
# Dimension formulas for spaces of Siegel modular forms of degree 2

Many thanks to Tomoyoshi Ibukiyama, who helped improve this page in several ways.
 Space Link Reference Remark $M_k(\Sp(4,\Z))$ LMFDB Igusa: On Siegel modular forms of genus 2, 1962 $\bigoplus M_{2k}(\Sp(4,\Z))=\C[\psi_4,\psi_6,\chi_{10},\chi_{12}]$ Igusa: On Siegel modular forms of genus 2 (II), 1964 $\bigoplus M_k(\Sp(4,\Z))=\C[\psi_4,\psi_6,\chi_{10},\chi_{12}]+\chi_{35}\C[\psi_4,\psi_6,\chi_{10},\chi_{12}]$ Igusa: Modular forms and projective invariants, 1967 $\chi_{35}^2$ expressed as a polynomial in $\psi_4,\psi_6,\chi_{10},\chi_{12}$ (page 849/850). Aoki, Ibukiyama: Simple graded rings of Siegel modular forms, differential operators and Borcherds products, 2015 Hecke eigenvalues at 2, 3, 5 for some cusp forms with weights from 35 to 45 (pages 275-278). $M_{k,j}(\Sp(4,\Z))$ LMFDB Freitag: Ein Verschwindungssatz für automorphe Formen zur Siegelschen Modulgruppe, 1979 Main result (Satz 1) implies that $M_{0,j}(\Sp(4,\Z))=0$ for $j>0$. Arakawa: Vector valued Siegel's modular forms of degree 2 and the associated Andrianov L-functions, 1983 Proves that $M_{k,j}(\Sp(4,\Z))=S_{k,j}(\Sp(4,\Z))$ for odd $k$, and $\dim M_{k,j}(\Sp(4,\Z))-\dim S_{k,j}(\Sp(4,\Z))=\dim S_{k+j}(\SL(2,\Z))$ for even $k\geq6$ (Proposition 1.3). Ibukiyama, Wakatsuki: Siegel modular forms of small weight and the Witt operator, 2009 Proves that $M_{2,j}(\Sp(4,\Z))=S_{2,j}(\Sp(4,\Z))$, and $\dim M_{4,j}(\Sp(4,\Z))-\dim S_{4,j}(\Sp(4,\Z))=\dim S_{4+j}(\SL(2,\Z))$ (Theorem 5.1). Ibukiyama: Lifting conjectures from vector valued Siegel modular forms of degree two, 2012 Generating series for $\dim M_{4,j}(\Sp(4,\Z))$ (page 8). $S_{k,j}(\Sp(4,\Z))$ Tsushima: An explicit dimension formula for the spaces of generalized automorphic forms with respect to $\Sp(2,\Z)$, 1983 Dimension formula for $j=0,k\geq4$ or $j\geq1,k\geq5$ (Theorem 4). Ibukiyama: A conjecture on a Shimura type correspondence for Siegel modular forms, and Harder's conjecture on congruences, 2008 Generating series for $\dim S_{k,j}(\Sp(4,\Z))$ for $k\geq3$ odd (pages 115/116). Examples of eigenforms for several $k,j$ (page 118). Wakatsuki: Dimension formulas for spaces of vector-valued Siegel cusp forms of degree $2$, 2012 Dimension formula for $k\geq5$ (Theorem 7.1). Ibukiyama: Lifting conjectures from vector valued Siegel modular forms of degree two, 2012 Generating series for $\dim S_{4,j}(\Sp(4,\Z))$ (page 8). Bergström, Faber, van der Geer: Siegel modular forms of degree three and the cohomology of local systems, 2014 Proves that Tsushima's formula holds for $k=4$ (and any $j$) as well (page 97). Petersen: Cohomology of local systems on the moduli of principally polarized abelian surfaces, 2015 Dimension formula for $k\geq3$ (explicit for $k=3$ on page 45). $M_{k,2}(\Sp(4,\Z))$ LMFDB Satoh: On certain vector valued Siegel modular forms of degree two, 2012 Structure of $M_{k,2}(\Sp(4,\Z))$ for even $k$. Ghitza, Ryan, Sulon: Computations of vector-valued Siegel modular forms, 2013 Eigenforms for $k\leq 30$. $M_{k,6}(\Sp(4,\Z))$ van Dorp: Vector-valued Siegel modular forms of genus 2 (Master's thesis), 2011 Module structure. van Dorp: Generators for a module of vector-valued Siegel modular forms of degree 2, 2018 $M_{k,8}(\Sp(4,\Z))$ Kiyuna: Vector-valued Siegel modular forms of weight $\det^k\otimes{\rm Sym}(8)$, 2015 Module structure (Theorem 3.1). $M_{k,10}(\Sp(4,\Z))$ Takemori: Structure theorems for vector-valued Siegel modular forms of degree 2 and weight $\det^k\otimes{\rm Sym}(10)$, 2016 Module structure (Theorem 1.1) and generating series for dimensions (Lemma 7.1). $M_{k,j}(\Sp(4,\Z))$, $j\in\{2,4,6\}$ Ibukiyama: Vector valued Siegel modular forms of symmetric tensor weight of small degree, 2012 Structure theorems for even and odd $k$. $S_{k,j}(\Sp(4,\Z),{\rm sgn})$ Ibukiyama, Wakatsuki: Siegel modular forms of small weight and the Witt operator, 2009 Dimension formula for $k\geq5$ (Theorem 6.2). $M_{2,j}(\Sp(4,\Z))$ Proves that $M_{2,j}(\Sp(4,\Z))=S_{2,j}(\Sp(4,\Z))$ (Theorem 5.1). $M_k(\Gamma(2))$ LMFDB Igusa: On Siegel modular forms of genus 2 (II), 1964 $S_6$-types (Theorem 2). Clery, van der Geer, Grushevsky: Siegel modular forms of genus 2 and level 2, 2015 Graded ring structure and $S_6$-types. $S_k(\Gamma(2))$ LMFDB Tsushima: On the spaces of Siegel cusp forms of degree 2, 1982 Dimension formula for $k\geq4$. $M_{k,j}(\Gamma(2))$ LMFDB ($j=2$) Clery, van der Geer, Grushevsky: Siegel modular forms of genus 2 and level 2, 2015 Dimension formula for $k+j\geq5$ or ($j=0$ and $k\geq0$) (Theorem 12.1). $S_{k,j}(\Gamma(2))$ LMFDB ($j=2$) Tsushima: An explicit dimension formula for the spaces of generalized automorphic forms with respect to $\Sp(2,\Z)$, 1983 Dimension formula for $j=0,k\geq4$ or $j\geq1,k\geq5$ (Theorem 3). Wakatsuki: Dimension formulas for spaces of vector-valued Siegel cusp forms of degree $2$, 2012 Dimension formula for $k\geq5$ (Theorem 7.2). Clery, van der Geer, Grushevsky: Siegel modular forms of genus 2 and level 2, 2015 Dimension formula for $k+j\geq5$ (Theorem 12.1 and Remark 12.1). SMF Bergström, Faber, van der Geer, 2017 $S_6$ types (no published reference except for $j=0$). Bergström, Faber, van der Geer: Siegel Modular Forms of Genus 2 and Level 2: Cohomological Computations and Conjectures, 2008 $S_6$ types for lifts (Conjectures in Sect. 6). Rösner: Parahoric restriction for GSp(4) and the inner cohomology of Siegel modular threefolds, 2016 Proves the conjectures of Bergström, Faber, van der Geer on lifts. $M_k(\Gamma(3))$ Freitag, Salvati-Manni: The Burkhardt group and modular forms, 2004 Graded ring structure (Theorem 3). Gunji: On the graded ring of Siegel modular forms of degree 2, level 3, 2004 Dimension formula for all $k\geq1$. $S_k(\Gamma(3))$ Gunji: On the graded ring of Siegel modular forms of degree 2, level 3, 2004 Dimension formula for all $k\geq1$. $S_k(\Gamma(N))$ Morita: An explicit formula for the dimension of spaces of Siegel modular forms of degree two, 1974 Dimension formula for $k\geq7$, $N\geq3$. Yamazaki: On Siegel modular forms of degree two, 1976 Dimension formula for $k\geq4$, $N\geq3$. Christian: Zur Berechnung des Ranges der Schar der Spitzenformen zur Modulgruppe zweiten Grades und Stufe $q>2$, 1977 Dimension formula for $k\geq5$, $N\geq3$. $S_{k,j}(\Gamma(N))$ Tsushima: An explicit dimension formula for the spaces of generalized automorphic forms with respect to $\Sp(2,\Z)$, 1983 Dimension formula for $j=0,k\geq4$ or $j\geq1,k\geq5$, and any $N$ (Theorems 2 and 3). Wakatsuki: Dimension formulas for spaces of vector-valued Siegel cusp forms of degree $2$, 2012 Dimension formula for $k\geq5$ and any $N$ (Theorems 7.2, 7.3). $M_k(\Gamma_1(2))$ LMFDB Clery, van der Geer, Grushevsky: Siegel modular forms of genus 2 and level 2, 2015 Graded ring structure (Theorem 9.3). $M_{k,j}(\Gamma_1(2))$ LMFDB Clery, van der Geer, Grushevsky: Siegel modular forms of genus 2 and level 2, 2015 Dimension formulas for $j>0$ (Theorem 14.1). $S_{k,j}(\Gamma_1(2))$ LMFDB Clery, van der Geer, Grushevsky: Siegel modular forms of genus 2 and level 2, 2015 Dimension formulas for $j>0$ (Theorem 14.1). $M_k(\Gamma_0(2))$ LMFDB Ibukiyama: On Siegel modular varieties of level 3, 1991 Graded ring structure and ideal of cusp forms (Theorem A, C). Ibukiyama, Wakatsuki: Siegel modular forms of small weight and the Witt operator, 2009 Determines $\dim M_k(\Gamma_0(p))-\dim S_k(\Gamma_0(p))$ for $k\geq2$ (Theorem 3.2). Clery, van der Geer, Grushevsky: Siegel modular forms of genus 2 and level 2, 2015 Graded ring structure (Theorem 9.2). $S_k(\Gamma_0(2))$ Ibukiyama: On symplectic Euler factors of genus $2$, 1984 Explicit eigenforms for $k\in\{6,8,10\}$ (Theorem 3.3). $M_k(\Gamma_0(3))$ LMFDB Ibukiyama: On Siegel modular varieties of level 3, 1991 Graded ring structure and dimension formulas. Aoki, Ibukiyama: Simple graded rings of Siegel modular forms, differential operators and Borcherds products, 2015 Graded ring structure (Theorem 4.1). $M_k(\Gamma_0(3),\psi_3)$ LMFDB Aoki, Ibukiyama: Simple graded rings of Siegel modular forms, differential operators and Borcherds products, 2015 Graded ring structure (Theorem 4.1). $S_k(\Gamma_0(3))$ Hashimoto: The dimension of the spaces of cusp forms on Siegel upper half-plane of degree two. I, 1983 Dimensions for $k\geq5$ (and any odd prime level) (Theorem 7-1). Yoshida: On representations of finite groups in the space of Siegel modular forms and theta series, 1988 Dimensions for $k=2$ and $k=4$. Ibukiyama: On Siegel modular varieties of level 3, 1991 Ideal of cusp forms in the graded ring is given. $M_k(\Gamma_0(4))$ LMFDB Tsushima: Dimension formula for the spaces of Siegel cusp forms of half integral weight and degree two, 2003 Dimension formula (Proposition 5.4). $M_k(\Gamma_0(4),\psi_4)$ LMFDB Hayashida, Ibukiyama: Siegel modular forms of half integral weight and a lifting conjecture, 2005 Graded ring structure (Theorem 1.1, 1.2). $S_k(\Gamma_0(p))$ Hashimoto: The dimension of the spaces of cusp forms on Siegel upper half-plane of degree two. I, 1983 Dimensions for $k\geq5$ and any odd prime $p$ (Theorem 7-1). Poor, Yuen: Dimensions of cusp forms for $\Gamma_0(p)$ in degree two and small weights, 2007 Dimensions for $k=2,3,4$ and small primes $p$. Ibukiyama: Dimension formulas of Siegel modular forms of weight 3 and supersingular abelian surfaces (revised version), 2007 (2018) Dimensions for $k=3,4$ and all primes $p$. $S_{k,j}(\Gamma_0(p),\chi)$ Tsushima: Dimension formula for the spaces of Siegel cusp forms and a certain exponential sum, 1997 Dimensions for (($k\geq5$, $j>0$) or ($k\geq4$, $j=0$)), any prime $p$, and a Dirichlet character mod $p$ (Corollary 4.12). Wakatsuki: Dimension formulas for spaces of vector-valued Siegel cusp forms of degree $2$, 2012 Dimensions for $k\geq5$, any prime $p$, and a Dirichlet character mod $p$ (Theorem 7.4). $M_k(\Gamma_0(N))$ Böcherer, Ibukiyama: Surjectivity of Siegel Phi operator for square free level and small weight, 2012 Codimension of the space of cusp forms for square-free $N$ (Corollary 1.2). $M_k(K(2))$ Ibukiyama, Onodera: On the graded ring of modular forms of the Siegel paramodular group of level 2, 1997 Graded ring structure, ideal of cusp forms, dimension formula. Ibukiyama, Poor, Yuen: Jacobi forms that characterize paramodular forms, 2013 Dimension formula for $M_k(K(2))^\pm$. Aoki: On Siegel paramodular forms of degree 2 with small levels, 2016 Dimension formula for a subgroup of index $2$. $M_k(K(3))$ Dern: Paramodular forms of degree 2 and level 3, 2002 Ring structure and dimension formula. Ibukiyama, Poor, Yuen: Jacobi forms that characterize paramodular forms, 2013 Dimension formula, also for $M_k(K(3))^\pm$. Aoki: On Siegel paramodular forms of degree 2 with small levels, 2016 Dimension formula for a subgroup of index $3$. $M_k(K(4))$ Ibukiyama, Poor, Yuen: Jacobi forms that characterize paramodular forms, 2013 Dimension formula, also for $M_k(K(4))^\pm$. Aoki: On Siegel paramodular forms of degree 2 with small levels, 2016 Dimension formula for a subgroup of index $4$. $S_k(K(4))$ Poor, Yuen: The cusp structure of the paramodular groups for degree two, 2013 Dimension formula (Theorem 1.1). $S_k(K(8))$ Poor, Schmidt, Yuen: Paramodular forms of level 8 and weights 10 and 12, 2018 Dimension and eigenforms for $k=10$ and $k=12$. $S_k(K(16))$ Poor, Schmidt, Yuen: Paramodular forms of level 16 and supercuspidal representations, 2018 Dimension and eigenforms for $k\leq14$. $S_2(K(p))$ LMFDB Poor, Yuen: Paramodular cusp forms, 2015 All cusp forms for prime level $p<600$. $S_k(K(p))$ Ibukiyama: On relations of dimensions of automorphic forms of $\Sp(2,\R)$ and its compact twist $\Sp(2)$. I, 1985 Dimension formula for $k\geq5$ and any prime $p$. Ibukiyama: Dimension formulas of Siegel modular forms of weight 3 and supersingular abelian surfaces (revised version), 2007 (2018) Dimensions for $k=3,4$ and all primes $p$. $S_2(K(N))$ Poor, Shurman, Yuen: Siegel paramodular forms of weight 2 and squarefree level, 2017 All cusp forms for square-free level $N<300$. $S_{k,j}(K(p))$ Ibukiyama: Paramodular forms and compact twist, 2006 Dimension formula for prime $p$ and $k\geq5$ (Theorem 6.1). Wakatsuki: Multiplicity formulas for discrete series representations in $L^2(\Gamma\backslash{\rm Sp}(2,\R))$, 2013 Dimension formula for prime $p$ and $k\geq5$ (Theorem 4.2 and Sect. 5.1). $S_{k,j}(K(N))$ Ibukiyama, Kitayama: Dimension formulas of paramodular forms of squarefree level and comparison with inner twist, 2017 Dimension formula for square-free $N$ and ($k\geq5$ or $(k\geq3,j=0)$). $S_k(\Gamma_0'(2))$ Ibukiyama: On symplectic Euler factors of genus $2$, 1984 Explicit eigenforms for $k\in\{6,8,10,12\}$ (Theorem 3.3, Theorem 3.4). $M_k(\Gamma_0'(3))$ Ibukiyama: Conjectures on correspondence of symplectic modular forms of middle parahoric type and Ihara lifts, 2018 Generating series for $\dim M_k(\Gamma_0'(3))$ (Sect. 5.3, p. 33). $S_k(\Gamma_0'(3))$ Generating series for $\dim S_k(\Gamma_0'(3))$ (Sect. 5.3, p. 34). $S_{11}(\Gamma_0'(3))$ Explicit eigenforms (Sect. 4.3). $S_k(\Gamma_0'(p))$ Hashimoto, Ibukiyama: On relations of dimensions of automorphic forms of $\Sp(2,\R)$ and its compact twist $\Sp(2)$. II, 1985 Dimension formula for $k\geq5$ and primes $p\neq2,3$ (Theorem 3.3). Ibukiyama: Dimension formulas of Siegel modular forms of weight 3 and supersingular abelian surfaces (revised version), 2007 (2018) Dimensions for $k=3,4$ and all primes $p$. $S_{k,j}(\Gamma_0'(p))$ Wakatsuki: Multiplicity formulas for discrete series representations in $L^2(\Gamma\backslash{\rm Sp}(2,\R))$, 2013 Dimension formula for $k\geq5$ and any prime $p$ (Theorem A.1). $M_k(B(2))$ Ibukiyama: On Siegel modular varieties of level 3, 1991 Graded ring structure and ideal of cusp forms (Theorem B, C). $S_k(B(2))$ Ibukiyama: On symplectic Euler factors of genus $2$, 1984 Explicit eigenforms for $k\in\{6,8,10,12\}$ (Theorem 3.3, Theorem 3.4). $S_k(B(p))$ Hashimoto, Ibukiyama: On relations of dimensions of automorphic forms of $\Sp(2,\R)$ and its compact twist $\Sp(2)$. II, 1985 Dimension formula for $k\geq5$ and primes $p\neq2,3$ (Theorem 3.2). Ibukiyama: Dimension formulas of Siegel modular forms of weight 3 and supersingular abelian surfaces (revised version), 2007 (2018) Dimensions for $k=3,4$ and all primes $p$. $S_{k,j}(B(p))$ Wakatsuki: Multiplicity formulas for discrete series representations in $L^2(\Gamma\backslash{\rm Sp}(2,\R))$, 2013 Dimension formula for $k\geq5$ and any prime $p$ (Theorem A.2). $M_k(\Gamma(1,2))$ Kogiso, Tsushima: On an algebra of Siegel modular forms associated with the theta group $\Gamma_2(1,2)$, 1988 Graded ring structure. $S_3(\Gamma(4,8))$ von Geemen, van Straten: The cusp forms of weight $3$ on $\Gamma_2(2,4,8)$, 1993 Explicit eigenforms. Okazaki: $L$-functions of $S_3(\Gamma_2(4,8))$, 2007 Spin $L$-functions of eigenforms. $S_3(\Gamma(2,4,8))$ von Geemen, van Straten: The cusp forms of weight $3$ on $\Gamma_2(2,4,8)$, 1993 Explicit eigenforms. Okazaki: $L$-functions of $S_3(\Gamma_2(2,4,8))$, 2012 Spin $L$-functions of eigenforms.
Last update: January 2019.