Startseite Mathematik Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ
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Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ

  • Dohoon Choi und Youngmin Lee EMAIL logo
Veröffentlicht/Copyright: 28. Oktober 2022
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Open Mathematics
Aus der Zeitschrift Open Mathematics Band 20 Heft 1

Abstract

Let k be a nonnegative integer. Let K be a number field and O K be the ring of integers of K . Let ℓ ≥ 5 be a prime and v be a prime ideal of O K over ℓ . Let f be a modular form of weight k + 1 2 on Γ 0 (4) such that its Fourier coefficients are in O K . In this article, we study sufficient conditions that if f has the form

f ( z ) ≡ ∑ n = 1 ∞ ∑ i = 1 t a f ( s i n 2 ) q s i n 2 ( mod v )

with square-free integers s i , then f is congruent to a linear combination of iterated derivatives of a single theta function modulo v .

Keywords: Fourier coefficients of modular forms; Galois representations; modular forms of half-integral weight; theta functions
MSC 2010: 11F33; 11F80

1 Introduction

The Fourier coefficients of modular forms of half-integral weight are related to various objects in number theory and combinatorics such as the algebraic parts of the central critical values of modular L-functions, orders of Tate-Shafarevich groups of elliptic curves, the number of partitions of a positive integer, and so on. With a lot of application to these objects, Bruinier [1], Bruinier and Ono [2], Ono and Skinner [3], Ahlgren and Boylan [4,5], and the others studied congruence properties modulo a power of a prime for Fourier coefficients of modular forms of half-integral weight. Many of them considered modular forms of half-integral weight whose the Fourier coefficients are supported on only finitely many square classes modulo a prime ℓ .

Let f be a modular form of half-integral weight on Γ 1 ( 4 N ) . Vignéras [6] proved that if the q -expansion of f has the form

f ( z ) = a f ( 0 ) + ∑ n = 1 ∞ ∑ i = 1 t a f ( s i n 2 ) q s i n 2 , q ≔ e 2 π i z

with a positive integer t and square-free integers s i , then f is a linear combination of single variable theta functions (a different proof of this result was given by Bruinier [1]). Many of the aforementioned results can be considered as positive characteristic extensions of Vignéras’ result on classification of modular forms of half-integral weight such that their nonvanishing Fourier coefficients lie in only finitely many square classes. Especially, Ahlgren et al. [7] obtained an explicit mod ℓ analog of the result of Vignéras for modular forms of half-integral weight on Γ 0 ( 4 ) satisfying the Kohnen-plus condition.

Let K be a number field and O K be the ring of integers of K . Let M k + 1 2 ( Γ 0 ( 4 ) ; O K ) (resp. S k + 1 2 ( Γ 0 ( 4 ) ; O K ) ) be the space of modular forms (resp. cusp forms) of weight k + 1 2 on Γ 0 ( 4 ) such that their Fourier coefficients are in O K and S k + 1 2 + ( Γ 0 ( 4 ) ; O K ) be the subspace of S k + 1 2 ( Γ 0 ( 4 ) ; O K ) consisting of f ∈ S k + 1 2 ( Γ 0 ( 4 ) ; O K ) satisfying the Kohnen-plus condition.

Let ℓ ≥ 5 be a prime and v be a prime ideal of O K over ℓ . For f ∈ S k + 1 2 + ( Γ 0 ( 4 ) ; O K ) , Ahlgren et al. [7] proved that if

(1.1) k + 1 2 < ℓ ℓ + 3 2

and

(1.2) f ( z ) ≡ ∑ n = 1 ∞ ∑ i = 1 t a f ( s i n 2 ) q s i n 2 ( mod v )

with square-free integers s i , then k is even and

f ( z ) ≡ a f ( 1 ) ∑ n = 1 ∞ n k q n 2 ( mod v ) .

In this article, we study sufficient conditions that if f has the form (1.2), then f is congruent to a linear combination of iterated derivatives of a single theta function modulo v .

For a positive number ε , let P ε be the set of primes ℓ such that for every f ∈ S k + 1 2 + ( Γ 0 ( 4 ) ; O K ) with k + 1 2 < ℓ 2 ( log ℓ ) 2 − ε , if

f ( z ) ≡ ∑ n = 1 ∞ ∑ i = 1 t a f ( s i n 2 ) q s i n 2 ( mod v )

with square-free integers s i , then

f ( z ) ≡ a f ( 1 ) ∑ n = 1 ∞ n k q n 2 + a f ( ℓ ) ∑ n = 1 ∞ n k + ℓ − 1 2 q ℓ n 2 ( mod v ) .

The following theorem proves that the portion of P ε in the set of primes is one.

Theorem 1.1

For a positive integer X, there is an absolute constant C such that

# { ℓ : ℓ ∉ P ε and ℓ ≤ X } ≤ C 0 X ( log X ) 1 + ε 2 1 + C log log X log X ,

where C 0 ≔ 2 2 π 2 3 ∏ p > 2 p 2 p 2 − 1 .

For a nonnegative real number r , we define an operator Θ r on C [ [ q ] ] by

Θ r ∑ n = 0 ∞ a ( n ) q n ≔ ∑ n = 0 ∞ n r a ( n ) q n if r ∈ Z > 0 , 0 elsewhere.

For convenience, we let Θ ≔ Θ 1 . As in Theorem 1.1, the previous results on modular forms of half-integral weight having the form (1.2) such as [1,2,4,5,7] and so on imply that in many cases, if f has the form (1.2), then Θ ( f ) is congruent to a linear combination of iterated derivatives of a single theta function modulo v . These lead us to the following conjecture on modular forms f of half-integral weight having the form (1.2).

Conjecture 1.2

Let K be a number field and O K be the ring of integers of K . Let ℓ ≥ 5 be a prime and v be a prime ideal of O K over ℓ . Assume that f ∈ S k + 1 2 ( Γ 0 ( 4 ) ; O K ) has the form

Θ ( f ) ( z ) ≡ ∑ n = 1 ∞ s n 2 a f ( s n 2 ) q s n 2 ( mod v )

with a square-free integer s, then

Θ ( f ) ( z ) ≡ 1 2 a f ( 1 ) ∑ n ∈ Z ℓ ∤ n n k + 2 q n 2 ( mod v ) .

Assume that ℓ is a prime and m is a nonnegative integer. Let r ℓ ( m ) be the least positive integer such that

r ℓ ( m ) ≡ m ( mod ℓ − 1 ) .

Let α ( ℓ , m ) be the smallest nonnegative integer i such that

m + 1 2 < ℓ 2 i r ℓ ( m ) ℓ + 1 2 + 1 2 ,

and β ( ℓ , m ) be the smallest nonnegative integer i such that

m + 1 2 < ℓ 2 i + 1 r ℓ m + ℓ − 1 2 ℓ + 1 2 + 1 2 .

Let

T ( z ) ≔ 1 + 2 ∑ n = 1 ∞ q n 2 .

For convenience, let

∑ n = a b ′ a n ≔ ∑ n = a b a n if a ≤ b , 0 if a > b .

By using Conjecture 1.2, we have an explicit formula for modular forms of half-integral weight having the form (1.2).

Theorem 1.3

Let K , O K , ℓ , and v be as in Conjecture 1.2. Assume that f ∈ M k + 1 2 ( Γ 0 ( 4 ) ; O K ) . Conjecture 1.2  implies that if f has the form

(1.3) f ( z ) ≡ a f ( 0 ) + ∑ n = 1 ∞ ∑ i = 1 t a f ( s i n 2 ) q s i n 2 ( mod v )

with square-free integers s i , then the following statements are true.

  1. If r ℓ ( k ) ≠ ℓ − 1 and r ℓ ( k ) ≠ ℓ − 1 2 , then

    f ( z ) ≡ 1 2 ∑ i = 0 α ( ℓ , k ) − 1 ′ a f ( ℓ 2 i ) Θ k ∕ 2 ( T ) ( ℓ 2 i z ) + 1 2 ∑ i = 0 β ( ℓ , k ) − 1 ′ a f ( ℓ 2 i + 1 ) Θ ( 2 k + ℓ − 1 ) ∕ 4 ( T ) ( ℓ 2 i + 1 z ) ( mod v ) .

  2. If r ℓ ( k ) = ℓ − 1 , then

    f ( z ) ≡ a f ( 0 ) T ( z ) + 1 2 ∑ i = 0 α ( ℓ , k ) − 1 ′ ( a f ( ℓ 2 i ) − 2 a f ( 0 ) ) Θ k ∕ 2 ( T ) ( ℓ 2 i z ) + 1 2 ∑ i = 0 β ( ℓ , k ) − 1 ′ a f ( ℓ 2 i + 1 ) Θ ( 2 k + ℓ − 1 ) ∕ 4 ( T ) ( ℓ 2 i + 1 z ) ( mod v ) .

  3. If r ℓ ( k ) = ℓ − 1 2 , then

    f ( z ) ≡ a f ( 0 ) T ( ℓ z ) + 1 2 ∑ i = 0 α ( ℓ , k ) − 1 ′ a f ( ℓ 2 i ) Θ k ∕ 2 ( T ) ( ℓ 2 i z ) + 1 2 ∑ i = 0 β ( ℓ , k ) − 1 ′ ( a f ( ℓ 2 i + 1 ) − 2 a f ( 0 ) ) Θ ( 2 k + ℓ − 1 ) ∕ 4 ( T ) ( ℓ 2 i + 1 z ) ( mod v ) .

To give numerical evidence for Conjecture 1.2, we consider a basis of the space of modular forms of weight k + 1 2 on Γ 0 ( 4 ) . Let F 2 ( z ) = ∑ n = 0 ∞ σ ( 2 n + 1 ) q 2 n + 1 be the modular form of weight 2 on Γ 0 ( 4 ) , where σ ( n ) is the sum of positive divisors of n . Then

{ F 2 j T 2 k + 1 − 4 j } 0 ≤ j ≤ k 2

is a C -basis of the space of modular forms of weight k + 1 2 on Γ 0 ( 4 ) . Let A k , m be an m × k 2 + 1 matrix such that the ( i , j ) -entry of A k , m is the ( i − 1 )th Fourier coefficient of F 2 j − 1 T 2 k + 5 − 4 j modulo ℓ . Let B k , m be a submatrix of A k , m obtained by removing n 2 + 1 th rows for all nonnegative integers n with ( ℓ , n ) = 1 . Let Null ( B k , m ) be the null space of B k , m . With this notation, we give the following conjecture.

Conjecture 1.4

Let ℓ ≥ 5 be a prime. Let 1 + be the characteristic function of the set of positive real numbers. Then, for a positive even integer k, we have

lim m → ∞ dim Null ( B k , m ) = 1 + ( α ( ℓ , k ) ) .

By comparing the intersection of the null spaces of B k , m and the space of mod v modular forms of weight k + 1 2 on Γ 0 ( 4 ) having the form

f ( z ) ≡ ∑ ℓ ∤ n a ( n 2 ) q n 2 ( mod v ) ,

we have the following theorem.

Theorem 1.5

Conjecture 1.2  is equivalent to Conjecture 1.4.

Let us note that Null ( B k , m ) is stable for sufficiently large m . In the proof of Theorem 1.5, we prove that dim Null ( B k , m ) is larger than or equal to 1 + ( α ( ℓ , k ) ) for all positive integers m . Hence, if there is a positive integer m such that dim Null ( B k , m ) = 1 + ( α ( ℓ , k ) ) , then Conjecture 1.2 is true. To compute dim Null ( B k , m ) , we consider the row echelon form of B k , m . We use C++ in this process. Then we have the following theorem.

Theorem 1.6

Assume that k ≤ 1,000 , or that ℓ ∈ { 5 , 7 , 11 , 13 , 17 , 19 } and k ≤ 10,000 . Then, Conjecture 1.2  is true.

The remainder of this article is organized as follows. In Section 2, we review some properties of f having the form (1.3) and the filtration for modular forms. In Section 3, we prove Theorems 1.1, 1.3, 1.5, and 1.6.

2 Preliminaries

In this section, we review some notions and properties of the filtration for modular forms, and then we introduce some properties about modular forms of half-integral weight on Γ 0 ( 4 ) such that their Fourier coefficients are supported on finitely many square classes modulo a prime ℓ . For further details, see [8].

Throughout the rest of this article, we fix the following notation. For a congruence subgroup Γ and w ∈ 1 2 Z , let M w ( Γ ) (resp. S w ( Γ ) ) be the space of modular forms (resp. cusp forms) of weight w on Γ . For a Dirichlet character χ modulo N , let M w ( Γ 0 ( N ) , χ ) (resp. S w ( Γ 0 ( N ) , χ ) ) be the space of modular forms (resp. cusp forms) of weight w on Γ 0 ( N ) with character χ .

Let k be a nonnegative integer and ℓ ≥ 5 be a prime. Let K be a number field and O K be the ring of integers of K . Let v be a prime ideal of O K over ℓ . Let M k + 1 2 ( Γ 0 ( 4 N ) ; O K ) (resp. S k + 1 2 ( Γ 0 ( 4 N ) ; O K ) ) be the space of modular forms (resp. cusp forms) of weight k + 1 2 on Γ 0 ( 4 N ) such that their Fourier coefficients are in O K and S k + 1 2 + ( Γ 0 ( 4 ) ; O K ) be the subspace of S k + 1 2 ( Γ 0 ( 4 ) ; O K ) consisting of f ∈ S k + 1 2 ( Γ 0 ( 4 ) ; O K ) satisfying the Kohnen-plus condition.

Now, we review the basic notions and properties about the Shimura correspondence. Assume that f is a cusp form of weight k + 1 2 on Γ 0 ( 4 ) . For a square-free integer t , we define A t ( n ) by

∑ n = 1 ∞ A t ( n ) n s ≔ ∑ n = 1 ∞ ( − 1 ) k t n 1 n s − k + 1 ∑ n = 1 ∞ a t n 2 ( f ) n s .

Then, the Shimura lift Sh t ( f ) of f is defined by

Sh t ( f ) ( z ) ≔ ∑ n = 1 ∞ A t ( n ) q n .

Note that Sh t ( f ) ∈ S 2 k ( Γ 0 ( 2 ) ) . In particular, if f ∈ S k + 1 2 + ( Γ 0 ( 4 ) ) , then Sh t ( f ) ∈ S 2 k ( Γ 0 ( 1 ) ) . For each odd prime p with p ∤ t , we have

Sh t f ∣ T p 2 , k + 1 2 = Sh t ( f ) ∣ T p , 2 k ,

where T n , w denotes the n th Hecke operator on the space of modular forms of weight w . For each prime ℓ , operators U ℓ and V ℓ on formal power series are defined by

∑ n = 0 ∞ a ( n ) q n ∣ U ℓ ≔ ∑ n = 0 ∞ a ( ℓ n ) q n

and

∑ n = 0 ∞ a ( n ) q n ∣ V ℓ ≔ ∑ n = 0 ∞ a ( n ) q ℓ n .

2.1 Filtration for modular forms of half integral weight modulo a prime ℓ

The theory of filtration for modular forms of integral weight was developed by Serre [9], Swinnerton-Dyer [10], Katz [11], and Gross [12]. From this, the theory of filtration for modular forms of half-integral weight on Γ 0 ( 4 ) was studied. In this section, we review some properties of filtration for modular forms of half-integral weight on Γ 0 ( 4 ) . For the details, we refer to [13, Section 2].

We say that ∑ n = 0 ∞ a ( n ) q n is congruent to ∑ n = 0 ∞ b ( n ) q n modulo v , i.e.,

∑ n = 0 ∞ a ( n ) q n ≡ ∑ n = 0 ∞ b ( n ) q n ( mod v ) ,

if a ( n ) ≡ b ( n ) ( mod v ) for all nonnegative integers n . For f ∈ M k + 1 2 ( Γ 0 ( 4 ) ; O K ) , we define a filtration ω ( f ) of f modulo v by

ω ( f ) ≔ inf k ′ + 1 2 : there is f ′ ∈ M k ′ + 1 2 ( Γ 0 ( 4 ) ; O K ) such that f ′ ≡ f ( mod v ) .

For convenience, if f ≡ 0 ( mod v ) , then let ω ( f ) = − ∞ . We summarize the properties of ω ( f ) in the following lemma.

Lemma 2.1

Let f ∈ M k + 1 2 ( Γ 0 ( 4 ) ; O K ) . Then, the following statements are true.

  1. k ≡ ω ( f ) − 1 2 ( mod ℓ − 1 ) .

  2. ω ( f ℓ ) = ℓ ⋅ ω ( f ) .

  3. There is a nonnegative integer k ′ such that

    k ′ ≡ k + ℓ − 1 2 ( mod ℓ − 1 ) ,

    and there is g ∈ M k ′ + 1 2 ( Γ 0 ( 4 ) ; O K ) such that g ≡ f ∣ U ℓ ( mod v ) . Moreover, if f ( z ) ≡ ∑ n = 0 ∞ a f ( ℓ n ) q ℓ n ( mod v ) , then there is a nonnegative integer k ′ such that

    k ′ ≡ k + ℓ − 1 2 ( mod ℓ − 1 ) and k ′ + 1 2 ≤ 1 ℓ k + 1 2 ,

    and there is g ∈ M k ′ + 1 2 ( Γ 0 ( 4 ) ; O K ) such that g ≡ f ∣ U ℓ ( mod v ) .

  4. There is h ∈ S k + ℓ + 3 2 ( Γ 0 ( 4 ) ) such that h ≡ Θ ( f ) ( mod v ) . In particular, if f ∈ S k + 1 2 + ( Γ 0 ( 4 ) ) , then h ∈ S k + ℓ + 3 2 + ( Γ 0 ( 4 ) ) .

Proof

The proofs of (1) and (2) are in [13, Proposition 2.2]. The proof of (3) is obtained by combining [7, Lemma 4.2] and [13, Proposition 2.2]. To prove (4), let

h ≔ k + 1 2 Θ ( E ℓ − 1 ) f − ( ℓ − 1 ) E ℓ − 1 Θ ( f ) ,

where E ℓ − 1 denotes the Eisenstein series of weight ℓ − 1 . Since E ℓ − 1 ≡ 1 ( mod v ) , we have h ≡ Θ ( f ) ( mod v ) . By [14, Corollary 7.2], we obtain h ∈ S k + ℓ + 3 2 ( Γ 0 ( 4 ) ) . When f satisfies the Kohnen-plus condition, the proof of ( 4 ) is in [7, Lemma 4.1].□

2.2 Modular forms of half-integral weight such that their Fourier coefficients are supported on finitely many square classes modulo ℓ

In this section, we introduce some properties of modular forms of half-integral weight on Γ 0 ( 4 ) such that their Fourier coefficients are supported on finitely many square classes modulo v .

Ahlgren and Boylan [4] obtained the necessary conditions for the weight of f ∈ M k + 1 2 ( Γ 0 ( 4 ) ) such that their Fourier coefficients are supported on finitely many square classes modulo v by using the theory of Galois representations. This was reproved in [15] by using only the theory of filtration for modular forms of integral weight. The Choi and Kilbourn [16] improved the necessary conditions for the weight by using only the theory of filtration for modular forms of integral weight. We review the results [4,16] in the following theorem.

Theorem 2.2

Let N be a positive integer and ℓ ≥ 5 be a prime with ( ℓ , N ) = 1 . Assume that f ( z ) ∈ M k + 1 2 ( Γ 1 ( 4 N ) ) ∩ O K [ [ q ] ] has the form

f ( z ) ≡ a f ( 0 ) + ∑ n = 1 ∞ ∑ i = 1 t a f ( s i n 2 ) q s i n 2 ( mod v )

with square-free integers s i . Let k ¯ and i k be nonnegative integers, which satisfy k = ( ℓ − 1 ) i k + k ¯ and k ¯ < ℓ − 1 . Then, the following statements are true.

  1. If ℓ ∤ n i for some i , then

    k ¯ ≤ 2 i k + 1 .

  2. If ℓ ∣ n i for all i and k ¯ ≤ ℓ − 3 2 , then

    k ¯ ≤ i k − ℓ + 1 2 .

  3. If ℓ ∣ n i for all i and k ¯ > ℓ − 3 2 , then

    k ¯ ≤ i k + ℓ − 1 2 .

Bruinier and Ono [2, Theorem 3.1] proved the following theorem by using an argument in [1].

Theorem 2.3

Let N be a positive integer and ℓ ≥ 5 be a prime with ( ℓ , N ) = 1 . Let χ be a real Dirichlet character modulo 4 N and f ( z ) ∈ S k + 1 2 ( Γ 0 ( 4 N ) , χ ) ∩ O K [ [ q ] ] . For each prime p with ( p , 4 N ℓ ) = 1 , if there exists ε p ∈ { ± 1 } such that

f ( z ) ≡ ∑ n p ∈ { 0 , ε p } a f ( n ) q n ( mod v ) ,

then

( p − 1 ) f ∣ T p 2 , k + 1 2 ≡ ε p ( − 1 ) k p χ ( p ) ( p k + p k − 1 ) ( p − 1 ) f ( mod v ) .

Ahlgren et al. [7] proved that if f ∈ S k + 1 2 + ( Γ 0 ( 4 ) ; O K ) and the Fourier coefficients of f are supported on finitely many square classes modulo v , then f has the form

f ( z ) ≡ ∑ n = 1 ∞ a f ( n 2 ) q n 2 + ∑ n = 1 ∞ a f ( ℓ n 2 ) q ℓ n 2 ( mod v ) .

By using the theory of Galois representations, we extend the result [7] to cusp forms of half-integral weight on Γ 0 ( 4 ) without the Kohnen-plus condition.

Proposition 2.4

Assume that f ∈ S k + 1 2 ( Γ 0 ( 4 ) ; O K ) has the form

(2.1) f ( z ) ≡ ∑ n = 1 ∞ ∑ i = 1 t a f ( s i n 2 ) q s i n 2 ( mod v )

with square-free integers s i . Then, the following statements are true.

  1. If 2 ∣ k and ℓ ≡ 1 ( mod 4 ) , then

    f ( z ) ≡ ∑ n = 1 ∞ a f ( n 2 ) q n 2 + ∑ n = 1 ∞ a f ( ℓ n 2 ) q ℓ n 2 ( mod v ) .

  2. If 2 ∣ k and ℓ ≡ 3 ( mod 4 ) , then

    f ( z ) ≡ ∑ n = 1 ∞ a f ( n 2 ) q n 2 ( mod v ) .

  3. If 2 ∤ k and ℓ ≡ 3 ( mod 4 ) , then

    f ( z ) ≡ ∑ n = 1 ∞ a f ( ℓ n 2 ) q ℓ n 2 ( mod v ) .

  4. If 2 ∤ k and ℓ ≡ 1 ( mod 4 ) , then

    f ( z ) ≡ 0 ( mod v ) .

Proof

Assume that for each i ∈ { 1 , … , t } , there is a positive integer n i such that a f ( s i n i 2 ) ≢ 0 ( mod v ) . Following the proof of Lemma 4.1 in [4], there exist distinct odd primes p i , 1 , … , p i , r i , each relatively to n i s i ℓ , and a modular form f i ∈ S k + 1 2 ( Γ 0 ( 4 ∏ j = 1 r i p i , j 2 ) ; O K ) such that

f i ( z ) ≡ ∑ n = 1 gcd ( n , ∏ j = 1 r i p i , j ) = 1 ∞ a f i ( s i n 2 ) q s i n 2 ≢ 0 ( mod v ) .

By Theorem 2.3, for each prime p with p ∤ 2 s i ℓ ∏ j = 1 r i p i , j and p ≢ 1 ( mod ℓ ) , we have

f i ∣ T p 2 , k + 1 2 ≡ ( − 1 ) k s i p ( p k + p k − 1 ) f i ( mod v ) .

Since S 1 2 ( Γ 0 ( 4 ) ) = S 3 2 ( Γ 0 ( 4 ) ) = { 0 } , we may assume that k ≥ 2 . Let F i ≔ Sh s i ( f i ) ∈ S 2 k ( Γ 0 ( 2 ∏ j = 1 r i p i , j 2 ) ) be the Shimura lift of f i . Since the Shimura correspondence commutes with the Hecke operators, for each prime p with p ∤ 2 s i ℓ ∏ j = 1 r i p i , j and p ≢ 1 ( mod ℓ ) , we obtain

F i ∣ T p , 2 k ≡ ( − 1 ) k s i p ( p k + p k − 1 ) F i ( mod v ) .

Then, there is an integer N i such that N i ∣ 2 ∏ j = 1 r i p i , j 2 , and there is a newform G i ∈ S 2 k ( Γ 0 ( N i ) ) such that for each prime p with p ∤ 2 s i ℓ ∏ j = 1 r i p i , j and p ≢ 1 ( mod ℓ ) ,

λ i ( p ) ≡ ( − 1 ) k s i p ( p k + p k − 1 ) ( mod v ) .

Here, λ i ( p ) denotes the p th Hecke eigenvalue of G i . Let F v ≔ O K ∕ v . Note that there is a semi-simple Galois representation

ρ i : Gal ( Q ¯ ∕ Q ) → GL 2 ( F v ) ,

such that for each prime p with p ∤ N i ℓ

tr ( ρ i ( Frob p ) ) ≡ λ i ( p ) ( mod v ) and det ( ρ i ( Frob p ) ) ≡ p 2 k − 1 ( mod v ) ,

where Frob p denotes any Frobenius element at p . Let χ ℓ : Gal ( Q ¯ ∕ Q ) → F ℓ ∗ be the mod- ℓ cyclotomic character. Following the argument of the proof of [5, Proposition 4.3], we have

(2.2) ρ i ≅ ( − 1 ) k s i ⋅ χ ℓ k 0 0 ( − 1 ) k s i ⋅ χ ℓ k − 1 if ℓ ∤ s i , ( − 1 ) k + ℓ − 1 2 s i ′ ⋅ χ ℓ k + ℓ − 1 2 0 0 ( − 1 ) k + ℓ − 1 2 s i ′ ⋅ χ ℓ k + ℓ − 3 2 if ℓ ∣ s i ,

where ℓ s i ′ = s i .

By the result of Carayol [17], the conductor of ρ i divides N i . By (2.2), we obtain that if ℓ ∤ s i , then s i 2 divides the conductor of ρ i , and if ℓ ∣ s i , then ( s i ′ ) 2 divides the conductor of ρ i . Since N i ∣ 2 ∏ j = 1 r i p i , j 2 and gcd ( s i , ∏ j = 1 r i p i , j ) = 1 , we have s i ∈ { 1 , ℓ } . Moreover, the conductor of ρ i is not divided by 4. Therefore, we conclude that if k is odd, then s i ≠ 1 and if k + ℓ − 1 2 is odd, then s i ≠ ℓ .□

We extend Proposition 2.4 to general modular forms of half-integral weight including noncusp forms in the following proposition.

Proposition 2.5

Assume that f ∈ M k + 1 2 ( Γ 0 ( 4 ) ; O K ) has the form

(2.3) f ( z ) ≡ a f ( 0 ) + ∑ n = 1 ∞ ∑ i = 1 t a f ( s i n 2 ) q s i n 2 ( mod v )

with square-free integers s i . Then,

f ( z ) ≡ a f ( 0 ) + ∑ n = 1 ∞ a f ( n 2 ) q n 2 + ∑ n = 1 ∞ a f ( ℓ n 2 ) q ℓ n 2 ( mod v ) .

Proof

Without loss of generality, we assume that there is a positive integer n 1 such that a f ( s 1 n 1 2 ) ≢ 0 ( mod v ) . Let a be the exponent of ℓ in s 1 n 1 2 . Then, there is a unique square-free integer s 1 ′ such that s 1 n 1 2 = ℓ a s 1 ′ m 1 2 for some positive integer m 1 . By Lemma 2.1 (3), there is an integer k ′ and a modular form g ∈ M k ′ + 1 2 ( Γ 0 ( 4 ) ) such that g ≡ f ∣ U ℓ a ( mod v ) . By Lemma 2.1 (4), there is h ∈ S k ′ + ℓ + 3 2 ( Γ 0 ( 4 ) ) such that h ≡ Θ ( g ) ( mod v ) . Since a f ( s 1 n 1 2 ) ≢ 0 ( mod v ) , we have a h ( s 1 ′ m 1 2 ) ≢ 0 ( mod v ) and then h has the form (2.1). Then, s 1 ′ = 1 by Proposition 2.4. This implies that s 1 ∈ { 1 , ℓ } . Therefore, Proposition 2.5 is proved.□

Combining Theorem 2.2 and Proposition 2.5, we obtain an explicit formula of f ∈ M k + 1 2 ( Γ 0 ( 4 ) ) having the form (2.3) when k < ℓ − 1 .

Lemma 2.6

Assume that f ∈ M k + 1 2 ( Γ 0 ( 4 ) ; O K ) has the form (2.3) and f ≢ 0 ( mod v ) . If k < ℓ − 1 , then k ∈ { 0 , ℓ − 1 2 } . Moreover,

f ( z ) ≡ a f ( 0 ) 1 + 2 ∑ n = 1 ∞ q n 2 ( mod v ) if k = 0

and

f ( z ) ≡ a f ( 0 ) 1 + 2 ∑ n = 1 ∞ q ℓ n 2 ( mod v ) if k = ℓ − 1 2 .

Proof

We assume that k < ℓ − 1 . By Theorem 2.2, we have k ∈ { 0 , 1 , ℓ − 1 2 } . Note that M 1 2 ( Γ 0 ( 4 ) ) is generated by T . Thus, when k = 0 , we obtain that f is a constant multiple of T . If f has the form (2.3), then a f ( 2 ) ≡ 0 ( mod v ) by Proposition 2.5. Note that M 3 2 ( Γ 0 ( 4 ) ) is generated by T 3 and a T 3 ( 2 ) = 3 . Thus, when k = 1 , we have f ≡ 0 ( mod v ) . When k = ℓ − 1 2 , we have by Theorem 2.2

f ( z ) ≡ ∑ n = 0 ∞ a f ( ℓ n ) q ℓ n ( mod v ) .

By Lemma 2.1 (3), there is g ∈ M 1 2 ( Γ 0 ( 4 ) ) such that g ≡ f ∣ U ℓ ( mod v ) . Since g is a constant multiple of T , f is congruent to a constant multiple of T ∣ V ℓ modulo v .□

3 Proof of Theorems

In this section, we prove Theorems 1.1, 1.3, 1.5, and 1.6. First, we prove Theorem 1.3.

Proof of Theorem 1.3

We fix a prime ℓ ≥ 5 . We prove Theorem 1.3 by induction on k . When k < ℓ − 1 , Theorem 1.3 is true by Lemma 2.6. Thus, we assume that Theorem 1.3 is true when k < k 0 with a fixed positive integer k 0 , where k 0 is a positive integer larger than ℓ − 1 .

To prove Theorem 1.3, it is enough to show that Theorem 1.3 is true when k = k 0 by induction on k . Assume that f ∈ M k 0 + 1 2 ( Γ 0 ( 4 ) ; O K ) has the form (1.3). Then by Lemma 2.5, f has the form

f ( z ) ≡ a f ( 0 ) + ∑ n = 1 ∞ a f ( n 2 ) q n 2 + ∑ n = 1 ∞ a f ( ℓ n 2 ) q ℓ n 2 ( mod v ) ,

and

Θ ( ℓ − 1 ) ∕ 2 ( f ) ( z ) ≡ 1 2 ∑ n ∈ Z ℓ ∤ n a f ( n 2 ) q n 2 ( mod v ) .

By Lemma 2.1 (4), there is g 0 ∈ S k 0 + ℓ 2 2 ( Γ 0 ( 4 ) ) such that

g 0 ≡ Θ ( ℓ − 1 ) ∕ 2 ( f ) ( mod v ) .

Let k 1 ≔ max k 0 + 1 2 , ω ( g 0 ) − 1 2 . Then, there is g 1 ∈ M k 1 + 1 2 ( Γ 0 ( 4 ) ; O K ) such that

g 1 ( z ) ≡ ( f − Θ ( ℓ − 1 ) ∕ 2 ( f ) ) ( z ) ≡ a f ( 0 ) + ∑ n = 1 ∞ a f ( ℓ n 2 ) q ℓ n 2 + ∑ n = 1 ∞ a f ( ℓ 2 n 2 ) q ℓ 2 n 2 ( mod v ) .

Let k 2 be the largest integer satisfying

(3.1) k 2 + 1 2 ≤ 1 ℓ k 1 + 1 2 and k 2 ≡ ℓ − 1 2 + k 1 ≡ ℓ − 1 2 + k 0 ( mod ℓ − 1 ) .

By Lemma 2.1 (3), there is g 2 ∈ M k 2 + 1 2 ( Γ 0 ( 4 ) ; O K ) such that

g 2 ( z ) ≡ g 1 ∣ U ℓ ( z ) ≡ a f ( 0 ) + ∑ n = 1 ∞ a f ( ℓ n 2 ) q n 2 + ∑ n = 1 ∞ a f ( ℓ 2 n 2 ) q ℓ n 2 ( mod v ) .

Since k 0 > ℓ 2 , we have

k 2 + 1 2 ≤ 1 ℓ k 1 + 1 2 ≤ 1 ℓ k 0 + ℓ 2 2 < k 0 + 1 2 .

For a nonnegative integer k , we define a subset ℬ k of M k + 1 2 ( Γ 0 ( 4 ) ) by

ℬ k ≔ { Θ k ∕ 2 ( T ) ∣ V ℓ 2 i } 0 ≤ i < α ( ℓ , k ) ∪ { Θ ( 2 k + ℓ − 1 ) ∕ 4 ( T ) ∣ V ℓ 2 i + 1 } 0 ≤ i < β ( ℓ , k ) ∪ { T } if r ℓ ( k ) = ℓ − 1 , { Θ k ∕ 2 ( T ) ∣ V ℓ 2 i } 0 ≤ i < α ( ℓ , k ) ∪ { Θ ( 2 k + ℓ − 1 ) ∕ 4 ( T ) ∣ V ℓ 2 i + 1 } 0 ≤ i < β ( ℓ , k ) ∪ { T ∣ V ℓ } if r ℓ ( k ) = ℓ − 1 2 , { Θ k ∕ 2 ( T ) ∣ V ℓ 2 i } 0 ≤ i < α ( ℓ , k ) ∪ { Θ ( 2 k + ℓ − 1 ) ∕ 4 ( T ) ∣ V ℓ 2 i + 1 } 0 ≤ i < β ( ℓ , k ) otherwise.

To prove Theorem 1.3, it is enough to show that if f ∈ M k 0 + 1 2 ( Γ 0 ( 4 ) ; O K ) has the form (1.3), then f is congruent to a linear combination of ℬ k 0 modulo v .

By Proposition 2.4, if k 0 is odd, then g 0 ≡ 0 ( mod v ) . Combining the assumption that Conjecture 1.2 is true, we have

g 0 ≡ a f ( 1 ) 2 Θ k 0 ∕ 2 ( T ) ( mod v ) .

Since k 2 ≡ k 0 + ℓ − 1 2 ( mod ℓ − 1 ) , it follows that Θ k 0 ∕ 2 ( T ) ≡ Θ ( 2 k 2 + ℓ − 1 ) ∕ 4 ( T ) ( mod v ) . By the induction hypothesis, g 2 is congruent to a linear combination of ℬ k 2 . Since

f ≡ f − Θ ℓ − 1 2 ( f ) + Θ ℓ − 1 2 ( f ) ≡ g 2 ∣ V ℓ + g 0 ( mod v ) ,

we deduce that f is congruent to a linear combination of

{ Θ k 2 ∕ 2 ( T ) ∣ V ℓ 2 i + 1 } 0 ≤ i < α ( ℓ , k 2 ) ∪ { Θ ( 2 k 2 + ℓ − 1 ) ∕ 4 ( T ) ∣ V ℓ 2 i } 0 ≤ i < β ( ℓ , k 2 ) + 1 ∪ { T ∣ V ℓ } if r ℓ ( k 2 ) = ℓ − 1 , { Θ k 2 ∕ 2 ( T ) ∣ V ℓ 2 i + 1 } 0 ≤ i < α ( ℓ , k 2 ) ∪ { Θ ( 2 k 2 + ℓ − 1 ) ∕ 4 ( T ) ∣ V ℓ 2 i } 0 ≤ i < β ( ℓ , k 2 ) + 1 ∪ { T ∣ V ℓ 2 } if r ℓ ( k 2 ) = ℓ − 1 2 , { Θ k 2 ∕ 2 ( T ) ∣ V ℓ 2 i + 1 } 0 ≤ i < α ( ℓ , k 2 ) ∪ { Θ ( 2 k 2 + ℓ − 1 ) ∕ 4 ( T ) ∣ V ℓ 2 i } 0 ≤ i < β ( ℓ , k 2 ) + 1 otherwise.

If r ℓ ( k 2 ) = ℓ − 1 2 , then

T ∣ V ℓ 2 ≡ T − Θ ( ℓ − 1 ) ∕ 2 ( T ) ≡ T − Θ ( 2 k 2 + ℓ − 1 ) ∕ 4 ( T ) ( mod v ) .

Thus, f is congruent to a linear combination of

{ Θ k 2 ∕ 2 ( T ) ∣ V ℓ 2 i + 1 } 0 ≤ i < α ( ℓ , k 2 ) ∪ { Θ ( 2 k 2 + ℓ − 1 ) ∕ 4 ( T ) ∣ V ℓ 2 i } 0 ≤ i < β ( ℓ , k 2 ) + 1 ∪ { T ∣ V ℓ } if r ℓ ( k 2 ) = ℓ − 1 , { Θ k 2 ∕ 2 ( T ) ∣ V ℓ 2 i + 1 } 0 ≤ i < α ( ℓ , k 2 ) ∪ { Θ ( 2 k 2 + ℓ − 1 ) ∕ 4 ( T ) ∣ V ℓ 2 i } 0 ≤ i < β ( ℓ , k 2 ) + 1 ∪ { T } if r ℓ ( k 2 ) = ℓ − 1 2 , { Θ k 2 ∕ 2 ( T ) ∣ V ℓ 2 i + 1 } 0 ≤ i < α ( ℓ , k 2 ) ∪ { Θ ( 2 k 2 + ℓ − 1 ) ∕ 4 ( T ) ∣ V ℓ 2 i } 0 ≤ i < β ( ℓ , k 2 ) + 1 otherwise.

To complete the proof, it is sufficient to show that

(3.2) α ( ℓ , k 2 ) ≤ β ( ℓ , k 0 ) and β ( ℓ , k 2 ) + 1 ≤ α ( ℓ , k 0 ) .

First, we assume that k 0 + 1 2 ≥ ℓ 2 2 . Since Θ m ( T ) ≡ Θ ( 2 m + ℓ − 1 ) ∕ 2 ( T ) for any positive integer m , we have ω ( g 0 ) ≤ ω ( Θ k 0 ∕ 2 ( T ) ) ≤ ℓ 2 2 . This implies that

k 1 = max k 0 , ω ( g 0 ) − 1 2 = k 0 .

Then by (3.1), we obtain (3.2).

Now, we assume that k 0 + 1 2 < ℓ 2 2 . In this case, we have

k 2 + 1 2 ≤ 1 ℓ k 1 + 1 2 ≤ 1 ℓ ⋅ max k 0 + 1 2 , ω ( g 0 ) ≤ ℓ 2 .

Further, assume that k 2 ≠ 0 and k 2 ≠ ℓ − 1 2 . Then α ( ℓ , k 2 ) = β ( ℓ , k 2 ) = β ( ℓ , k 0 ) = 0 . By Lemma 2.6, we have g 2 ≡ 0 ( mod v ) , and then

f ≡ Θ ℓ − 1 2 ( f ) ≡ a f ( 1 ) 2 Θ k 0 ∕ 2 ( T ) ( mod v ) .

Note that Θ ( ℓ − 1 ) ∕ 2 ( T ) ≡ T − T ℓ 2 ( mod v ) , we have ω ( Θ ( ℓ − 1 ) ∕ 2 ( T ) ) = ℓ 2 2 . Then, for a positive integer m with m ≤ ℓ − 1 2 , we have

(3.3) ω ( Θ m ( T ) ) = ( ℓ + 1 ) m + 1 2 .

By (3.3), we have

ω ( Θ k 0 ∕ 2 ( T ) ) = r ℓ ( k 0 ) ⋅ ℓ + 1 2 + 1 2 ≤ k 0 + 1 2 .

It implies that α ( ℓ , k 0 ) = 1 . Hence, α ( ℓ , k 2 ) = β ( ℓ , k 0 ) and β ( ℓ , k 2 ) + 1 = α ( ℓ , k 0 ) . For the cases when k 0 = 0 and k 0 = ℓ − 1 2 , we obtain (3.2) by direct computation. Thus, we conclude that if f ∈ M k 0 + 1 2 ( Γ 0 ( 4 ) ; O K ) has the form (1.3), then f is congruent to a linear combination of ℬ k 0 modulo v . Therefore, Theorem 1.3 is proved by induction on k .□

To prove Theorem 1.1, we use the following theorem which gives a sufficient condition for the weight k + 1 2 that Conjecture 1.2 holds for f ∈ S k + 1 2 + ( Γ 0 ( 4 ) ; O K ) . It was proved in the proof of [7, Theorem 5.2].

Theorem 3.1

Assume that f ∈ S k + 1 2 + ( Γ 0 ( 4 ) ; O K ) has the form

(3.4) f ( z ) ≡ 1 2 ∑ n ∈ Z ℓ ∤ n a f ( n 2 ) q n 2 ( mod v )

and f ≢ 0 ( mod v ) . Let p ℓ be the smallest positive prime p such that p ≡ 1 ( mod ℓ ) . If 2 k + 1 < p ℓ 2 , then k is even and

f ≡ 1 2 a f ( 1 ) Θ k ∕ 2 ( T ) ( mod v ) .

Proof

We follow the proof of [7, Theorem 5.2]. By Proposition 2.4, we obtain that k is even. By Theorem 2.3, for each odd prime p with p ≢ 0 , 1 ( mod ℓ ) , we have

f ∣ T p 2 , k + 1 2 ≡ ( p k + p k − 1 ) f ( mod v ) .

Hence, for any positive odd integer m which is not divisible by any prime p with p ≡ 0 , 1 ( mod v ) , we have

a f ( m 2 ) ≡ a f ( 1 ) m k ( mod v ) .

Let k 1 ≔ max k , r ℓ ( k ) 2 ( ℓ + 1 ) . Then, there is g 1 ∈ S k 1 + 1 2 + ( Γ 0 ( 4 ) ; O K ) such that

g 1 ≡ f − 1 2 a f ( 1 ) Θ r ℓ ( k ) ∕ 2 ( T ) ( mod v ) .

Let h ≔ g 1 − g 1 ∣ U 4 ∣ V 4 ∈ S k 1 + 1 2 + ( Γ 0 ( 16 ) ) . Then, a h ( n ) ≡ 0 ( mod v ) for n < p ℓ 2 . Since

1 12 k 1 + 1 2 ⋅ [ SL 2 ( Z ) : Γ 0 ( 16 ) ] = 2 k 1 + 1 < p ℓ 2 ,

we have h ≡ 0 ( mod v ) by the result of Sturm [18] called the Sturm bound. Then,

g 1 ( z ) ≡ g 1 ∣ U 4 ∣ V 4 ( z ) ≡ ∑ m = 1 ∞ a g 1 ( 4 m 2 ) q 4 m 2 ( mod v ) .

From the proof of [7, Theorem 5.2], we have g 1 ≡ 0 ( mod v ) . Then,

□ f ( z ) ≡ 1 2 a f ( 1 ) Θ k ∕ 2 ( T ) ( z ) ≡ 1 2 a f ( 1 ) ∑ n ∈ Z ℓ ∤ n n k q n 2 ( mod v ) .

The following proposition is a refinement of Theorem 1.1.

Proposition 3.2

Let g : R → R be a function such that g ( x ) log x is an increasing function and lim x → ∞ g ( x ) = 0 . Let P be a set of primes ℓ such that for every f ∈ S k + 1 2 + ( Γ 0 ( 4 ) ; O K ) with k + 1 2 < g ( ℓ ) ℓ 2 ( log ℓ ) 2 , if f has the form (1.2), then

f ( z ) ≡ 1 2 ∑ i = 0 α ( ℓ , k ) − 1 ′ a f ( ℓ 2 i ) Θ k ∕ 2 ( T ) ( ℓ 2 i z ) + 1 2 ∑ i = 0 β ( ℓ , k ) − 1 ′ a f ( ℓ 2 i + 1 ) Θ ( 2 k + ℓ − 1 ) ∕ 4 ( T ) ( ℓ 2 i + 1 z ) ( mod v ) .

Then, there is an absolute constant C such that

# { ℓ : ℓ ∉ P and ℓ ≤ X } ≤ C 0 g ( X ) X log X 1 + C log log X log X ,

where C 0 ≔ 2 2 π 2 3 ∏ p > 2 p 2 p 2 − 1 .

Proof

Let p ℓ be the smallest positive prime p with p ≡ 1 ( mod ℓ ) . By using Theorem 3.1 to follow the proof of Theorem 1.3, we deduce that if p ℓ 2 > 2 g ( ℓ ) ℓ 2 ( log ℓ ) 2 , then ℓ ∈ P . From this, for a positive number X , we have

# { ℓ : ℓ ∉ P and ℓ ≤ X } ≤ # { ℓ : p ℓ 2 ≤ 2 g ( ℓ ) ℓ 2 ( log ℓ ) 2 and ℓ ≤ X } .

For convenience, let h ( x ) ≔ g ( x ) 2 . Then, we have

(3.5) # { ℓ : p ℓ 2 ≤ 2 g ( ℓ ) ℓ 2 ( log ℓ ) 2 and ℓ ≤ X } = # { ℓ : p ℓ ≤ 2 h ( ℓ ) ℓ log ℓ and ℓ ≤ X } ≤ ∑ n = 1 ∞ # { ℓ : p ℓ = 2 n ℓ + 1 , n < h ( ℓ ) log ℓ and ℓ ≤ X } ≤ ∑ n = 1 ∞ # { ℓ : p ℓ = 2 n ℓ + 1 , n < h ( X ) log X and ℓ ≤ X } ≤ ∑ n = 1 ⌊ h ( X ) log X ⌋ # { ℓ : p ℓ = 2 n ℓ + 1 and ℓ ≤ X } ≤ ∑ n = 1 ⌊ h ( X ) log X ⌋ # { ℓ : 2 n ℓ + 1 is a prime and ℓ ≤ X } .

By [19, Theorem 3.12], for any positive integer n , there is an absolute constant C such that

# { ℓ : 2 n ℓ + 1 is a prime and ℓ ≤ X } ≤ A ∏ 2 < p ∣ n p − 1 p − 2 X ( log X ) 2 1 + C log log X log X ,

where

A ≔ 8 ∏ 2 < p 1 − 1 ( p − 1 ) 2 .

Note that for any positive integer n , we have

∏ 2 < p ∣ n p − 1 p − 2 ≤ ∏ 2 < p p ( p − 1 ) ( p + 1 ) ( p − 2 ) ∏ p ∣ n p + 1 p ≤ ∏ 2 < p p ( p − 1 ) ( p + 1 ) ( p − 2 ) σ ( n ) n .

From this, we have

∑ n = 1 ⌊ h ( X ) log X ⌋ ∏ 2 < p ∣ n p − 1 p − 2 ≤ ∏ 2 < p p ( p − 1 ) ( p + 1 ) ( p − 2 ) ∑ n = 1 ⌊ h ( X ) log X ⌋ σ ( n ) n = ∏ 2 < p p ( p − 1 ) ( p + 1 ) ( p − 2 ) ∑ n = 1 ⌊ h ( X ) log X ⌋ ∑ d ∣ n 1 d ≤ ∏ 2 < p p ( p − 1 ) ( p + 1 ) ( p − 2 ) ∑ d = 1 ⌊ h ( X ) log X ⌋ 1 d ⋅ h ( X ) log X d ≤ π 2 6 ∏ 2 < p p ( p − 1 ) ( p + 1 ) ( p − 2 ) h ( X ) log X .

Thus, (3.5) becomes

# { ℓ : p ℓ ≤ 2 h ( ℓ ) ℓ log ℓ and ℓ ≤ X } ≤ 4 π 2 3 ∏ 2 < p p 2 p 2 − 1 ⋅ h ( X ) X log X 1 + C log log X log X .

Therefore, we conclude that

# { ℓ : ℓ ∉ P and ℓ ≤ X } ≤ 2 2 π 2 3 ∏ 2 < p p 2 p 2 − 1 ⋅ g ( X ) X log X 1 + C log log X log X . □

By using Proposition 3.2, we prove Theorem 1.1.

Proof of Theorem 1.1

Let g ( x ) = ( log x ) − ε . When 0 ≤ ε ≤ 2 , we obtain Theorem 1.1 by Proposition 3.2. If ε > 2 , then there is no prime ℓ satisfying p ℓ 2 ≤ 2 g ( ℓ ) ℓ 2 ( log ℓ ) 2 . Therefore, Theorem 1.1 is proved.□

Now, we prove Theorem 1.5.

Proof of Theorem 1.5

To prove Theorem 1.5, first, we prove that if α ( ℓ , k ) ≥ 1 and k is even, then dim Null ( B k , m ) ≥ 1 for any positive integer m . Since α ( ℓ , k ) ≥ 1 , we have

ω ( Θ r ℓ ( k ) ∕ 2 ( T ) ) = r ℓ ( k ) 2 ⋅ ( ℓ + 1 ) + 1 2 ≤ k + 1 2 .

Then, there is h ∈ M k + 1 2 ( Γ 0 ( 4 ) ; Z ) such that h ≡ Θ r ℓ ( k ) ∕ 2 ( T ) ( mod v ) . Let ( c ( 0 ) , … , c ( k ∕ 2 ) ) ∈ Z ( k ∕ 2 ) + 1 such that

h = ∑ j = 0 k ∕ 2 c ( j ) F 2 j T 2 k + 1 − 4 j .

Then, ( c ( 0 ) ¯ , … , c ( k ∕ 2 ) ¯ ) ∈ Null ( B k , m ) for any positive integer m since h has the form

h ( z ) ≡ 1 2 ∑ n ∈ Z ℓ ∤ n a h ( n ) q n 2 ( mod v ) .

Here, c ( j ) ¯ is the reduction of c ( j ) modulo ℓ . Thus, we conclude that dim Null ( B k , m ) ≥ 1 for any positive integer m , when α ( ℓ , k ) ≥ 1 and k is even.

Now, we assume that Conjecture 1.2 is true. Let v ≔ ( v ( 0 ) ¯ , … , v ( ⌊ k 2 ⌋ ) ¯ ) ∈ Null ( B k , m ) for all positive integers m , and let v ( j ) be an integer such that the reduction of v ( j ) modulo ℓ is equal to v ( j ) ¯ . Let

f v ≔ ∑ j = 0 ⌊ k 2 ⌋ v ( j ) F 2 j T 2 k + 1 − 4 j ∈ M k + 1 2 ( Γ 0 ( 4 ) ) .

Then f v has the form

f v ( z ) ≡ 1 2 ∑ n ∈ Z ℓ ∤ n a f v ( n 2 ) q n 2 ( mod v ) .

Note that f v ≡ Θ ( ℓ − 1 ) ∕ 2 ( f v ) ( mod v ) . We assume that k is even. By the assumption that Conjecture 1.2 is true, we have

f v ≡ a f v ( 1 ) 2 Θ r ℓ ( k ) ∕ 2 ( T ) ( mod v ) .

Thus, lim m → ∞ dim Null ( B k , m ) is less than or equal to 1. If lim m → ∞ dim Null ( B k , m ) = 1 , then there is f ∈ S k + 1 2 ( Γ 0 ( 4 ) ; Z ) such that

f ≡ Θ r ℓ ( k ) ∕ 2 ( T ) ( mod v ) .

This implies that

r ℓ ( k ) ⋅ ℓ + 1 2 + 1 2 = ω ( Θ r ℓ ( k ) ∕ 2 ( T ) ) ≤ k + 1 2 .

By the definition of α ( ℓ , k ) , we have α ( ℓ , k ) ≥ 1 . Hence, we conclude that Conjecture 1.4 is true.

To complete the proof of Theorem 1.5, we assume that Conjecture 1.4 is true. Further, assume that f ∈ S k + 1 2 ( Γ 0 ( 4 ) ; O K ) has the form

Θ ( f ) ≡ 1 2 ∑ n ∈ Z ℓ ∤ n s n 2 a f ( s n 2 ) q s n 2 ( mod v )

with a square-free integer s and Θ ( f ) ≢ 0 ( mod v ) . Then, k is even and s = 1 by Proposition 2.4. By Lemma 2.1, there is f 0 ∈ S k + ℓ + 3 2 ( Γ 0 ( 4 ) ) such that f 0 ≡ Θ ( f ) ( mod v ) . Let ( d ( 0 ) , … , d ( ( k + ℓ + 1 ) ∕ 2 ) ) ∈ O K ( k + ℓ + 3 ) ∕ 2 satisfying

f 0 = ∑ j = 0 ( k + ℓ + 1 ) ∕ 2 d ( j ) F 2 j T 2 k + 2 ℓ + 3 − 4 j .

Let F v ≔ O K ∕ v . Then, for any positive integer m , we have

( d ( 0 ) ¯ , … , d ( ( k + ℓ + 1 ) ∕ 2 ) ¯ ) ∈ Null ( B k + ℓ + 1 , m ) ⊗ F ℓ F v ,

where d ( j ) ¯ is the reduction of d ( j ) modulo v . By the assumption that Conjecture 1.4 is true, the dimension of Null ( B k + ℓ + 1 , m ) is 1 for a sufficiently large m . Hence, f 0 is congruent to a constant multiple of Θ r ℓ ( k + ℓ + 1 ) ∕ 2 ( T ) modulo v . Since r ℓ ( k + ℓ + 1 ) = r ℓ ( k + 2 ) , we conclude that Θ ( f ) is congruent to a constant multiple of Θ r ℓ ( k + 2 ) ∕ 2 ( T ) modulo v .□

We confirm Conjecture 1.2 under the assumption that k ≤ 1,000 , or that ℓ ∈ { 5 , 7 , 11 , 13 , 17 , 19 } and k ≤ 10,000.

Proof of Theorem 1.6

Note that if Θ ( f ) ≡ 0 ( mod v ) , then Conjecture 1.2 is true since a f ( 1 ) ≡ 0 ( mod v ) . Thus, we may assume that Θ ( f ) ≢ 0 ( mod v ) . By Proposition 2.4, s = 1 and k is even. Then, f has the form

f ( z ) ≡ 1 2 ∑ n ∈ Z ℓ ∤ n a f ( n 2 ) q n 2 + ∑ n = 1 ∞ a f ( ℓ n ) q ℓ n ( mod v ) .

From this, we have

( f − Θ ( ℓ − 1 ) ∕ 2 ( f ) ) ( z ) ≡ ∑ n = 1 ∞ a f ( ℓ n ) q ℓ n ( mod v ) .

Assume that k < ℓ − 1 2 . By Lemma 2.1 (3), if f ≢ Θ ( ℓ − 1 ) ∕ 2 ( f ) ( mod v ) , then there is a nonnegative integer k 0 such that

k 0 ≡ k + ℓ − 1 2 ( mod ℓ − 1 ) and k 0 + 1 2 ≤ 1 ℓ k + ℓ 2 2 ,

and there is g 0 ∈ S k 0 + 1 2 ( Γ 0 ( 4 ) ) such that

g 0 ( z ) ≡ ( f − Θ ( ℓ − 1 ) ∕ 2 ( f ) ) ∣ U ℓ ( z ) ≡ ∑ n = 1 ∞ a f ( ℓ n ) q n ( mod v ) .

Since k < ℓ − 1 2 , we have k 0 = 0 and then g 0 = 0 . Thus, f ≡ Θ ( ℓ − 1 ) ∕ 2 ( f ) ( mod v ) when k < ℓ − 1 2 . This implies that f ≡ 0 ( mod v ) by Lemma 2.6. Hence, we conclude that Conjecture 1.2 is true when k < ℓ − 1 2 .

We fix a prime ℓ with 5 ≤ ℓ ≤ 2001 . Assume that there is f ∈ S k + 1 2 ( Γ 0 ( 4 ) ; O K ) having the form

Θ ( f ) ( z ) ≡ 1 2 ∑ n ∈ Z ℓ ∤ n n 2 a f ( n 2 ) q n 2 ( mod v )

such that

Θ ( f ) ≢ a f ( 1 ) 2 Θ r ℓ ( k + 2 ) ∕ 2 ( T ) ( mod v ) .

Then, f ⋅ E ℓ − 1 ∈ S k + ℓ − 1 2 ( Γ 0 ( 4 ) ; O K ) satisfies

Θ ( f ⋅ E ℓ − 1 ) ≢ a f ( 1 ) 2 Θ r ℓ ( k + ℓ + 1 ) ∕ 2 ( T ) ( mod v ) .

Thus, for a positive integer m 0 , confirming Conjecture 1.2 for positive integers k such that k ≤ m 0 reduces to confirming Conjecture 1.2 for positive integers k such that m 0 + 2 − ℓ ≤ k ≤ m 0 .

When max ( 0 , 1,002 − ℓ ) ≤ k ≤ 1,000 and k is even, we obtain by numerical method

dim Null ( B k , 1,000 ) = 1 + ( α ( ℓ , k ) ) .

In the proof of Theorem 1.5, we have dim Null ( B k , m ) ≥ 1 + ( α ( ℓ , k ) ) for any positive integer m . Since dim Null ( B k , m ) ≤ dim Null ( B k , 1,000 ) for m ≥ 1,000 , we have

lim m → ∞ dim Null ( B k , m ) = 1 + ( α ( ℓ , k ) )

when max ( 0 , 1,002 − ℓ ) ≤ k ≤ 1,000 . By Theorem 1.5, we conclude that Conjecture 1.2 is true when k ≤ 1,000 .

The proofs for the cases when ℓ ∈ { 5 , 7 , 11 , 13 , 17 , 19 } and k ≤ 10,000 are similar to the proof of the previous case. So, we skip it.□

Acknowledgements

The authors appreciate referees for careful reading and useful comments. These comments improved the previous version of this article.

  1. Funding information: Dohoon Choi was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2019R1A2C1007517). Youngmin Lee was supported by a KIAS Individual Grant (MG086301) at Korea Institute for Advanced Study.

  2. Conflict of interest: The authors state no conflict of interest.

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Received: 2022-06-15
Revised: 2022-08-31
Accepted: 2022-09-21
Published Online: 2022-10-28

© 2022 Dohoon Choi and Youngmin Lee, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Artikel in diesem Heft

  1. Regular Articles
  2. A random von Neumann theorem for uniformly distributed sequences of partitions
  3. Note on structural properties of graphs
  4. Mean-field formulation for mean-variance asset-liability management with cash flow under an uncertain exit time
  5. The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients
  6. The intersection graph of graded submodules of a graded module
  7. Isoperimetric and Brunn-Minkowski inequalities for the (p, q)-mixed geominimal surface areas
  8. On second-order fuzzy discrete population model
  9. On certain functional equation in prime rings
  10. General complex Lp projection bodies and complex Lp mixed projection bodies
  11. Some results on the total proper k-connection number
  12. The stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects
  13. Well posedness of magnetohydrodynamic equations in 3D mixed-norm Lebesgue space
  14. Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems
  15. Generic uniqueness of saddle point for two-person zero-sum differential games
  16. Relational representations of algebraic lattices and their applications
  17. Explicit construction of mock modular forms from weakly holomorphic Hecke eigenforms
  18. The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property
  19. Arithmetic convolution sums derived from eta quotients related to divisors of 6
  20. Dynamical behaviors of a k-order fuzzy difference equation
  21. The transfer ideal under the action of orthogonal group in modular case
  22. The multinomial convolution sum of a generalized divisor function
  23. Extensions of Gronwall-Bellman type integral inequalities with two independent variables
  24. Unicity of meromorphic functions concerning differences and small functions
  25. Solutions to problems about potentially Ks,t-bigraphic pair
  26. Monotonicity of solutions for fractional p-equations with a gradient term
  27. Data smoothing with applications to edge detection
  28. An ℋ-tensor-based criteria for testing the positive definiteness of multivariate homogeneous forms
  29. Characterizations of *-antiderivable mappings on operator algebras
  30. Initial-boundary value problem of fifth-order Korteweg-de Vries equation posed on half line with nonlinear boundary values
  31. On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions
  32. On split twisted inner derivation triple systems with no restrictions on their 0-root spaces
  33. Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry
  34. Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting
  35. A posteriori error estimates of characteristic mixed finite elements for convection-diffusion control problems
  36. Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect
  37. An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem
  38. Connectivity with respect to α-discrete closure operators
  39. Khasminskii-type theorem for a class of stochastic functional differential equations
  40. On some new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in (p, q)-calculus with applications
  41. New properties for the Ramanujan R-function
  42. Shooting method in the application of boundary value problems for differential equations with sign-changing weight function
  43. Ground state solution for some new Kirchhoff-type equations with Hartree-type nonlinearities and critical or supercritical growth
  44. Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays
  45. Ambrosetti-Prodi-type results for a class of difference equations with nonlinearities indefinite in sign
  46. Research of cooperation strategy of government-enterprise digital transformation based on differential game
  47. Malmquist-type theorems on some complex differential-difference equations
  48. Disjoint diskcyclicity of weighted shifts
  49. Construction of special soliton solutions to the stochastic Riccati equation
  50. Remarks on the generalized interpolative contractions and some fixed-point theorems with application
  51. Analysis of a deteriorating system with delayed repair and unreliable repair equipment
  52. On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields
  53. The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods
  54. Regularity of models associated with Markov jump processes
  55. Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods
  56. Minimal period problem for second-order Hamiltonian systems with asymptotically linear nonlinearities
  57. Convergence rate of the modified Levenberg-Marquardt method under Hölderian local error bound
  58. Non-binary quantum codes from constacyclic codes over 𝔽q[u1, u2,…,uk]/⟨ui3 = ui, uiuj = ujui⟩
  59. On the general position number of two classes of graphs
  60. A posteriori regularization method for the two-dimensional inverse heat conduction problem
  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
  63. A note on commutators of strongly singular Calderón-Zygmund operators
  64. Generalized Munn rings
  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
  67. On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation
  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
  70. Class fields generated by coordinates of elliptic curves
  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
  74. Displacement structure of the DMP inverse
  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
  81. Eigenvalues of transition weight matrix for a family of weighted networks
  82. A modified Tikhonov regularization for unknown source in space fractional diffusion equation
  83. Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ
  84. Some estimates for commutators of bilinear pseudo-differential operators
  85. Extension of isometries in real Hilbert spaces
  86. Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays
  87. B-Fredholm elements in primitive C*-algebras
  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
  89. An algebraic semigroup method for discovering maximal frequent itemsets
  90. Class-preserving Coleman automorphisms of some classes of finite groups
  91. Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay
  92. Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses
  93. The transitivity of primary conjugacy in regular ω-semigroups
  94. Stability estimation of some Markov controlled processes
  95. On nonnil-coherent modules and nonnil-Noetherian modules
  96. N-Tuples of weighted noncommutative Orlicz space and some geometrical properties
  97. The dimension-free estimate for the truncated maximal operator
  98. A human error risk priority number calculation methodology using fuzzy and TOPSIS grey
  99. Compact mappings and s-mappings at subsets
  100. The structural properties of the Gompertz-two-parameter-Lindley distribution and associated inference
  101. A monotone iteration for a nonlinear Euler-Bernoulli beam equation with indefinite weight and Neumann boundary conditions
  102. Delta waves of the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas
  103. Multiplicity and minimality of periodic solutions to fourth-order super-quadratic difference systems
  104. On the reciprocal sum of the fourth power of Fibonacci numbers
  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
  107. Study on r-truncated degenerate Stirling numbers of the second kind
  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
  111. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
https://doi.org/10.1515/math-2022-0512
Schlagwörter für diesen Artikel
Fourier coefficients of modular forms; Galois representations; modular forms of half-integral weight; theta functions
Creative Commons
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Artikel in diesem Heft

  1. Regular Articles
  2. A random von Neumann theorem for uniformly distributed sequences of partitions
  3. Note on structural properties of graphs
  4. Mean-field formulation for mean-variance asset-liability management with cash flow under an uncertain exit time
  5. The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients
  6. The intersection graph of graded submodules of a graded module
  7. Isoperimetric and Brunn-Minkowski inequalities for the (p, q)-mixed geominimal surface areas
  8. On second-order fuzzy discrete population model
  9. On certain functional equation in prime rings
  10. General complex Lp projection bodies and complex Lp mixed projection bodies
  11. Some results on the total proper k-connection number
  12. The stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects
  13. Well posedness of magnetohydrodynamic equations in 3D mixed-norm Lebesgue space
  14. Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems
  15. Generic uniqueness of saddle point for two-person zero-sum differential games
  16. Relational representations of algebraic lattices and their applications
  17. Explicit construction of mock modular forms from weakly holomorphic Hecke eigenforms
  18. The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property
  19. Arithmetic convolution sums derived from eta quotients related to divisors of 6
  20. Dynamical behaviors of a k-order fuzzy difference equation
  21. The transfer ideal under the action of orthogonal group in modular case
  22. The multinomial convolution sum of a generalized divisor function
  23. Extensions of Gronwall-Bellman type integral inequalities with two independent variables
  24. Unicity of meromorphic functions concerning differences and small functions
  25. Solutions to problems about potentially Ks,t-bigraphic pair
  26. Monotonicity of solutions for fractional p-equations with a gradient term
  27. Data smoothing with applications to edge detection
  28. An ℋ-tensor-based criteria for testing the positive definiteness of multivariate homogeneous forms
  29. Characterizations of *-antiderivable mappings on operator algebras
  30. Initial-boundary value problem of fifth-order Korteweg-de Vries equation posed on half line with nonlinear boundary values
  31. On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions
  32. On split twisted inner derivation triple systems with no restrictions on their 0-root spaces
  33. Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry
  34. Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting
  35. A posteriori error estimates of characteristic mixed finite elements for convection-diffusion control problems
  36. Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect
  37. An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem
  38. Connectivity with respect to α-discrete closure operators
  39. Khasminskii-type theorem for a class of stochastic functional differential equations
  40. On some new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in (p, q)-calculus with applications
  41. New properties for the Ramanujan R-function
  42. Shooting method in the application of boundary value problems for differential equations with sign-changing weight function
  43. Ground state solution for some new Kirchhoff-type equations with Hartree-type nonlinearities and critical or supercritical growth
  44. Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays
  45. Ambrosetti-Prodi-type results for a class of difference equations with nonlinearities indefinite in sign
  46. Research of cooperation strategy of government-enterprise digital transformation based on differential game
  47. Malmquist-type theorems on some complex differential-difference equations
  48. Disjoint diskcyclicity of weighted shifts
  49. Construction of special soliton solutions to the stochastic Riccati equation
  50. Remarks on the generalized interpolative contractions and some fixed-point theorems with application
  51. Analysis of a deteriorating system with delayed repair and unreliable repair equipment
  52. On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields
  53. The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods
  54. Regularity of models associated with Markov jump processes
  55. Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods
  56. Minimal period problem for second-order Hamiltonian systems with asymptotically linear nonlinearities
  57. Convergence rate of the modified Levenberg-Marquardt method under Hölderian local error bound
  58. Non-binary quantum codes from constacyclic codes over 𝔽q[u1, u2,…,uk]/⟨ui3 = ui, uiuj = ujui⟩
  59. On the general position number of two classes of graphs
  60. A posteriori regularization method for the two-dimensional inverse heat conduction problem
  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
  63. A note on commutators of strongly singular Calderón-Zygmund operators
  64. Generalized Munn rings
  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
  67. On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation
  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
  70. Class fields generated by coordinates of elliptic curves
  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
  74. Displacement structure of the DMP inverse
  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
  81. Eigenvalues of transition weight matrix for a family of weighted networks
  82. A modified Tikhonov regularization for unknown source in space fractional diffusion equation
  83. Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ
  84. Some estimates for commutators of bilinear pseudo-differential operators
  85. Extension of isometries in real Hilbert spaces
  86. Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays
  87. B-Fredholm elements in primitive C*-algebras
  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
  89. An algebraic semigroup method for discovering maximal frequent itemsets
  90. Class-preserving Coleman automorphisms of some classes of finite groups
  91. Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay
  92. Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses
  93. The transitivity of primary conjugacy in regular ω-semigroups
  94. Stability estimation of some Markov controlled processes
  95. On nonnil-coherent modules and nonnil-Noetherian modules
  96. N-Tuples of weighted noncommutative Orlicz space and some geometrical properties
  97. The dimension-free estimate for the truncated maximal operator
  98. A human error risk priority number calculation methodology using fuzzy and TOPSIS grey
  99. Compact mappings and s-mappings at subsets
  100. The structural properties of the Gompertz-two-parameter-Lindley distribution and associated inference
  101. A monotone iteration for a nonlinear Euler-Bernoulli beam equation with indefinite weight and Neumann boundary conditions
  102. Delta waves of the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas
  103. Multiplicity and minimality of periodic solutions to fourth-order super-quadratic difference systems
  104. On the reciprocal sum of the fourth power of Fibonacci numbers
  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
  107. Study on r-truncated degenerate Stirling numbers of the second kind
  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
  111. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
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