问题

Please output this exact text, with no changes at all except for fixing the misspellings. Please leave all other stylistic decisions like commas and US vs British spellings as in the original text.

This paper is a complement of the modularity result of Bruinier, Howard, Kudla, Rapoport and Yang (BHKRY) for the special case $U(1,1)$ not considered ther. The main idea is to embed a $U(1, 1)$ Shimura curve to many $U(n-1, 1)$ Shimura varieties for big $n$, andd prove a precise pullback formula ofther generating series of arithmetic divisors. Afterwards, we uise the modularity result of BHKRY together iwth the existince of non-vanishing of clasical theta series at any given point inhten upper half plane to proovehten modularity result on $U(1, 1)$ Shimura curves.

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金声玉亮2.0 This paper is a complement to the modularity result of Bruinier, Howard, Kudla, Rapoport, and Yang (BHKRY) for the special case U(1,1) not considered there. The main idea is to embed a U(1, 1) Shimura curve into many U(n-1, 1) Shimura varieties for large n, and prove a precise pullback formula of their generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with the existence of non-vanishing of classical theta series at any given point in the upper half plane to prove the modularity result on U(1, 1) Shimura curves. This paper is a complement of the modularity result of Bruinier, Howard, Kudla, Rapoport and Yang (BHKRY) for the special case $U(1,1)$ not considered there. The main idea is to embed a $U(1, 1)$ Shimura curve to many $U(n-1, 1)$ Shimura varieties for big $n$, and prove a precise pullback formula of the generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with the existence of non-vanishing of classical theta series at any given point in the upper half plane to prove the modularity result on $U(1, 1)$ Shimura curves.
deepseek-r1

This paper is a complement of the modularity result of Bruinier, Howard, Kudla, Rapoport and Yang (BHKRY) for the special case $U(1,1)$ not considered there. The main idea is to embed a $U(1, 1)$ Shimura curve to many $U(n-1, 1)$ Shimura varieties for big $n$, and prove a precise pullback formula of their generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with the existence of non-vanishing of classical theta series at any given point in the upper half-plane to prove the modularity result on $U(1, 1)$ Shimura curves.		 This paper is a complement of the modularity result of Bruinier, Howard, Kudla, Rapoport and Yang (BHKRY) for the special case $U(1,1)$ not considered there. The main idea is to embed a $U(1, 1)$ Shimura curve to many $U(n-1, 1)$ Shimura varieties for big $n$, and prove a precise pullback formula of the generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with the existence of non-vanishing of classical theta series at any given point in the upper half plane to prove the modularity result on $U(1, 1)$ Shimura curves.
gpto1 This paper is a complement of the modularity result of Bruinier, Howard, Kudla, Rapoport and Yang (BHKRY) for the special case $U(1,1)$ not considered there. The main idea is to embed a $U(1, 1)$ Shimura curve to many $U(n-1, 1)$ Shimura varieties for big $n$, and prove a precise pullback formula of the generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with the existence of non-vanishing of classical theta series at any given point in the upper half plane to prove the modularity result on $U(1, 1)$ Shimura curves. This paper is a complement of the modularity result of Bruinier, Howard, Kudla, Rapoport and Yang (BHKRY) for the special case $U(1,1)$ not considered there. The main idea is to embed a $U(1, 1)$ Shimura curve to many $U(n-1, 1)$ Shimura varieties for big $n$, and prove a precise pullback formula of the generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with the existence of non-vanishing of classical theta series at any given point in the upper half plane to prove the modularity result on $U(1, 1)$ Shimura curves.
gpt-4o-2024-08-06 This paper is a complement of the modularity result of Bruinier, Howard, Kudla, Rapoport and Yang (BHKRY) for the special case $U(1,1)$ not considered there. The main idea is to embed a $U(1, 1)$ Shimura curve to many $U(n-1, 1)$ Shimura varieties for big $n$, and prove a precise pullback formula of their generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with the existence of non-vanishing of classical theta series at any given point in the upper half plane to prove the modularity result on $U(1, 1)$ Shimura curves. This paper is a complement of the modularity result of Bruinier, Howard, Kudla, Rapoport and Yang (BHKRY) for the special case $U(1,1)$ not considered there. The main idea is to embed a $U(1, 1)$ Shimura curve to many $U(n-1, 1)$ Shimura varieties for big $n$, and prove a precise pullback formula of the generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with the existence of non-vanishing of classical theta series at any given point in the upper half plane to prove the modularity result on $U(1, 1)$ Shimura curves.
DeepSeek-Coder-V2-Lite-Instruct This paper is a complement of the modularity result of Bruinier, Howard, Kudla, Rapoport and Yang (BHKRY) for the special case $U(1,1)$ not considered there. The main idea is to embed a $U(1, 1)$ Shimura curve to many $U(n-1, 1)$ Shimura varieties for big $n$, and prove a precise pullback formula of their generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with the existence of non-vanishing of classical theta series at any given point in the upper half plane to prove their modularity result on $U(1, 1)$ Shimura curves. This paper is a complement of the modularity result of Bruinier, Howard, Kudla, Rapoport and Yang (BHKRY) for the special case $U(1,1)$ not considered there. The main idea is to embed a $U(1, 1)$ Shimura curve to many $U(n-1, 1)$ Shimura varieties for big $n$, and prove a precise pullback formula of the generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with the existence of non-vanishing of classical theta series at any given point in the upper half plane to prove the modularity result on $U(1, 1)$ Shimura curves.
Meta-Llama-3.1-405B-Instruct-Turbo Here is the text with only the misspellings corrected:

This paper is a complement of the modularity result of Bruinier, Howard, Kudla, Rapoport and Yang (BHKRY) for the special case $U(1,1)$ not considered there. The main idea is to embed a $U(1, 1)$ Shimura curve to many $U(n-1, 1)$ Shimura varieties for big $n$, and prove a precise pullback formula of the generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with the existence of non-vanishing of classical theta series at any given point in the upper half plane to prove the modularity result on $U(1, 1)$ Shimura curves.		 This paper is a complement of the modularity result of Bruinier, Howard, Kudla, Rapoport and Yang (BHKRY) for the special case $U(1,1)$ not considered there. The main idea is to embed a $U(1, 1)$ Shimura curve to many $U(n-1, 1)$ Shimura varieties for big $n$, and prove a precise pullback formula of the generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with the existence of non-vanishing of classical theta series at any given point in the upper half plane to prove the modularity result on $U(1, 1)$ Shimura curves.
Meta-Llama-3.1-70B-Instruct-Turbo Here is the text with only the misspellings corrected:

This paper is a complement of the modularity result of Bruinier, Howard, Kudla, Rapoport and Yang (BHKRY) for the special case $U(1,1)$ not considered there. The main idea is to embed a $U(1, 1)$ Shimura curve to many $U(n-1, 1)$ Shimura varieties for big $n$, and prove a precise pullback formula of the generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with the existence of non-vanishing of classical theta series at any given point in the upper half plane to prove the modularity result on $U(1, 1)$ Shimura curves.		 This paper is a complement of the modularity result of Bruinier, Howard, Kudla, Rapoport and Yang (BHKRY) for the special case $U(1,1)$ not considered there. The main idea is to embed a $U(1, 1)$ Shimura curve to many $U(n-1, 1)$ Shimura varieties for big $n$, and prove a precise pullback formula of the generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with the existence of non-vanishing of classical theta series at any given point in the upper half plane to prove the modularity result on $U(1, 1)$ Shimura curves.
Qwen2-72B-Instruct This paper is a complement to the modularity result of Bruinier, Howard, Kudla, Rapoport, and Yang (BHKRY) for the special case U(1,1) not considered there. The main idea is to embed a U(1, 1) Shimura curve into many U(n-1, 1) Shimura varieties for large n, and prove a precise pullback formula for their generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with the existence of non-vanishing of classical theta series at any given point in the upper half-plane to prove the modularity result on U(1, 1) Shimura curves. This paper is a complement of the modularity result of Bruinier, Howard, Kudla, Rapoport and Yang (BHKRY) for the special case $U(1,1)$ not considered there. The main idea is to embed a $U(1, 1)$ Shimura curve to many $U(n-1, 1)$ Shimura varieties for big $n$, and prove a precise pullback formula of the generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with the existence of non-vanishing of classical theta series at any given point in the upper half plane to prove the modularity result on $U(1, 1)$ Shimura curves.
baichuan4 This paper is a complement of the modularity result of Bruinier, Howard, Kudla, Rapoport and Yang (BHKRY) for the special case $U(1,1)$ not considered there. The main idea is to embed a $U(1, 1)$ Shimura curve to many $U(n-1, 1)$ Shimura varieties for big $n$, and prove a precise pullback formula of their generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with the existence of non-vanishing of classical theta series at any given point in the upper half plane to prove the modularity result on $U(1, 1)$ Shimura curves. This paper is a complement of the modularity result of Bruinier, Howard, Kudla, Rapoport and Yang (BHKRY) for the special case $U(1,1)$ not considered there. The main idea is to embed a $U(1, 1)$ Shimura curve to many $U(n-1, 1)$ Shimura varieties for big $n$, and prove a precise pullback formula of the generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with the existence of non-vanishing of classical theta series at any given point in the upper half plane to prove the modularity result on $U(1, 1)$ Shimura curves.
doubao This paper is a complement of the modularity result of Bruinier, Howard, Kudla, Rapoport and Yang (BHKRY) for the special case $U(1,1)$ not considered there. The main idea is to embed a $U(1, 1)$ Shimura curve to many $U(n-1, 1)$ Shimura varieties for big $n$, and prove a precise pullback formula of the generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with the existence of non-vanishing of classical theta series at any given point in the upper half plane to prove the modularity result on $U(1, 1)$ Shimura curves. This paper is a complement of the modularity result of Bruinier, Howard, Kudla, Rapoport and Yang (BHKRY) for the special case $U(1,1)$ not considered there. The main idea is to embed a $U(1, 1)$ Shimura curve to many $U(n-1, 1)$ Shimura varieties for big $n$, and prove a precise pullback formula of the generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with the existence of non-vanishing of classical theta series at any given point in the upper half plane to prove the modularity result on $U(1, 1)$ Shimura curves.
gpto1 This paper is a complement of the modularity result of Bruinier, Howard, Kudla, Rapoport and Yang (BHKRY) for the special case $U(1,1)$ not considered there. The main idea is to embed a $U(1, 1)$ Shimura curve to many $U(n-1, 1)$ Shimura varieties for big $n$, and prove a precise pullback formula of the generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with the existence of non-vanishing of classical theta series at any given point in the upper half plane to prove the modularity result on $U(1, 1)$ Shimura curves. This paper is a complement of the modularity result of Bruinier, Howard, Kudla, Rapoport and Yang (BHKRY) for the special case $U(1,1)$ not considered there. The main idea is to embed a $U(1, 1)$ Shimura curve to many $U(n-1, 1)$ Shimura varieties for big $n$, and prove a precise pullback formula of the generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with the existence of non-vanishing of classical theta series at any given point in the upper half plane to prove the modularity result on $U(1, 1)$ Shimura curves.
kimi This paper is a complement of the modularity result of Bruinier, Howard, Kudla, Rapoport, and Yang (BHKRY) for the special case $U(1,1)$ not considered there. The main idea is to embed a $U(1, 1)$ Shimura curve into many $U(n-1, 1)$ Shimura varieties for large $n$, and prove a precise pullback formula for their generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with the existence of non-vanishing of classical theta series at any given point in the upper half-plane to prove the modularity result on $U(1, 1)$ Shimura curves. This paper is a complement of the modularity result of Bruinier, Howard, Kudla, Rapoport and Yang (BHKRY) for the special case $U(1,1)$ not considered there. The main idea is to embed a $U(1, 1)$ Shimura curve to many $U(n-1, 1)$ Shimura varieties for big $n$, and prove a precise pullback formula of the generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with the existence of non-vanishing of classical theta series at any given point in the upper half plane to prove the modularity result on $U(1, 1)$ Shimura curves.
minimax This paper is a complement of the modularity result of Bruinier, Howard, Kudla, Rapoport and Yang (BHKRY) for the special case $U(1,1)$ not considered there. The main idea is to embed a $U(1, 1)$ Shimura curve into many $U(n-1, 1)$ Shimura varieties for big $n$, and prove a precise pullback formula for their generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with the existence of non-vanishing of classical theta series at any given point in the upper half plane to prove the modularity result on $U(1, 1)$ Shimura curves. This paper is a complement of the modularity result of Bruinier, Howard, Kudla, Rapoport and Yang (BHKRY) for the special case $U(1,1)$ not considered there. The main idea is to embed a $U(1, 1)$ Shimura curve to many $U(n-1, 1)$ Shimura varieties for big $n$, and prove a precise pullback formula of the generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with the existence of non-vanishing of classical theta series at any given point in the upper half plane to prove the modularity result on $U(1, 1)$ Shimura curves.
wenxinyiyan This paper is a complement of the modularity result of Bruinier, Howard, Kudla, Rapoport and Yang (BHKRY) for the special case $U(1,1)$ not considered there. The main idea is to embed a $U(1, 1)$ Shimura curve into many $U(n-1, 1)$ Shimura varieties for big $n$, and prove a precise pullback formula of their generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with the existence of non-vanishing of classical theta series at any given point in the upper half plane to prove the modularity result on $U(1, 1)$ Shimura curves. This paper is a complement of the modularity result of Bruinier, Howard, Kudla, Rapoport and Yang (BHKRY) for the special case $U(1,1)$ not considered there. The main idea is to embed a $U(1, 1)$ Shimura curve to many $U(n-1, 1)$ Shimura varieties for big $n$, and prove a precise pullback formula of the generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with the existence of non-vanishing of classical theta series at any given point in the upper half plane to prove the modularity result on $U(1, 1)$ Shimura curves.
zhipu This paper is a complement to the modularity result of Bruinier, Howard, Kudla, Rapoport, and Yang (BHKRY) for the special case \( U(1,1) \) not considered there. The main idea is to embed a \( U(1, 1) \) Shimura curve into many \( U(n-1, 1) \) Shimura varieties for big \( n \), and prove a precise pullback formula for their generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with the existence of non-vanishing of classical theta series at any given point in the upper half-plane to prove the modularity result on \( U(1, 1) \) Shimura curves. This paper is a complement of the modularity result of Bruinier, Howard, Kudla, Rapoport and Yang (BHKRY) for the special case $U(1,1)$ not considered there. The main idea is to embed a $U(1, 1)$ Shimura curve to many $U(n-1, 1)$ Shimura varieties for big $n$, and prove a precise pullback formula of the generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with the existence of non-vanishing of classical theta series at any given point in the upper half plane to prove the modularity result on $U(1, 1)$ Shimura curves.
deepseek-v3 This paper is a complement to the modularity result of Bruinier, Howard, Kudla, Rapoport and Yang (BHKRY) for the special case $U(1,1)$ not considered there. The main idea is to embed a $U(1, 1)$ Shimura curve into many $U(n-1, 1)$ Shimura varieties for big $n$, and prove a precise pullback formula of their generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with the existence of non-vanishing of classical theta series at any given point in the upper half plane to prove the modularity result on $U(1, 1)$ Shimura curves. This paper is a complement of the modularity result of Bruinier, Howard, Kudla, Rapoport and Yang (BHKRY) for the special case $U(1,1)$ not considered there. The main idea is to embed a $U(1, 1)$ Shimura curve to many $U(n-1, 1)$ Shimura varieties for big $n$, and prove a precise pullback formula of the generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with the existence of non-vanishing of classical theta series at any given point in the upper half plane to prove the modularity result on $U(1, 1)$ Shimura curves.