Harder Congruences and Yoshida-Type Siegel Modular Liftings

Definitions and mathematical statement

The source bundles Conjecture 3 (Harder) and Conjecture 4 (Yoshida-type lifting). Write $S_k(\Gamma_0(N))^{\mathrm{new}}$ for elliptic new cusp forms, with $\Gamma_0(N)\subseteq\operatorname{SL}_2({\mathbb{Z}})$ given by the lower-left entry divisible by $N$. In genus two, weight $(j,k)$ conventionally means the representation $\operatorname{Sym}^j({\mathbb{C}}^2)\otimes\det^k$; write $S_{j,k}(\Gamma_2[2])$ for Siegel cusp forms at $\Gamma_2[2]=\ker(\operatorname{Sp}_4({\mathbb{Z}})\to\operatorname{Sp}_4({\mathbb{F}}_2))$. The allowed integers $l,m$ must make the displayed weights meaningful, but their exact range and parity hypotheses are not supplied.

1. Let $f=\sum_{n\geq1}a(n)q^n$ be a normalized eigenform of weight $l+m+4$ for $\operatorname{SL}_2({\mathbb{Z}})$, so $a(1)=1$. Let $\ell$ be a sufficiently large prime in its coefficient field dividing the source's critical value $\Lambda(f,l+3)$. Is there a Siegel modular eigenform $F$ of weight $(l-m,m+3)$ such that \[\lambda_F(p)\equiv p^{m+1}+a(p)+p^{l+2}\pmod{\ell'}\] for the intended rational primes $p$, where $\ell'$ is a prime above $\ell$ in the compositum of the eigenvalue fields? Divisibility requires an algebraic, period-normalized critical value and an integrality convention; the source does not define $\Lambda$ or sufficiently large. The level of $F$, normalization of its Hecke operator, and possible exclusions among $p$ are not stated in the extract.
2. If $f\in S_{l+m+4}(\Gamma_0(2))^{\mathrm{new}}$ and $g\in S_{l-m+2}(\Gamma_0(2))^{\mathrm{new}}$ are newforms, is there a Siegel modular Hecke eigenform $F\in S_{l-m,m+3}(\Gamma_2[2])$ with spinor $L$-function \[L(F,s)=L(f,s)L(g,s-m-1)?\] The predicted multiplicity is $5$ when $f$ and $g$ have the same $w_2$-eigenvalue, and $1$ otherwise, where $w_2$ is the source's level-two Atkin--Lehner involution. The source also asks for the analogous lifting from level-four newforms of the same respective weights, with multiplicity $5$ in $S_{l-m,m+3}(\Gamma_2[2])$. The level-four assertion is not replaced by a level-four target space.

The meaning of multiplicity, including whether it concerns a Hecke eigenspace or an additional finite-group representation, is not defined in the extract. The $L$-function normalizations and bad-place factors are also unspecified. The supplied 2026-08-21 triage labels the record partially solved, reports many verified cases and some proved families of these congruences and liftings, and says the all-weights/all-levels existence assertion remains open; it does not supply a precise range of established cases.

Short English statement

Ask for Harder's congruence triggered by a critical-value prime divisor, and for the stated Yoshida-type spinor-$L$-function factorizations from level two and four. Keep the distinct multiplicities, weight shifts, and level-two Siegel target, while flagging missing normalization and parameter conventions.

Sources

• Local UnsolvedMath registry, permanent ID 30000916, source problem catalogue entry, Harder Congruences and Yoshida-Type Siegel Modular Liftings, proposed year 2008. The original labels Conjecture 3 as Harder's and Conjecture 4 as Yoshida-type lifting; no individual proposer metadata is supplied.
• Automorphic Forms, Geometry and Arithmetic (2008), Oberwolfach Reports, DOI 10.4171/owr/2008/05, pp. 243--296: https://doi.org/10.4171/owr/2008/05. Supplied literature label: original OWR report.
• Supplied literature label: Bergstrom--Faber--van der Geer, https://arxiv.org/abs/0803.0917.
• Supplied literature label: verified Harder congruences, https://doi.org/10.1007/s00208-016-1510-2.