Paramodular Siegel Eisenstein Series

• Paramodular Siegel Eisenstein series are holomorphic genus‑2 modular forms on paramodular groups built as specific linear combinations of Eisenstein series attached to 0‑dimensional cusps.
• They are constructed using both classical and adelic methods that integrate induced representations for GSp₄, yielding explicit Fourier expansions and newform properties at level N².
• The arithmetic study demonstrates that all Fourier coefficients lie in a precise number field, linking Dirichlet characters, Gauss sums, and special L‑values to ensure algebraicity.

Paramodular Siegel Eisenstein series are Siegel modular forms attached to paramodular groups, most explicitly in genus $2$ as holomorphic Eisenstein series of weight $k\ge 4$ and paramodular level $N^2$ built from a primitive Dirichlet character $\eta$ of conductor $N$. In the genus-$2$ construction developed in 2025, the series $E_{k,\eta}(Z)$ is defined both classically and adelically for $\mathrm{GSp}_4$, its Fourier expansion is computed, and it is shown to be a paramodular newform whose adelization generates an irreducible automorphic representation; a subsequent arithmetic study proves that all of its Fourier coefficients lie in an explicit number field (Pierce et al., 4 Sep 2025, Pierce, 26 Oct 2025).

1. Definition in genus $2$ and the paramodular group

For degree $2$, the ambient Hermitian symmetric domain is the Siegel upper half space

$k\ge 4$0

with

$k\ge 4$1

If

$k\ge 4$2

then

$k\ge 4$3

A scalar Siegel modular form of degree $k\ge 4$4 and weight $k\ge 4$5 for a discrete subgroup $k\ge 4$6 is a holomorphic function $k\ge 4$7 satisfying

$k\ge 4$8

with slash operator

$k\ge 4$9

The relevant level subgroup is the paramodular group $N^2$0, in the genus-$N^2$1 setting. In the 2025 construction, the pertinent level is $N^2$2. These groups are described as arithmetic subgroups of $N^2$3 or $N^2$4, depending on the chosen model, and are natural from the viewpoint of abelian surfaces with polarization of type $N^2$5. They also feature prominently in the paramodular conjecture (Pierce et al., 4 Sep 2025).

Fix an even integer $N^2$6 and a primitive Dirichlet character $N^2$7 of conductor $N^2$8. For

$N^2$9

the cusp Eisenstein series attached to $\eta$0 is

$\eta$1

The paramodular Siegel Eisenstein series attached to $\eta$2 is then

$\eta$3

In this sense, a paramodular Siegel Eisenstein series of level $\eta$4 is a holomorphic genus-$\eta$5 Siegel modular form obtained as a specific linear combination of Eisenstein series attached to $\eta$6-dimensional cusps.

2. Adelic construction and the origin of the level $\eta$7

The construction begins with the primitive Dirichlet character $\eta$8 and its associated idele class character

$\eta$9

The parity condition is

$N$0

From $N$1 one forms the normalized induced representation

$N$2

In the flat model, for

$N$3

one has

$N$4

At the archimedean place, $N$5 contains a unique scalar $N$6-type $N$7, denoted $N$8, and one chooses a section $N$9 spanning that type, normalized by $2$0. For $2$1, this vector lies in the holomorphic discrete series $2$2. At finite places, one chooses local paramodular newforms and assembles them into the global section

$2$3

The adelic Eisenstein series is

$2$4

It is right-invariant under $2$5, and the resulting global level is exactly $2$6. The explanation given in the construction is representation-theoretic: the conductor of the induced representation is $2$7, and this is the smallest paramodular level at which a non-zero paramodular vector exists (Pierce et al., 4 Sep 2025).

To descend from the adelic setting to $2$8, one uses the standard point $2$9 and, for $E_{k,\eta}(Z)$0, the unique upper triangular element $E_{k,\eta}(Z)$1 with positive diagonal such that $E_{k,\eta}(Z)$2. If $E_{k,\eta}(Z)$3 satisfies the $E_{k,\eta}(Z)$4-equivariance determined by $E_{k,\eta}(Z)$5, then

$E_{k,\eta}(Z)$6

Applied to $E_{k,\eta}(Z)$7, this yields the classical form $E_{k,\eta}(Z)$8. The identification relies on a double coset decomposition of $E_{k,\eta}(Z)$9 relative to $\mathrm{GSp}_4$0 and $\mathrm{GSp}_4$1, using Poor–Yuen’s classification of $\mathrm{GSp}_4$2-dimensional cusps.

3. Fourier expansion and local–global coefficient formulas

The Fourier expansion of the genus-$\mathrm{GSp}_4$3 paramodular Eisenstein series has the form

$\mathrm{GSp}_4$4

where

$\mathrm{GSp}_4$5

and the support condition

$\mathrm{GSp}_4$6

is built into the expansion. The constant term is $\mathrm{GSp}_4$7 when $\mathrm{GSp}_4$8 is trivial and $\mathrm{GSp}_4$9 otherwise (Pierce et al., 4 Sep 2025).

On the adelic side, the coefficient attached to $2$0 is obtained from the Fourier integral

$2$1

and factors as

$2$2

For rank $2$3, the Bruhat decomposition yields four terms $2$4, but only $2$5 is non-zero. At the real place, for $2$6,

$2$7

For $2$8, this becomes

$2$9

This is the point at which the Koecher principle appears in the calculation: the archimedean integral vanishes unless $2$0 is positive semidefinite.

For non-archimedean places not dividing $2$1, the coefficients are expressed in terms of the discriminant factorization

$2$2

with $2$3 a fundamental discriminant. The multiplicative arithmetic factor is packaged into

$2$4

For primes $2$5, the ramified local integral has the form

$2$6

where $2$7 is a local integral. In certain cases, specifically for odd $2$8 and quadratic $2$9, this local factor can be expressed in terms of point counts on elliptic curves over $k\ge 4$00 (Pierce et al., 4 Sep 2025).

The final coefficient formulas separate rank $k\ge 4$01 and rank $k\ge 4$02. In rank $k\ge 4$03, the coefficient is piecewise: the full-level term $k\ge 4$04 occurs only when $k\ge 4$05 and

$k\ge 4$06

while the paramodular case involves $k\ge 4$07, the condition $k\ge 4$08, and

$k\ge 4$09

otherwise the coefficient is zero.

In rank $k\ge 4$10, one obtains

$k\ge 4$11

with

$k\ge 4$12

For $k\ge 4$13, these formulas reduce to the classical Maass and Eichler–Zagier formulas for the degree-$k\ge 4$14 Eisenstein series on $k\ge 4$15.

4. Paramodular newforms and automorphic representation theory

The genus-$k\ge 4$16 paramodular Eisenstein series $k\ge 4$17 is not merely an Eisenstein series with a prescribed level. It is shown to be a paramodular newform in the sense of the local newform theory for $k\ge 4$18 developed by Roberts–Schmidt. For each finite prime $k\ge 4$19, if

$k\ge 4$20

then $k\ge 4$21 is irreducible of type IIb, and its chosen $k\ge 4$22-fixed vector is the local paramodular newform (Pierce et al., 4 Sep 2025).

Globally, the automorphic representation generated by the adelization of $k\ge 4$23 is

$k\ge 4$24

with

$k\ge 4$25

The representation $k\ge 4$26 is irreducible, and the adelization of the classical form $k\ge 4$27 is exactly the adelic Eisenstein series $k\ge 4$28.

The conductor statement is equally precise: each local factor $k\ge 4$29 has minimal paramodular level $k\ge 4$30, hence the global paramodular conductor is $k\ge 4$31. Consequently, $k\ge 4$32 is new at level $k\ge 4$33: any trace, or oldform projection, from level $k\ge 4$34 down to a paramodular subgroup $k\ge 4$35 with

$k\ge 4$36

is zero. This distinguishes the series from the older full-level degree-$k\ge 4$37 Siegel Eisenstein series and places it as a genuine newform inside the Eisenstein spectrum.

A common conflation in the literature is between “paramodular Eisenstein series” as a level condition and “paramodular Eisenstein newforms” as an oldform–newform statement. In the present genus-$k\ge 4$38 theory, the latter is proved rather than assumed.

5. Arithmeticity and rationality of Fourier coefficients

A separate 2025 paper studies the arithmetic of the same genus-$k\ge 4$39 series and proves a rationality theorem for its Fourier expansion. For even $k\ge 4$40 and primitive $k\ge 4$41 of conductor $k\ge 4$42, all Fourier coefficients of $k\ge 4$43 lie in the explicit number field

$k\ge 4$44

More precisely, if

$k\ge 4$45

then the coefficient fields refine as follows: $k\ge 4$46 and

$k\ge 4$47

where $k\ge 4$48 is the primitive Dirichlet character associated to $k\ge 4$49 and $k\ge 4$50 is the primitive Dirichlet character associated to $k\ge 4$51 (Pierce, 26 Oct 2025).

The proof is local–global. The local ramified integrals are shown to satisfy

$k\ge 4$52

so the genuinely local correction terms are already algebraic in the character field. The global transcendental factors are controlled by special-value formulas for $k\ge 4$53 and $k\ge 4$54, together with Gauss sums and $k\ge 4$55-factors. One key identity is

$k\ge 4$56

Another is the control of the mixed quadratic twist: $k\ge 4$57

The arithmetic conclusion is that no transcendental constants besides $k\ge 4$58 and $k\ge 4$59 appear in the coefficient formulas, and the powers of $k\ge 4$60 cancel after substituting the algebraicity formulas for the relevant Dirichlet $k\ge 4$61-values. In particular, $k\ge 4$62 admits a model over $k\ge 4$63. When $k\ge 4$64, the coefficient field collapses to $k\ge 4$65, recovering the rationality of the full-level classical Siegel Eisenstein series.

6. Related variants, comparisons, and broader theory

The 2025 genus-$k\ge 4$66 series $k\ge 4$67 is a Siegel-parabolic Eisenstein series built from $k\ge 4$68-dimensional cusps of $k\ge 4$69. This should be distinguished from the paramodular Klingen Eisenstein series studied earlier. If $k\ge 4$70 is an elliptic newform, the paramodular Klingen Eisenstein series is

$k\ge 4$71

a Siegel modular form on $k\ge 4$72. Its pullback along the embedded $k\ge 4$73 has the form

$k\ge 4$74

with $k\ge 4$75. The level-raising phenomenon is the same: starting from level $k\ge 4$76 on $k\ge 4$77, the paramodular Klingen Eisenstein series lives naturally at level $k\ge 4$78 (Shukla, 2020).

A second broad generalization concerns higher degree and squarefree level. For a squarefree matrix level $k\ge 4$79, Böcherer–Schulze-Pillot define the paramodular Siegel Eisenstein series

$k\ge 4$80

and prove that for $k\ge 4$81 the Eisenstein space is one-dimensional. In this setting the normalized Eisenstein series is identified, via the Siegel–Weil formula, with the genus theta series attached to the unique genus of paramodular lattice chains. This higher-degree theory is part of a larger program involving Hecke algebras, boundary components, Garrett’s doubling method, and the basis problem for paramodular cusp forms (Böcherer et al., 2020).

Comparison with the full-level genus-$k\ge 4$82 series clarifies what is genuinely paramodular in the 2025 construction. The differences are the level $k\ge 4$83 rather than $k\ge 4$84, the twist by a primitive Dirichlet character $k\ge 4$85, the support condition $k\ge 4$86 in the Fourier expansion, and the ramified local factors $k\ge 4$87 at primes dividing $k\ge 4$88. For $k\ge 4$89 and $k\ge 4$90, the formulas reduce exactly to the classical full-level formulas (Pierce et al., 4 Sep 2025).

A further analytic perspective comes from the study of real-analytic Siegel Eisenstein series at full level. Nagaoka gives an explicit residue formula at $k\ge 4$91 in terms of a constant term built from Koecher–Maass zeta data and a rank-$k\ge 4$92 Fourier expansion with confluent hypergeometric factors. The accompanying exposition states that many of these structures are local or independent of level. This suggests a template for future paramodular residue formulas: the archimedean hypergeometric factors should remain unchanged, while the finite local factors should be modified at primes dividing the paramodular level (Nagaoka, 2021).

Two misconceptions are thereby avoided. First, not every “paramodular Eisenstein series” is of the same parabolic type: Siegel-parabolic constructions such as $k\ge 4$93 and Klingen-type constructions such as $k\ge 4$94 are distinct objects. Second, in the constructions presently available, the natural paramodular level is often $k\ge 4$95, not $k\ge 4$96. This is not an artifact of normalization, but a reflection of the conductor of the underlying induced representation and of the local paramodular invariants.