Nivolumab in Biochemically Recurrent dMMR Prostate Cancer

Inclusion Criteria: * Willing and able to provide signed informed consent and HIPAA authorization for the release of personal health information * Males aged 18 years and above * Prior local therapy with prostatectomy or EBRT/brachytherapy is required * Prior salvage or adjuvant radiation therapy is allowed but not mandated. Radiation therapy must have been completed for at least 6 months. * Absolute PSA \>=1.0 ng/mL at screening * Must have at least one of the following genetic alterations identified using archival tissue (i.e. prostate needle biopsy prior to radiation therapy or prostatectomy specimen): * Microsatellite instability (MSI-high) status by clinical grade testing * MMR protein loss (MSH2, MSH6, MLH1, PMS2) by immunohistochemistry * Inactivating mutation of MSH2, MSH6, MLH1 or PSM2 by clinical grade genomic testing * Tumor mutational burden \>= 20 mutations/megabase (TMB \>=20 muts/Mb) by clinical grade testing * Inactivating mutation (at least monoallelic of CDK12 by clinical grade testing * Serum testosterone \>= 150 ng/dL * No radiographic evidence of metastatic disease by CT scan and bone scan, performed within the prior 4 weeks. * Karnofsky Performance Status (KPS) \>= 70% within 14 days before start of study treatment (ECOG \<=1) * Participants must have normal organ and bone marrow function measured within 28 days prior to administration of study treatment as defined below: * Hemoglobin \>= 9.0 g/dL with no blood transfusion in the past 28 days * Absolute neutrophil count (ANC) \>= 1.0x10\^9 / L * Platelet count \>= 100 x 10\^9 /L * Total bilirubin within institutional upper limit of normal (ULN) (in patients with Gilbert's syndrome, total bilirubin \<1.5x institutional ULN will be acceptable) * Aspartate aminotransferase (AST), Serum Glutamic Oxaloacetic Transaminase (SGOT) / Alanine aminotransferase (ALT), Serum Glutamic Pyruvate Transaminase (SGPT) within institutional ULN * Participants must have creatinine clearance estimated using the Cockcroft-Gault equation of \>=40 mL/min: Estimated creatinine clearance = \[(140 - age (years)) x weight (kg)\] / \[serum creatinine (mg/dL) x 72\] * Participants must have a life expectancy of \>= 6 months * Male participants and their partners who are sexually active and of childbearing potential must agree to the use of two highly effective forms of contraception in combination, throughout the period of taking study treatment and for 7 months after the last dose of nivolumab to prevent pregnancy in a partner. * No evidence (within 5 years) of prior malignancies (except successfully treated basal cell or squamous cell carcinoma of the skin) Exclusion Criteria: * Metastatic disease or currently active second malignancy * Prior androgen deprivation therapy (ADT) in the past 6 months. Prior ADT in context of neoadjuvant/adjuvant primary; prior ADT for biochemical recurrence is allowed, as long as no ADT has been administered in past 6 months and testosterone has recovered (\>150 ng/dL) * Prior oral anti-androgen (e.g. bicalutamide, nilutamide, enzalutamide, apalutamide), or androgen synthesis inhibitor (e.g. abiraterone, orteronel) within the past 2 weeks is not permitted. 5-alpha reductase inhibitor therapy (e.g. finasteride, dutasteride) is allowed, as long as subject has been stable on medication for past 6 months. * Involvement in the planning and/or conduct of the study (applies to both BMS staff and/or staff at the study site) * Participation in another clinical study with an investigational product during the last 4 weeks/28 days * Patients should be excluded if they have had prior systemic treatment with an anti-PD-1, anti-PD-L1, anti-PD-L2, anti-CTLA-4 antibody, or any other antibody or drug specifically targeting T-cell costimulation or immune checkpoint pathways * Patients should be excluded if they have an active, known or suspected autoimmune disease (e.g. inflammatory bowel disease, rheumatoid arthritis, autoimmune hepatitis, lupus, celiac disease). Subjects are permitted to enroll if they have vitiligo, type I diabetes mellitus, residual hypothyroidism due to autoimmune condition only requiring hormone replacement, psoriasis not requiring systemic treatment, or conditions not expected to recure in the absence of an external trigger. * Patients should be excluded if they have a condition requiring systemic treatment with either corticosteroids (\>10 mg daily prednisone daily equivalents) or other immunosuppressive medications within 14 days of study drug administration. Inhaled or topical steroids and adrenal replacement doses \>10mg daily prednisone equivalents are permitted in the absence of active autoimmune disease * Permitted therapies include topical, ocular, intra-articular, intranasal, and inhalational corticosteroids (with minimal systemic absorption). Physiologic replacement doses of systemic corticosteroids are permitted, even if \> 10 mg/day prednisone equivalents. A brief course of corticosteroids for prophylaxis (e.g. contrast dye allergy) or for treatment of nonautoimmune conditions (e.g. delayed-type hypersensitivity reaction caused by contact allergen) is permitted. * As there is potential for hepatic toxicity with nivolumab, drugs with a predisposition to hepatotoxicity should be used with caution in patients treated with nivolumab-containing regimen. * Patients should be excluded if they have a positive test for hepatitis B virus surface antigen (HBVsAg) or hepatitis C virus ribonucleic acid (HCV antibody) indicating acute or chronic infection * Patients should be excluded if they have known history of testing positive for human immunodeficiency virus (HIV) or known acquired immunodeficiency syndrome (AIDS). * History of allergy to study drug components * History of severe hypersensitivity reaction to any monoclonal antibody * Any other serious illness or medical condition that would, in the opinion of the investigator, make this protocol unreasonably hazardous, including but not limited to: * Any uncontrolled major infection * Cardiac failure NYHA (New York Heart Association) III or IV * Crohn's disease or ulcerative colitis * Bone marrow dysplasia * Known allergy to any of the compounds under investigation * Unmanageable fecal incontinence * Poor medical risk due to a serious, uncontrolled medical disorder, non-malignant systemic disease or active, uncontrolled infection. Examples include, but are not limited to, uncontrolled ventricular arrhythmia, recent (within 6 months) myocardial infarction, uncontrolled major seizure disorder, extensive interstitial bilateral lung disease, or any psychiatric disorder that prohibits obtaining informed consent.