Paramodular Newform Overview

• Paramodular newforms are degree-2 Siegel modular forms on the paramodular group K(N) with the minimal local conductor, serving as new vectors for GSp(4) automorphic representations.
• They play a crucial role in arithmetic by connecting typically simple abelian surfaces over Q with spinor L-functions, as predicted by the Brumer–Kramer conjecture.
• Their construction and analysis involve methods such as Gritsenko lifts, Borcherds products, and refined Hecke operator techniques, which enable efficient eigenvalue computations.

A paramodular newform is a degree-$2$ Siegel modular form on the paramodular group $K(N)$ that is new at level $N$, equivalently a cuspidal Siegel eigenform whose attached automorphic representation of $\mathrm{GSp}(4)$ has minimal local paramodular conductor $N$; in the local representation-theoretic formulation, a paramodular newform at a finite place $v$ is a vector fixed by $K(\mathfrak p_v^{N_v})$ with $N_v$ minimal among compact subgroups fixing a nonzero vector (Brumer et al., 2018, Johnson-Leung et al., 15 Jan 2025). In weight $2$, paramodular newforms are central to the Brumer–Kramer paramodularity conjecture, which relates typical abelian surfaces over $\mathbb Q$ to nonlift paramodular newforms through equality of spinor and Hasse–Weil $K(N)$0-functions (Brumer et al., 2010).

1. Ambient groups, level structure, and classical realization

The ambient algebraic group is

$K(N)$1

with $K(N)$2. For a global level $K(N)$3, the global paramodular subgroup $K(N)$4 is the subgroup of $K(N)$5 consisting of matrices with the standard paramodular entry pattern

$K(N)$6

and at a finite place $K(N)$7 the local subgroup $K(N)$8 is defined by the analogous $K(N)$9-integrality conditions in $N$0 (Johnson-Leung et al., 15 Jan 2025). In the rational setting, $N$1 is also described as the natural modular group for $N$2-polarized abelian surfaces and as a stabilizer of a paramodular lattice (Brumer et al., 2010).

A scalar-weight Siegel paramodular form is a holomorphic function $N$3 on the Siegel upper half-space $N$4 satisfying

$N$5

or, in slash notation, $N$6. For arithmetic subgroups commensurable with $N$7, the resulting spaces $N$8 and $N$9 are the spaces of Siegel modular forms and cusp forms of degree $\mathrm{GSp}(4)$0 and level $\mathrm{GSp}(4)$1 (Brumer et al., 2018, Johnson-Leung et al., 15 Jan 2025).

The paramodular group is normalized by a Fricke involution. In the notation of the 2018 verification paper, $\mathrm{GSp}(4)$2 is normalized by the paramodular Fricke involution $\mathrm{GSp}(4)$3, and the Fricke action gives a plus/minus decomposition

$\mathrm{GSp}(4)$4

This decomposition is fundamental in explicit constructions, especially for distinguishing lift and nonlift components and for organizing Atkin–Lehner eigenvalues (Brumer et al., 2018).

2. Local newness, Hecke operators, and operator algebras

The local theory attaches to an irreducible admissible representation $\mathrm{GSp}(4)$5 of $\mathrm{GSp}(4)$6 the spaces

$\mathrm{GSp}(4)$7

A representation is paramodular if $\mathrm{GSp}(4)$8 for some $\mathrm{GSp}(4)$9, and its minimal such $N$0 is the paramodular level $N$1; the paramodular newform is a nonzero vector in $N$2. For paramodular $N$3, one has $N$4, and for $N$5 the higher-level spaces are generated from the newvector by level-raising operators $N$6 (Johnson-Leung et al., 2022). In the global cuspidal setting, Roberts–Schmidt newform theory implies that if $N$7 is new, then $N$8 is a Hecke eigenform at all $N$9 and all paramodular Atkin–Lehner involutions (Brumer et al., 2018).

For primes $v$0, the principal Hecke operators are

$v$1

and in weight $v$2 one also uses the integral combination $v$3. If $v$4 is a common eigenform with eigenvalues $v$5 and $v$6, then the spinor Hecke polynomial is

$v$7

which in weight $v$8 becomes

$v$9

These polynomials are the Euler factors entering the spin $K(\mathfrak p_v^{N_v})$0-function of the form (Brumer et al., 2018).

The local structure at primes dividing the level is subtler. A paramodular newform at $K(\mathfrak p_v^{N_v})$1 is fixed by $K(\mathfrak p_v^{N_v})$2 with $K(\mathfrak p_v^{N_v})$3 minimal, and stable Klingen congruence subgroups provide an auxiliary filtration $K(\mathfrak p_v^{N_v})$4 relating local paramodular vectors to three upper-block operators: two stable Klingen Hecke operators and one level-lowering operator. This yields a partition of paramodular representations into two classes, with category $K(\mathfrak p_v^{N_v})$5 characterized by the absence of unramified one-dimensional factors in the $K(\mathfrak p_v^{N_v})$6-parameter decomposition; for generic $K(\mathfrak p_v^{N_v})$7, category $K(\mathfrak p_v^{N_v})$8 is equivalent to $K(\mathfrak p_v^{N_v})$9 (Johnson-Leung et al., 2022).

A structural distinction also appears at the level of Hecke algebras. For squarefree $N_v$0, the Hecke algebra of the maximal discrete normal extension $N_v$1 is commutative and each local primary component is a polynomial ring in two algebraically independent generators, whereas the Hecke algebra of the non-maximal paramodular group $N_v$2 fails to be commutative if $N_v$3. In particular, at $N_v$4 the Atkin–Lehner double coset does not commute with one of the standard local Hecke operators, and zero-divisors appear in the non-maximal setting (Gallenkämper et al., 2017). This clarifies why local newform theory is typically formulated either with the local $N_v$5-adic paramodular subgroup or with the maximal extension rather than with the global non-maximal Hecke algebra.

3. Paramodularity, $N_v$6-functions, and Galois representations

The arithmetic significance of paramodular newforms is encoded in the Brumer–Kramer conjecture. In its weight-$N_v$7 genus-$N_v$8 form, the conjecture asserts that if $N_v$9 is an abelian surface of conductor $2$0 with $2$1, then there exists a cuspidal paramodular newform

$2$2

that is not a Gritsenko lift, has rational Hecke eigenvalues, is unique up to scaling, and satisfies

$2$3

At a good prime $2$4, the local factor of $2$5 has the shape

$2$6

while the spinor polynomial of $2$7 has the shape

$2$8

and the conjectural identity becomes $2$9 (Brumer et al., 2018, Brumer et al., 2010).

The associated Galois representations are likewise central. For a polarized abelian surface $\mathbb Q$0, the $\mathbb Q$1-adic Tate module gives

$\mathbb Q$2

and for a paramodular newform $\mathbb Q$3 of type $\mathbb Q$4, one has a continuous semisimple representation

$\mathbb Q$5

with characteristic polynomial at good Frobenius equal to $\mathbb Q$6. Serre’s appendix on reduction of $\mathbb Q$7-covariant bilinear forms ensures that residual representations retain the symplectic structure after semisimplification, which is decisive for comparison arguments in $\mathbb Q$8 (Brumer et al., 2018).

The comparison method used in rigorous cases generalizes the Faltings–Serre method from $\mathbb Q$9 to general $K(N)$00. In the 2018 verification paper, a deterministic algorithm based only on Frobenius traces decides equivalence of two $K(N)$01-adic representations and produces a witness prime if they differ. The method combines residual comparison, deformation cocycles, class field theory, exact core-free subextensions, and local conductor control at primes dividing $K(N)$02 (Brumer et al., 2018).

The conjectural arithmetic class is the class of typical abelian surfaces, meaning $K(N)$03 with $K(N)$04. It is shown that $K(N)$05 is typical if $K(N)$06 is simple, semistable, and of nonsquare conductor, and in particular if the conductor is prime (Brumer et al., 2018). This typicality hypothesis excludes extra endomorphisms and isolates the genuinely degree-$K(N)$07 case relevant to nonlift paramodular forms.

4. Construction mechanisms

Several independent constructions produce paramodular newforms or spaces in which paramodular newforms live. The basic additive construction is the Gritsenko lift from Jacobi cusp forms

$K(N)$08

which is Hecke-equivariant and furnishes the lift subspace. In weight $K(N)$09, nonlifts are defined by lying outside this image, and the prime-level construction paper completed the construction of all nonlift weight-two cusp paramodular Hecke eigenforms for prime levels $K(N)$10, using Borcherds products and, at levels $K(N)$11 and $K(N)$12, trace-down from level $K(N)$13 (Poor et al., 2018).

Borcherds products provide the dominant explicit nonlift mechanism in low weight. In the antisymmetric setting, meromorphic weight-$K(N)$14 Jacobi forms yield holomorphic antisymmetric paramodular Borcherds products whenever the Humbert multiplicities are nonnegative. This construction produces the antisymmetric nonlift in $K(N)$15, and further weight-$K(N)$16 examples at levels $K(N)$17 and $K(N)$18, together with weight-$K(N)$19 examples furnishing antisymmetric canonical differential forms on Siegel modular threefolds (Poor et al., 2016).

A distinct automorphic construction comes from Hilbert modular forms over real quadratic fields. Johnson–Leung and Roberts prove that if $K(N)$20 is real quadratic and $K(N)$21 is a cuspidal irreducible automorphic representation of $K(N)$22 with trivial central character, infinity type $K(N)$23, and not Galois invariant, then there exists a non-zero holomorphic Siegel paramodular newform $K(N)$24 of weight $K(N)$25 and explicitly determined paramodular level $K(N)$26, with

$K(N)$27

for every prime $K(N)$28 (Johnson-Leung et al., 2010).

The 2025 theta-lift paper makes this construction explicit at the local level. For a real quadratic extension $K(N)$29, it constructs local test data that produce a nonzero $K(N)$30-fixed vector in the local theta lift $K(N)$31 at every finite place except when the local extension has wild ramification. The conductor exponent is determined directly from the $K(N)$32 conductor data: $K(N)$33 and the paper states explicitly that no suitable Schwartz functions were found in the wildly ramified case (Johnson-Leung et al., 15 Jan 2025).

Another source is the symmetric cube lift from non-CM elliptic curves. If $K(N)$34 is a non-CM elliptic curve, then the symmetric cube transfer produces a holomorphic degree-$K(N)$35 Siegel cusp form $K(N)$36 of scalar weight $K(N)$37, paramodular at level

$K(N)$38

and satisfying

$K(N)$39

The local conductor exponents are computed case by case from the reduction type of $K(N)$40 and the local $K(N)$41 representation $K(N)$42 (Roy, 2019).

Finally, the term “paramodular newform” is not confined to cuspidal forms in every paper. A 2025 construction starts from a primitive Dirichlet character of conductor $K(N)$43, produces a holomorphic Siegel Eisenstein series of paramodular level $K(N)$44, computes its Fourier expansion, and states that the resulting function is a paramodular newform whose adelization generates an irreducible automorphic representation (Pierce et al., 4 Sep 2025). This shows that the newvector formalism extends beyond the cuspidal case, even though the arithmetic literature on abelian surfaces is mainly concerned with cuspidal nonlifts.

5. Computation, identification, and explicit extraction of Hecke data

The identification of a paramodular newform in explicit spaces depends on specialized computational frameworks. One such method is specialization to modular curves. For a suitable positive-definite symmetric matrix $K(N)$45, the pullback

$K(N)$46

doubles the weight and preserves cuspidality, converting a degree-$K(N)$47 Siegel form into a one-variable $K(N)$48-series. In the 2018 verification paper, key cancellation identities reduce the cost of computing $K(N)$49 from $K(N)$50 sums to $K(N)$51, making the Hecke eigenvalues $K(N)$52 and $K(N)$53 accessible by coefficient comparison (Brumer et al., 2018).

At primes with $K(N)$54, classical paramodular Hecke operators do not admit simple upper-block formulas. The stable Klingen framework replaces them by $K(N)$55, $K(N)$56, and a level-lowering operator $K(N)$57. For a newform $K(N)$58, these operators yield explicit criteria for $K(N)$59 or $K(N)$60, formulas recovering $K(N)$61 and $K(N)$62 from Fourier coefficients, and a rational generating function for radial coefficients whose denominator is $K(N)$63. In particular, when $K(N)$64, the paper shows

$K(N)$65

in the stable Klingen normalization (Johnson-Leung et al., 2022).

A different computational bridge is provided by algebraic modular forms and quinary lattices. For squarefree levels, the exact relationship between algebraic modular forms on a definite quaternionic unitary group and genera of positive-definite quinary lattices identifies the Hecke operator $K(N)$66 with the $K(N)$67-neighbor operator and $K(N)$68 with the $K(N)$69-neighbor operator. The resulting adjacency matrices can be diagonalized to produce Hecke eigenvalues for paramodular forms of general type, and the Atkin–Lehner signs are isolated by explicit sign characters $K(N)$70 (Dummigan et al., 2021).

In the low-weight, low-level computations of weight $K(N)$71, Jacobi restriction remains a principal tool. It organizes a paramodular form through finitely many Fourier–Jacobi coefficients, constructs finite-dimensional superspaces $K(N)$72 of truncated data, and, together with reduction modulo a prime and Atkin–Lehner “infilling,” certifies dimensions and nonlift status. This method underlies the determination of $K(N)$73 for squarefree $K(N)$74 and of nonlift prime-level forms below $K(N)$75 (Poor et al., 2016, Poor et al., 2018).

6. Explicit cases, congruences, and current extensions

The strongest complete verifications currently available are the prime-conductor cases $K(N)$76, $K(N)$77, and $K(N)$78. For $K(N)$79, there is a unique nonlift $K(N)$80, constructed as a rational function of ten Gritsenko lifts of weight-$K(N)$81 theta blocks; the paper computes Hecke data up to $K(N)$82, compares traces of the associated mod-$K(N)$83 Galois representations at a finite witness set, and proves

$K(N)$84

for the Jacobian $K(N)$85 of conductor $K(N)$86. The same strategy proves the analogous statements for $K(N)$87 and for the antisymmetric Borcherds product $K(N)$88 (Brumer et al., 2018).

At composite squarefree level below $K(N)$89, the structure is more rigid. The space $K(N)$90 equals the additive Gritsenko lift space for all composite squarefree $K(N)$91 except $K(N)$92 and $K(N)$93, and in each of these two exceptional cases there is exactly one additional nonlift newform line. For those levels, the spin $K(N)$94-Euler factors of the nonlift newform agree with the Hasse–Weil factors of the associated abelian surface at the first two good primes $K(N)$95 (Poor et al., 2016). A common simplification is therefore incorrect: a paramodular space need not contain nonlifts, and at many levels it is entirely accounted for by the Gritsenko lift space.

The role of congruences has also expanded. In the quinary-lattice framework, examples of Harder-type and Buzzard–Golyshev-type congruences connect paramodular eigenvalues to elliptic modular forms at levels $K(N)$96, $K(N)$97, $K(N)$98, $K(N)$99, $N$00, and $N$01 (Dummigan et al., 2021). In a more geometric direction, a 2024 paper proves residual paramodularity for a Calabi–Yau threefold: the unique nonlift Hecke eigenform

$N$02

is shown to satisfy congruences modulo a prime above $N$03 with a Johnson–Leung–Roberts lift $N$04, and the semisimplified mod-$N$05 Galois representation on $N$06 of the threefold is identified with the residual representation attached to $N$07 (Dummigan et al., 2024).

These examples suggest two complementary themes. First, the classical weight-$N$08 nonlift case tied to abelian surfaces remains the core arithmetic application. Second, the same local-newvector and Hecke-theoretic formalism now supports weight-$N$09 lifts, residual modularity for four-dimensional motives, and even paramodular Eisenstein newforms. The modern theory of paramodular newforms therefore sits simultaneously in explicit Siegel modular form computation, local representation theory on $N$10, and the arithmetic of degree-$N$11 $N$12-functions.