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In the special case $G=\\mathbb{Z}_p^d$ and $A$ a constant ordinary variety, using Akashi series, we show how the Euler characteristic of the dual of $Sel_A(K)_p$ is related to special values of a $p$-adic $\\mathcal{L}$-function."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1705.04703","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-05-12T18:08:24Z","cross_cats_sorted":[],"title_canon_sha256":"f4987e8963e91ea651e2cdcc1a557657e3eaab976eadc6abb84c6763e8c80f42","abstract_canon_sha256":"7605b8c5fbc63916b1acb7be30d7d2e471451fb667782c4ed0094575654752bc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:44:34.864376Z","signature_b64":"vsQYYaeA4/0ntwZvKOCLgVmtoTwBtgyOs3u0uVGWUbNmLNDyAOLuO/mpCg3XWbC6l9bKv4T2C9AkXsD9DUDyDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d3728b3736f91117adf2d70790a4172688c4991d8485f2fe0002635470ec0aa6","last_reissued_at":"2026-05-18T00:44:34.863780Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:44:34.863780Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Euler characteristic and Akashi series for Selmer groups over global function fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andrea Bandini, Maria Valentino","submitted_at":"2017-05-12T18:08:24Z","abstract_excerpt":"Let $A$ be an abelian variety defined over a global function field $F$ of positive characteristic $p$ and let $K/F$ be a $p$-adic Lie extension with Galois group $G$. 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