Eichler–Shimura Theory for Families of Modular Forms

Definitions and mathematical statement

Does an extension of the Eichler--Shimura theorem directly relate the full family of modular forms to the corresponding family of modular symbols?

For terminology, given a congruence subgroup $\Gamma\subset SL(2,{\mathbb{Z}})$ and a coefficient module $V$ with a specified compatible $\Gamma$-action, a standard space of modular symbols is \[ \operatorname{Symb}_{\Gamma}(V) =\operatorname{Hom}_{\Gamma}(\Delta_0,V),\qquad \Delta_0=\ker\!\left({\mathbb{Z}}[{\mathbb{P}}^1({\mathbb{Q}})]\xrightarrow{\deg}{\mathbb{Z}}\right). \] Thus a modular symbol assigns coefficients to degree-zero divisors on rational cusps, subject to equivariance. This definition does not select the coefficient systems, cusp conditions, or variants intended in the report. Modular forms likewise require a weight, level, character, coefficient field, and holomorphy conditions.

The requested extension is a comparison at the level of entire families, rather than merely separate comparisons at individual parameter values. To formulate it, one must specify a parameter space $B$, a family of modular forms over $B$, a corresponding family of coefficient modules and modular symbols, and the desired comparison morphism or relation. Its specializations should recover the intended fixed-parameter Eichler--Shimura comparison, with any required conjugate, sign, cuspidal, or Eisenstein components made explicit. The source does not specify whether this relation should be an isomorphism, exact sequence, or another comparison, and none is imposed here.

The dated assessment discusses $p$-adic families and overconvergent Eichler--Shimura theory. These indicate relevant context but do not identify the report's "full family," its base, its prime, or its admissible slopes. In particular, no restriction to an ordinary or finite-slope locus is silently substituted for the full-family request. The extracted question remains underdetermined until these data and the meaning of "directly relate" are recovered.

Short English statement

The question asks for an Eichler--Shimura comparison that works across a whole family of modular forms and modular symbols. The supplied record does not define that family or the type of comparison, so existing overconvergent results cannot automatically be identified with the full requested statement.

Sources

• Local UnsolvedMath registry, permanent ID 30000251, code catalogue entry, "Eichler--Shimura Theory for Families of Modular Forms." No proposer is given; proposed year 2005.
• Algebraische Zahlentheorie (2006), Oberwolfach Reports, pp. 1547--1588, DOI 10.4171/owr/2005/28; original problem report https://doi.org/10.4171/owr/2005/28. The displayed year 2006 differs from the DOI and proposed year 2005.
• Supplied reference labeled "Stevens, Overconvergent Eichler--Shimura isomorphisms": https://arxiv.org/abs/1303.4878. The author attribution is reproduced as supplied, without independent verification.
• Supplied reference "Eichler--Shimura theory for general congruence subgroups": https://arxiv.org/abs/2410.20385.
• Supplied statement status corrected_verified, with a note that the official report was checked and the wording made self-contained with normalized notation. The missing family data remain unresolved in this draft.
• Literature triage checked 2026-08-21 labels the problem partially solved. It says overconvergent theory relates $p$-adic modular-symbol families to overconvergent modular-form families, but the report's direct entire-family formulation is not established in exactly that form. These claims require independent verification. Created 2026-08-25; updated 2026-09-16.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; underlying report and reference provenance remain as supplied. Unreviewed statement-only draft.