a consequence of their formula for N´eron-Tate height pairing of Heegner points. Such a result is … reading museum events

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Notes on gross-zagier formula pdf

WebThe purpose of this paper is to discuss some work on elliptic curves over function ﬁelds inspired by the Gross–Zagier theorem and to present new ideas about ranks of elliptic curves from the function ﬁeld case which I hope will inspire work over number ﬁelds. Weba higher analog of the formula of Gross and Zagier (see Corollary 0.3.2 of [Z]), and applying our argument below to this situation, one gets a variant of Theorem A determining (the motive of) f by the heights of the f-components of (homologically trivial) Heegner cycles, assuming one of them has a non-trivial height.

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WebToday we will define the objects on the two sides of the Gross–Zagier formula (one side is in terms of L-series of modular forms, the other side is in terms of Heegner points on X 0(N)). Then we can state the main theorem precisely. 1Heegner points Let x= (ϕ: E→E′) be an isogeny of elliptic curves over C, cyclic of degree N. By abuse we http://davidrenshawhansen.com/gztalk.pdf

WebGross-Zagier formula computes L!(1,E /K) as the value of a height function. Given a ﬁnite extension k/Q,letM k be the set of all places ofk and let · v be the corresponding … Webfor the entire study is a p-adic formula of Gross-Zagier type which relates the images of these diagonal cycles under the p-adic Abel-Jacobi map to special values of certain p-adic L-functions attached to the Garrett-Rankin triple convolution of three Hida families of modular forms. The main goal of this article is to describe and prove this ...

Webthe methods of Gross and Zagier and give a computable formula for v ‘(J(d 1;d 2)) for any pair of discriminants d 1 6= d 2 and any prime ‘ > 2. In the case that d 1 is squarefree and d 2 is the discriminant of any quadratic imaginary order, our formula can be stated in a simple closed form. We also give a conjectural closed formula when the ... WebSep 1, 2024 · The original proof of (1.3) given by Gross and Zagier [12] relies on deep connections between the modular -invariant and elliptic curves. In recent work [21], the author of the present work...

Webfor the entire study is a p-adic formula of Gross-Zagier type which relates the images of these diagonal cycles under the p-adic Abel-Jacobi map to special values of certain p-adic L-functions attached to the Garrett-Rankintriple convolution of three Hida families of modular forms. The main goal of this article is to describe and prove this ... how to subtract percentageWebSuch a non-critical slope result is precisely Theorem 5.1.1 (p-adic Gross-Zagier formula for non-ordinary eigenforms of arbitrary weight; which is work in progress by Kobayashi [Kob19]). More precisely, Kobayashi’s method only establishes a p-adic Gross{Zagier formula in the non-ordinary case for one of the two p-stabilization how to subtract polynomials calculatorWebJun 21, 2005 · A new proof of Howard’s Λ-adic Gross–Zagier formula is given, via Iwasawa theory, based on the connection between Heegner points, Beilinson–Flach elements, and their explicit reciprocity laws, to the context of indefinite Shimura curves over Q attached to nonsplit quaternion algebras. 32 PDF View 2 excerpts, cites background and methods ... 1 … how to subtract polynomials step by stephttp://library.msri.org/books/Book49/files/11ulmer.pdf how to subtract rational equationsWebtion formula. Theorem A. For any unramiﬁed cuspidal automorphic representation π, and for any even r≥0, h[Shtr T1] π,[Sht r T2] i= Cr(π). Theorem A is the function ﬁeld analogue of the Gross-Kohnen-Zagier formula [5, Theorem B], but for higher order derivatives. The original Gross-Kohnen-Zagier formula corresponds to the case r= 1 over ... reading museum silchester galleryhttp://math.columbia.edu/~goldfeld/GaussProblem.pdf how to subtract rows in rWebfollowing Zagier’s computations, we will instead consider the curve 61.a1 and do all the computations from scratch. The notes are structured around the computation of this … how to subtract rationals