Some cases of Kudla's modularity conjecture for unitary Shimura varieties

Xia, Jiacheng

doi:10.48550/arxiv.2101.06304

Cited by 3 publications

(3 citation statements)

References 27 publications

(52 reference statements)

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“…(iv) Conjecture 3.5.4 in the unitary case was formulated by Liu [Liu11a], who also proved the case n = 1 and reduce the n > 1 case to the convergence. Recently Xia [Xia21] proved the desired convergence when E = Q( √ −d) for d = 1, 2, 3, 7, 11, and thus established Conjecture 3.5.4 in these cases.…”

Section: O Omentioning

confidence: 92%

From sum of two squares to arithmetic Siegel-Weil formulas

Li¹

2021

Preprint

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The main goal of this expository article is to survey recent progress on the arithmetic Siegel-Weil formula and its applications. We begin with the classical sum of two squares problem and put it in the context of the Siegel-Weil formula. We then motivate the geometric and arithmetic Siegel-Weil formula using the classical example of the product of modular curves. After explaining the recent result on the arithmetic Siegel-Weil formula for Shimura varieties of arbitrary dimension, we discuss some aspects of the proof and its application to the arithmetic inner product formula and the Beilinson-Bloch conjecture. Rather than intended to be a complete survey of this vast field, this article focuses more on examples and background to provide easier access to the recent works [LZ, LZ21, LL, LL21]. Contents 1. Sum of two squares 1 2. Siegel-Weil formula 5 3. Geometric Siegel-Weil formula 11 4. Arithmetic Siegel-Weil formula 18 5. Local arithmetic Siegel-Weil formula 21 6. Arithmetic inner product formula 28 References 341. Sum of two squares 1.1. Which prime p can be written as the sum of two squares? For the first few primes we easily find that 5 = 1 2 + 2 2 , 13 = 2 2 + 3 2 , 17 = 1 2 + 4 2 , 29 = 2 2 + 5 2 are sum of two squares, while other primes like 3, 7, 11, 19, 23 are not. The answer seems to depend on the residue class of p modulo 4.Theorem 1.1.1. A prime p = 2 is the sum of two squares if and only if p ≡ 1 (mod 4).Theorem 1.1.1 is usually attributed to Fermat, and appeared in his letter to Mersenne dated Dec 25, 1640 (hence the name Fermat's Christmas Theorem), although the statement can already be found in the work of Girard in 1625. The "only if" direction is obvious, but the "if" direction is far from trivial. Fermat claimed that he had an irrefutable proof, but nobody was able to find the complete proof among his work -apparently the margin was often too narrow for Fermat. The Date: October 15, 2021. The author would like to thank Benedict Gross, Yifeng Liu, Murilo Zanarella and Wei Zhang for helpful comments.

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Section: O Omentioning

confidence: 92%

From sum of two squares to arithmetic Siegel-Weil formulas

Li¹

2021

Preprint

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“…Therefore we give a generalization of their work. On the other hand, Xia [15] showed the Liu's result, not assuming the absolute convergence of the generating series. He uses the formal Fourier-Jacobi series method similar to the work over Q of Bruinier-Westerholt-Raum [4].…”

Section: Introductionmentioning

confidence: 97%

“…The author would like to thank to his advisor, Tetsushi Ito, for useful discussions and warm encouragement. I would also like to thank Jiacheng Xia for letting me know his paper [15] and suggesting stating Corollary 1.13 explicitly.…”

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confidence: 99%

Modularity of special cycles on unitary Shimura varieties over CM-fields

Maeda

2021

Preprint

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We study the modularity of the generating series of special cycles on unitary Shimura varieties over CM-fields of degree 2d associated with a Hermitian form in n `1 variables whose signature is pn, 1q at e real places and pn `1, 0q at the remaining d ´e real places for 1 ď e ă d. For e " 1, Liu proved the modularity and Xia showed the absolute convergence of the generating series. On the other hand, Bruinier constructed regularized theta lifts on orthogonal groups over totally real fields and proved the modularity of special divisors on orthogonal Shimura varieties. By using Bruinier's result, we work on the problem for e " 1 and give an another proof of Liu's proof. For e ą 1, we prove that the generating series of special cycles of codimension er in the Chow group is a Hermitian modular form of weight n `1 and genus r, assuming the Beilinson-Bloch conjecture with respect to orthogonal Shimura varieties. Our result is a generalization of Kudla's modularity conjecture, solved by Liu unconditionally when e " 1.

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Modularity of special cycles on unitary Shimura varieties over CM-fields

Maeda

2022

Acta Arith.

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Some cases of Kudla's modularity conjecture for unitary Shimura varieties

Cited by 3 publications

References 27 publications

From sum of two squares to arithmetic Siegel-Weil formulas

From sum of two squares to arithmetic Siegel-Weil formulas

Modularity of special cycles on unitary Shimura varieties over CM-fields

Modularity of special cycles on unitary Shimura varieties over CM-fields

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