Pith Number

pith:GDRQPEYH

pith:2022:GDRQPEYHPZ6Y4CUMXLKPYY3THY

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On the Lang--Trotter conjecture for Siegel modular forms

Ariel Weiss, Arvind Kumar, Moni Kumari

An adelic open image theorem for Galois representations attached to genus two Siegel modular eigenforms implies upper bounds on the number of primes where a fixed Hecke eigenvalue occurs.

arxiv:2201.09278 v2 · 2022-01-23 · math.NT

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Record completeness

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4 Citations open

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Claims

C1strongest claim

We prove an adelic open image theorem for the compatible system of Galois representations associated to f, generalising the results of Ribet and Momose for elliptic modular forms. Using this result, we investigate the distribution of the Hecke eigenvalues a_p of f, and obtain upper bounds for the sizes of the sets {p ≤ x : a_p = a} for fixed a∈C.

C2weakest assumption

The existence and basic properties of a compatible system of Galois representations attached to a genus two cuspidal Siegel modular eigenform (invoked in the abstract to state the open image theorem).

C3one line summary

Proves an adelic open image theorem for Galois representations of genus two Siegel modular forms and obtains upper bounds on the size of sets where a_p equals a fixed complex number a.

References

5 extracted · 5 resolved · 0 Pith anchors

[1] MR 3135650 ↑2 [AS06] Mahdi Asgari and Freydoon Shahidi, Generic transfer from GSp(4) to GL(4), Compos 2006

[2] MR 0568299 ↑1 [Mit14] Howard H 1914

[3] MR 0453647 ↑3 [Rib80] , Twists of modular forms and endomorphisms of abelian variet ies, Math. Ann. 253 (1980), no. 1, 43–62. MR 594532 ↑3, 7 [Rib85] , On l-adic representations attached to modular fo 1980

[4] MR 819838 ↑2, 3, 7, 11, 14, 15 [Ser18] Jean-Pierre Serre, On the mod p reduction of orthogonal representations , Lie groups, geometry, and representation theory, 2018, pp. 527–540. MR 3890220 ↑6, 8 [S 2018

[5] MR 1484415 ↑2, 3, 4, 13, 14 [Tay91] Richard Taylor, Galois representations associated to Siegel modular forms of low weight , Duke Math 1968

Cited by

1 paper in Pith

Bounds for the distribution of the Frobenius traces associated to a generic abelian variety

Receipt and verification

First computed	2026-06-04T01:09:36.242666Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

30e30793077e7d8e0a8cbad4fc63733e3213f93b20071e18d52054ab97f24af2

Aliases

arxiv: 2201.09278 · arxiv_version: 2201.09278v2 · doi: 10.48550/arxiv.2201.09278 · pith_short_12: GDRQPEYHPZ6Y · pith_short_16: GDRQPEYHPZ6Y4CUM · pith_short_8: GDRQPEYH

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/GDRQPEYHPZ6Y4CUMXLKPYY3THY \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 30e30793077e7d8e0a8cbad4fc63733e3213f93b20071e18d52054ab97f24af2

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "6a5bf4476478f2b96c945de8406f9959a3c533cfdda574a31978b56578c2d172",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2022-01-23T14:25:11Z",
    "title_canon_sha256": "4e60d642f31278275c441eefa87b56431262f0c7ed2f50c85cd618765aec649d"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2201.09278",
    "kind": "arxiv",
    "version": 2
  }
}